On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials πn(z) with the quartic exponential weight exp[−N(1/2z²+1/4tz⁴)], where t∈C and N∈N, N→∞. Our goal is to describe these asymptotic behaviors globally for t∈C in different regions. We also describe the ''bre...

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Datum:	2016
Hauptverfasser:	Bertola, M., Tovbis, A.
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spelling	irk-123456789-1485472019-02-19T01:24:20Z On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight Bertola, M. Tovbis, A. We study the asymptotics of recurrence coefficients for monic orthogonal polynomials πn(z) with the quartic exponential weight exp[−N(1/2z²+1/4tz⁴)], where t∈C and N∈N, N→∞. Our goal is to describe these asymptotic behaviors globally for t∈C in different regions. We also describe the ''breaking'' curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic ''phase portrait'' of the recurrence coefficients of πn(z), which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the g-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex t-plane to all noncritical values of t. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation]. 2016 Article On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight / M. Bertola, A. Tovbis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D45; 33E17; 15B52 DOI:10.3842/SIGMA.2016.118 http://dspace.nbuv.gov.ua/handle/123456789/148547 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study the asymptotics of recurrence coefficients for monic orthogonal polynomials πn(z) with the quartic exponential weight exp[−N(1/2z²+1/4tz⁴)], where t∈C and N∈N, N→∞. Our goal is to describe these asymptotic behaviors globally for t∈C in different regions. We also describe the ''breaking'' curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic ''phase portrait'' of the recurrence coefficients of πn(z), which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the g-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex t-plane to all noncritical values of t. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation].
format	Article
author	Bertola, M. Tovbis, A.
spellingShingle	Bertola, M. Tovbis, A. On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Bertola, M. Tovbis, A.
author_sort	Bertola, M.
title	On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
title_short	On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
title_full	On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
title_fullStr	On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
title_full_unstemmed	On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
title_sort	on asymptotic regimes of orthogonal polynomials with complex varying quartic exponential weight
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/148547
citation_txt	On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight / M. Bertola, A. Tovbis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT bertolam onasymptoticregimesoforthogonalpolynomialswithcomplexvaryingquarticexponentialweight AT tovbisa onasymptoticregimesoforthogonalpolynomialswithcomplexvaryingquarticexponentialweight
first_indexed	2025-07-12T18:53:46Z
last_indexed	2025-07-12T18:53:46Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 118, 50 pages On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight∗ Marco BERTOLA † and Alexander TOVBIS ‡ † Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8 E-mail: marco.bertola@concordia.ca ‡ University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, USA E-mail: alexander.tovbis@ucf.edu Received March 11, 2016, in final form December 12, 2016; Published online December 27, 2016 http://dx.doi.org/10.3842/SIGMA.2016.118 Abstract. We study the asymptotics of recurrence coefficients for monic orthogonal poly- nomials πn(z) with the quartic exponential weight exp[−N( 1 2z 2 + 1 4 tz 4)], where t ∈ C and N ∈ N, N → ∞. Our goal is to describe these asymptotic behaviors globally for t ∈ C in different regions. We also describe the “breaking” curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic “phase portrait” of the recurrence coefficients of πn(z), which was stu- died numerically in [Constr. Approx. 41 (2015), 529–587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the g-function mechanism) of the cor- responding Riemann–Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex t-plane to all noncritical values of t. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order be- haviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529–587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation]. Key words: double scaling limit of orthogonal polynomials; asymptotics of recurrence coeffi- cients; method of Riemann–Hilbert problem; nonlinear steepest descent analysis 2010 Mathematics Subject Classification: 33D45; 33E17; 15B52 Contents 1 Introduction and main results 2 2 Steepest descent analysis of the RHP (2.1) 7 2.1 The RHP for recurrence coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 General requirements on the g-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Equality requirements and formula for g . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Inequality (sign) requirements (or sign distribution requirements) for h and the modulation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Modulation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Schematic conclusion of the steepest descent analysis . . . . . . . . . . . . . . . . . . . . . 14 2.5 Solution of the model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ∗This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica- tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html mailto:marco.bertola@concordia.ca mailto:alexander.tovbis@ucf.edu http://dx.doi.org/10.3842/SIGMA.2016.118 http://www.emis.de/journals/SIGMA/OPSFA2015.html 2 M. Bertola and A. Tovbis 3 Formal calculation of the g-function and h-function 21 3.1 Explicit form of g and h functions in the genus zero region . . . . . . . . . . . . . . . . . . 21 3.2 The g-function and h-function and their properties in the genus L case . . . . . . . . . . . 23 4 Existence of genus zero g and h functions, breaking curves and continuation principle 29 4.1 Sign requirements in the genus zero region . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Breaking curves. Regular and critical breaking points. Symmetries . . . . . . . . . . . . . 30 4.3 Continuation principle for Boutroux deformations . . . . . . . . . . . . . . . . . . . . . . . 32 5 Boundedness of the branchpoints 34 5.1 A general theorem on ODE’s for polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Properties of the polynomial (5.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6 Breaking curves for quartic potential f(z; t) 40 6.1 Existence of breaking curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7 Asymptotics of recurrence coeff icients in the higher genera regions 45 7.1 Genus 1 region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.2 Genus 2 solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 References 49 1 Introduction and main results In this paper we continue (see [4]) our study of monic polynomials πn(z), orthogonal with respect to the quartic exponential weight e−Nf(z,t), where f(z, t) = 1 2z 2+ 1 4 tz 4, t ∈ C and N ∈ N, and the contour of integration consists of four rays in the directions of the exponential decay of e−Nf(z,t). The exponent f(z, t) is also known as the potential. In [4] we obtained local (in t) asymptotics of the recurrence coefficients αn(N, t), βn(N, t), associated with πn(z), in the limit n = N →∞, in full neighborhoods of critical points t0 = − 1 12 and t1 = 1 15 through certain Painlevé I transcendents (note that we will not encounter t1 in the particular settings considered in this paper). Those results were a significant development of [13, 15], where these asymptotics were studied only “away from the poles” of the Painlevé I transcendents, that is, on certain subsets of full neighborhoods. The (only) remaining critical point t2 = 1 4 for f(z, t) = 1 2z 2 + 1 4 tz 4 was studied in [7], where the Painlevé II transcendents played the same role as Painlevé I transcendents in the case of t0, t1 (see [8] for analysis of t2 in the case of real orthogonal polynomials on R). In order to understand the role of the critical points tj , j = 0, 1, 2, in the global “asymptotic phase portrait”, in [4] we studied numerically the regions of different asymptotic regimes (in the limit n = N → ∞) in the complex t-plane (in fact, on the Riemann surface of t 1 4 ), see, for example, Fig. 2. We found there that there are six possible topologically different cases of global “asymptotic phase portraits”, which are determined by the constant weights %j , see (1.1), associated with each ray. The objective of the current paper is to give a rigorous foundation for the numerical results, obtained in [4]. We will restrict our attention only to one of the above mentioned six cases, called Generic case, which will be defined below. Our goal is to rigorously justify the asymptotic phase portrait for the Generic case (case (a1) in [4]) using the principle of continuation in the parameter space, developed in [1] and [24]. As it was stated in [1], this technique will work for any k-th degree polynomial f(z,~t) that depends on k complex parameters (t1, . . . , tk) = ~t that are the coefficients of f (it is obvious that the free term in f is irrelevant). Because of that consideration, a number of statements in this paper will be formulated and proved for general polynomial potentials f(z;~t). We also obtain explicit expressions for the recurrence coefficients and for the pseudonorms of πn(z) in terms of the Riemann theta functions. For the benefit of the reader, we repeat here some facts from [4] that are required for the exposition. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 3 − arg(t)/4 Ξ1 Ξ3 Ω1 Ω2 Ω3 Ξ2 Ω0 Ξ0 Figure 1. The contours of integration and the asymptotic directions. The contour Ξ0 is homologically equal to −Ξ1 − Ξ2 − Ξ3 and, thus, can be excluded from the definition of the pairing (1.3). As z → ∞, the weight function is exponentially decaying in four sectors Sj of width π/4, centered around the rays Ωj = { z : arg z = −arg t 4 + π(1−j) 2 } , j = 0, 1, 2, 3. Since the argument of t enters in this formula divided by four, we shall understand arg(t) ∈ [−π, 7π), so that, effectively, t lives on a four-sheeted Riemann surface. We consider the polynomials πn(z) to be integrated on the “cross” formed by the rays Ωj , where the rays Ω0,1 are oriented outwards (away from the origin) and the rays Ω2,3 – inwards. The corresponding bilinear form is 〈p, q〉~% = 3∑ j=0 %j ∫ Ωj p(z)q(z)e−Nf(z,t)dz, f(z, t) := 1 2 z2 + 1 4 tz4, (1.1) where ~% = {%0, %1, %2, %3} is a vector of fixed complex numbers chosen to satisfy the so-called “traffic condition”1 %1 + %0 = %3 + %2. (1.2) Moreover, since multiplying all the %j ’s by a common nonzero constant does not affect the families of orthogonal polynomials, these parameters are only defined modulo multiplication by C∗ = C \ {0}. Interpreting the coefficients %j as the traffics along the rays Ωj , j = 0, 1, 2, 3, we observe that the condition (1.2) implies that there is no source/sink of traffic at the origin. Equivalently, this condition means that all the contours of integration can be viewed as starting and ending at z =∞. Indeed [2], the bilinear form (1.1) can be represented as 〈p, q〉~ν = 3∑ k=1 νk ∫ Ξk p(z)q(z)e−Nf(z,t)dz, (1.3) ν1 = %1, ν2 = %3 − %0, ν3 = −%0, where Ξj , j = 1, 2, 3, are simple contours emanating from ∞ along Ωj+1 and returning to ∞ along Ωj with the indices taken modulo 4, see Fig. 1. The weights νj on the corresponding contours Ξj , j = 1, 2, 3, will be also called traffics. Contours with zero traffic are not included in the Ω = ∪3 j=1Ξj for the bilinear form (1.3). 1The naming is suggested by the fact that the sums of the %j associated with the incoming and with the outgoing rays are the same. 4 M. Bertola and A. Tovbis The generic configuration of the traffic that we consider in this paper is determined by %1 + %0 = %3 + %2 6= 0 and %j 6= 0 for any j = 0, 1, 2, 3. (1.4) (The “asymptotic phase portrait” of the generic case is shown on Fig. 2.) However, our methods are equally applicable to any of the five remaining cases. Because of (1.4), without any loss of generality we will assume that %1 + %0 = %3 + %2 = 1. (1.5) To obtain the asymptotics of the recurrence coefficients (and of the pseudonorm), associ- ated with the polynomials πn, we use the nonlinear steepest descent analysis of the associated Riemann–Hilbert problem (RHP), see Section 2.1. Our goal is to show that this method is applicable at any t ∈ C∗ \ {t0, t2}. (In this paper, we restrict our attention to the first sheet of the Riemann surface of t 1 4 only; the remaining sheets can be analyzed similarly.) As it was mentioned earlier, our approach is based on the method of continuation in the (external) param- eter space: once the validity of the nonlinear steepest descent method is established for some (regular) point t = t∗, say, for some t∗ ∈ (− 1 12 , 0), it will remain in place as t is continuously deformed in C∗ \ {t0, t2}. We now introduce the recurrence coefficients and the pseudonorms. The orthogonality con- dition for the monic polynomials πn(z) can be written as 〈πn, zk〉~ν = hnδnk, k = 0, 1, 2, . . . , n, ~ν = (ν1, ν2, ν3), (1.6) where δij denotes the Kronecker delta. The coefficient hn can also be written as hn = 〈πn, πn〉~ν and hence is the equivalent of the “square norm” of πn (in general, it is a complex number). We call hn the pseudonorms. The existence of orthogonal polynomials πn(z) is not a priori clear. However, if three consecutive monic polynomials exists, then they are related by a three-term recurrence relation πn+1(z) = (z − βn)πn(z)− αnπn−1(z), (1.7) where αn, βn are called recurrence coefficients. These recurrence coefficients depend on t, N . If the bilinear pairing is invariant under the map z 7→ −z then it follows immediately that the orthogonal polynomials are even or odd according to their degree and thus βn = 0, ∀n ∈ N. The remaining recurrence coefficients αn satisfy αn[1 + t(αn+1 + αn + αn−1)] = n N , which is known in literature as the string equation or the Freud equation [16]. We are interested in the asymptotic limit of αn, βn as N → ∞ and n N = 1, so we will use notations αn = αn(t), βn = βn(t). In the case of t ∈ (− 1 12 , 0), the asymptotics of αn, βn was obtained in [13] as αn(t) = √ 1 + 12t− 1 6t +O ( n−1 ) and βn decaying exponentially as n → ∞. In fact, Theorem 1.1 from [13] states that there exists some n0 = n0(t), such that αn(t), βn(t) exist for all n ≥ n0 and have the above mentioned asymptotics. We now outline the content of the paper. The Riemann–Hilbert problem (RHP) represen- tation of the orthogonal polynomials, introduced in [15], is briefly described in Section 2.1. In Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 5 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 − 1 12 1 4 Figure 2. Asymptotic regions on the complex t plane in the case of generic traffic configuration. The smaller contour plots, attached to a particular value of t (shown by the ∗ and connected with the plot by a dashed line), show zero level curves <h(z; t) = 0 in the complex z-plane. On these plots, the <h(z; t) > 0 regions are shown by green (darker) colour whereas the <h(z; t) < 0 regions are shown by white (lighter) colour; the solid red (dark) lines with darker colour on both sides show the main arcs, whereas the solid red (dark) lines with different colors on each side indicate bounded complementary arcs (actual complementary arcs, not shown here, can be obtained by deforming off the corresponding red (dark) line into the lighter region while keeping their endpoints fixed). Four unbounded complementary arcs, also not shown on the panels, are located in the lighter regions. Each of them connects one of the branchpoints with z = ∞. The asymptotic directions between the neighboring unbounded complementary arcs differ by the angle π/2. Complementary arcs for some of the panels are shown on Fig. 3 below. The region inside the curve joining 1 4 to 0 is of genus 1; inside the curve that joins t = − 1 12 and t = 0 it is of genus 0. Everywhere else the genus is 2, except for the degeneration to genus 0 occurring on the curve that joins t = − 1 12 and t = 1 4 , and to genus 1 on the ray [ 14 ,∞). Local behavior near t = t0 = − 1 12 is expressed through Painlevé I transcendents whereas local behavior near t = t2 = 1 4 is expressed through Painlevé II transcendents. the rest of Section 2, we use the nonlinear steepest descent method of Deift–Zhou and the g- function mechanism to find the large N asymptotics of the corresponding RHP. Most of the statements/results of this section are valid for general polynomial potentials f(z) = f(z;~t). In Section 2.2 we define the g function and the associated h function (also known as the modified external field) as solutions to certain scalar RHPs, whose jump contours are yet to be defined. 6 M. Bertola and A. Tovbis Figure 3. Four panels from Fig. 2 with complementary arcs shown as solid black curves and main arcs shown as solid red (dark) lines, surrounded by green (darker) regions. The panels correspond to the genus 0 (left), 1 (right), and 2 (center) regions, the latter is represented by two panels as there are two topologically different configurations of main and complementary arcs for genus two. Bounded complementary arcs connect branchpoints with each other whereas the unbounded complementary arcs connect some branchpoints with z =∞ along the four asymptotic directions in the corresponding lighter regions. The actual position of the contours is irrelevant (within the lighter regions). In addition, <g must satisfy a set of inequalities, known as sign conditions (sign requirements). Defining the jump contours for g starts with defining their end points, which are also the branchpoints of the hyperelliptic Riemann surface R = R(~t), associated with g. Various forms of modulation equations that define these branchpoints λj are discussed in Section 2.3. As the genus of R(~t) can change when we deform ~t, we can talk about regions of different genera in the space of parameters ~t. Schematic description of the steepest descent method and explicit formulae for the recurrence coefficients and the pseudonorms in the genus L region, L ∈ N, can be found in Section 2.4. Some basic facts about hyperelliptic Riemann surfaces and Riemann theta functions are also provided there. In Section 3 we calculate the g function in the regions of different genera. (Our approach of calculating g follows [18, 25], since the branchcuts of g are in C and the usual variational approach used for the equilibrium problem on the real line is not applicable.) This calculation is still formal, as we postpone the proof of the sign requirements until Section 4. In Section 3.1, following [4], we solve the modulation equations for the branchpoints and calculate the g and h functions in the genus zero region for our quartic potential f(z, t). In Section 3.2, we construct the g and h functions for higher genera regions in the determinantal form and derive new forms of the modulation equations. In particular, we proved Theorem 3.4, stating that the modulation equations for a branchpoint λ is equivalent to ∂g ∂λ ≡ 0 for all z ∈ C and is equivalent to the fact that real jump constants (that we will refer to as phases) in the RHP for g, see (2.8) and (2.9), are all independent of λ. This statement is valid for a larger class of RHPs, see, for example, [23], as applied to the semiclassical limit of the focusing Nonlinear Schrödinger equation (NLS). In Section 4 we prove the existence of g function for any t ∈ C∗ \ {t0, t2} for our quartic potential f(z; t) in the Generic case, that is, we prove that there is a formal g function satisfying the sign conditions. We also derive equations for the breaking curves on complex t plane, separating regions of different genera, and define critical (breaking) points. Following [4], in Section 4.1 we remind the basic steps in proving the existence of g-function in the genus zero region, which contains, in particular, the interval (− 1 12 , 0). That implies that the asymptotics of the recurrence coefficients is proven on the genus zero region. In Section 4.2 we derived the breaking curve equations for a general polynomial potential f(z;~t) and study properties of breaking curves. One of the central points of this paper, discussed in Section 4.3, is a proof that once the asymptotics of the recurrence coefficients is established in some region of the t-plane (or even just at one regular point t), the continuation principle for Boutroux deformations would establish the existence of the asymptotics of recurrence coefficients for any t ∈ C∗\{t0, t2}. That is to say Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 7 that if there exists the nonlinear steepest descent solution to the RHP with some regular value of t, then the nonlinear steepest descent solution to the RHP exists for any regular value of t ∈ C∗; moreover, it can be obtained by continuing the solution along any smooth curve (in the parameter space), connecting the original and the final values of t that avoids t = 0 as well as critical values of t = t0 and t = t2. Equivalently, one can say that if the g-function satisfying all the sign conditions (see Section 2.2) is established in some region of the t-plane, it can be continued to any regular value of t ∈ C∗ while preserving all the above-mentioned sign conditions. In the process of such a continuation in the t-plane, the genus of the underlying Riemann surface R(t) will undergo the appropriate changes. The continuation principle for Boutroux deformations was (independently) introduced in [1] in the context of general polynomial potential f(z,~t) and in [24] in the context of the semiclassical NLS, where f = f(z;x, t) is a Schwarz-symmetrical function in z that depends on real parameters x, t (space and time). Following [1, 24], we outline the proof of the continuation principle for general potentials f(z,~t). The essentially novel part in this proof, presented separately in Section 5, is the proof that the branchpoints λj of R(~t) are bounded under Boutroux deformations on any compact subset of ~t that preserves the degree of the polynomial f(z;~t). This particular issue was not entirely addressed in [1]. The problem discussed in Section 4.3 is closely related to the problem of existence of a curve with S-property for the polynomial external field that, using the approach of [21], was recently solved in [20]. In Section 6 we present explicit formulae for all the breaking curves for our quartic poten- tial f(z, t) and prove that they separate regions of particular genera, as shown on Fig. 2. In short, we prove the topology of regions on the “asymptotic phase portrait” from Fig. 2. Explicit formu- lae for the leading order behavior of the recurrence coefficients αn(t) and the pseudonorms hn(t) in the regions of genera one and two are given in Section 7. 2 Steepest descent analysis of the RHP (2.1) In this section we first (Section 2.1) express the recurrence coefficients αn(t) and the pseudo- norms hn(t) through the solution of a matrix RHP Y , and then (Sections 2.2 and 2.3) describe the nonlinear steepest descent approach to find the asymptotics of Y as n = N →∞. The leading order asymptotics for recurrence coefficients and pseudonorms can be found in Section 2.4. The results of this section can be extended for general potentials f(z;~t) in an obvious way. Our exposition in Section 2.1 and in some other parts of this section follow [4]. 2.1 The RHP for recurrence coefficients It is well known [11] that the existence of the above-mentioned orthogonal polynomials πn(z) is equivalent to the existence of the solution to the following RHP (2.1). More precisely, re- lation between the RHP (2.1) and the orthogonal polynomials πn(z) is given by the following proposition [13], which has the standard proof (see [11]). Proposition 2.1. Define ν : Ω→ C (where Ω = ⋃3 j=1 Ξj) by ν(z) = νj when z ∈ Ξj. Then the solution of the following RHP problem Y (z) is analytic in C \ Ω, Y+(z) = Y−(z) ( 1 ν(z)e−Nf(z,t) 0 1 ) , z ∈ Ω, (2.1) Y (z) = ( 1 +O ( z−1 ))(zn 0 0 z−n ) , z →∞, 8 M. Bertola and A. Tovbis exists (and it is unique) if and only if there exist a monic polynomial p(z) of degree n and a polynomial q(z) of degree ≤ n− 1 such that 〈p(z), zk〉~ν = 0, for all k = 0, 1, 2, . . . , n− 1, 〈q(z), zk〉~ν = 0, for all k = 0, 1, 2, . . . , n− 2, and 〈q(z), zn−1〉~ν = −2πi, where ~ν = (ν1, ν2, ν3). In that case the solution to the RHP (2.1) is given by Y (z) = ( p(z) CΩ [ p(z)ν(z)e−Nf(z,t) ] q(z) CΩ [ q(z)ν(z)e−Nf(z,t) ]) , z ∈ C \ Ω, where CΩ[φ] = 1 2πi ∫ Ω φ(ζ)dζ ζ − z is the Cauchy transform of φ(z). Remark 2.2. We remind that the contour Ω in (2.1) should coincide with the contour of integration in the bilinear form (1.3), so if any traffic νj = 0 the corresponding contour Ξj should not be included in Ω. If πn+1(z), πn(z), πn−1(z) are monic orthogonal polynomials then they satisfy the three term recurrence relation (1.7) for certain recurrence coefficients αn, βn [9, 22]. The following well known statements (see, for example, [11, 13, 15]) show the connection between the RHP (2.1), the orthogonal polynomials πn(z) and their recurrence coefficients. Proposition 2.3. Let Y (n)(z) denote the solution of the RHP (2.1). If we write Y (n)(z) = ( 1 + Y (n) 1 z + Y (n) 2 z2 +O(z−3) )( zn 0 0 z−n ) , z →∞, (2.2) then the pseudo-norms hn, appearing in (1.6), and the recurrence coefficients αn, appearing in (1.7), are given by hn = −2iπ ( Y (n) 1 ) 12 , αn = ( Y (n) 1 ) 12 · ( Y (n) 1 ) 21 , βn = ( Y (n) 2 ) 12( Y (n) 1 ) 12 − ( Y (n) 1 ) 22 . (2.3) Proposition 2.4. Suppose the RHP (2.1) has solution Y (n)(z). Let (2.2) be its expansion at ∞ and let αn, βn be given by (2.3). If αn 6= 0, then the monic orthogonal polynomials πn+1(z), πn(z), πn−1(z) exist and satisfy the three term recurrence relation (1.7). The Deift–Zhou steepest descent analysis is a powerful tool in finding the asymptotic behavior of a matrix RHP with respect to a given asymptotic parameter. In the case of the RHP (2.1), the asymptotic parameter is the large parameter N = n. The remaining parameters of the RHP will be called external parameters. In the case of the RHP (2.1), the external parameters are t, ~ν, however, we will consider ~ν to be fixed in the analysis below. The steepest descent analysis, as applied to the RHP (2.1), will be outlined here. As customary, the RHP undergoes a sequence of modifications into equivalent RHPs until it can be effectively solved (so-called model RHP) while keeping the error terms under control. • One starts with the RHP (2.1) for Y and seeks an auxiliary scalar function g = g(z; t), called the g-function, which is analytic in C except for a collection Σ of appropriate contours to be described in Section 2.2 and which behaves like ln z + O(z−1) near z = ∞. The contour Ω Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 9 of the RHP (2.1) can be deformed because the RHP (2.1) has an analytic in z jump matrix. If two or more parts of Ω are deformed into a single oriented arc, the traffic on this arc is a signed (according to the orientation) sum of the traffics on the parts of Ω that formed the arc. The parts, if any, of the deformed Ω carrying zero traffic (no integration upon) should be removed from the contour Ω. • Then we introduce a new matrix T (z) := e−N` σ3 2 Y (z)e−N(g(z,t)− ` 2 )σ3 , (2.4) where the constant ` ∈ C is to be defined. Direct calculations show that T (z) solves the RHP T (z) is analytic in C \ Ω, T+(z) = T−(z) ( e− N 2 (h+−h−) ν(z)e N 2 (h++h−) 0 e N 2 (h+−h−) ) , z ∈ Ω, (2.5) T (z) = ( 1 +O ( z−1 )) , z →∞, where h(z, t) := 2g(z, t)− f(z, t)− `. • At this point the Deift–Zhou method can proceed provided that the function g, the constant ` and the collection of contours Ω into which we have deformed the original contour of integration fulfill a rather long collection of equalities and – most importantly – inequalities (sign conditions) that we briefly describe in Section 2.2 (see, for example, [2, 25] for more details). If all these requirements are fulfilled, the leading order asymptotics for the problem (the solution to the model problem) can be obtained in terms of Riemann theta functions on a suitable hyperelliptic Riemann surface of a positive genus (with the case of zero genus not requiring the Riemann theta function). We construct the g-function (or, more precisely, the corresponding h-function, see (2.5)), not by using the variational principle, as often done for orthogonal polynomials on R, but by satisfying the collection of the above-mentioned equalities. These equalities can be considered as jump conditions in the corresponding scalar RHP for g (or for h). If the obtained g-function does not satisfy the inequality requirements, we need to change the genus of the scalar RHP for g. The continuation principle, see Section 4.3, enables us to trace the correct genus of g through the complex t-plane. 2.2 General requirements on the g-function The final configuration of the contour Ω can be partitioned into two subsets of oriented arcs that we shall denote by M and term main arcs (bands), and by C and term complementary arcs (gaps), Ω = M ∪ C, where all the main arcs are bounded and have a nonzero traffic. The final configuration of Ω must contain all the contours where g is not analytic. This partitioning defines the hyperelliptic Riemann surface R(t), whose branchcuts are the main arcs γm,j , where M = ∪Lj=0γm,j . It is subordinated to the list of requirements for the function g (or for the corresponding function h), associated with the Riemann surface R(t) of genus L, given in the following two subsections. For example, on Fig. 2, main arcs on each panel are the (red) arcs separating two blue (darker) regions. The genus of the problem is the number of the main arcs reduced by one. A finite complementary arc (these arcs are not usually shown on the panels) is a bounded curve within a white (lighter) region (or on its boundary), connecting the corresponding endpoints of the main arcs, see, for example, the panel in the low left corner. On the two panels in the middle of Fig. 2, finite complementary arcs are the (red) arcs, separating the blue (darker) and the white (lighter) regions. On some panels of positive genus (e.g., panels 2, 3 from the bottom on the left edge) there are no finite complementary arcs. 10 M. Bertola and A. Tovbis 2.2.1 Equality requirements and formula for g 1) g(z) (to shorten notations, we will often drop the t variable) is analytic in C \ Σ, where Σ = (M ∪ C), and has the asymptotic behaviour g(z) = ln z +O ( z−1 ) , z →∞; (2.6) 2) g(z) is analytic along all the unbounded complementary arcs except for exactly one un- bounded complementary arc which we will denote by γ0 (we assume γ0 is oriented away from infinity), where g+(z)− g−(z) = 2iπ, z ∈ γ0 ⊂ C (2.7) (note that the function eNg(z) from (2.4) is analytic across the unbounded complementary arc since N = n ∈ N by definition). In fact, γ0 could be any simple contour connecting ∞ with a point on M that has no other intersections with Ω; 3) on each bounded complementary arc γc,j , j = 1, 2, . . . , L, the function g(z) has a constant jump g+(z)− g−(z) = 2πiηj , ηj ∈ R, z ∈ γc,j ⊂ C (2.8) (note that there are no bounded complementary arcs in the genus zero case L = 0); 4) across each main arc γm,j , j = 0, 1, 2, . . . , L, the function g(z) has a jump g+(z) + g−(z) = f(z) + `+ 2πi$j , $j ∈ R, z ∈ γm,j ⊂M, (2.9) where $0 = 0. We stress that the constant ` is the same for all the main arcs; 5) the limiting values g±(z) along all the contours belong to L2 loc. Remark 2.5. The assumption that all the main and complementary arcs are simple smooth curves that can intersect each other only at the endpoints would have simplified the following discussion. In reality, though, main (complementary) arcs could intersect each other in a finite number of interior points forming, for example, crosses, “Y”-shaped contours (whiskers), etc., see, for example, various panels on Fig. 2. If an endpoint of a main arc γm,j−1 is in the interior of the neighboring main arc γm,j for some j = 1, . . . , L, then there is no complementary arc γc,j that connects these two main arc. Thus, the number of the complementary arcs in (2.8) can be less than L. However, in this case, the constant $j from (2.9) on γm,j will take different values on the different parts of this arc (separated by the intersection point). This jump condition will be equivalent to the situation where the constant $j is the same on all of the γm,j , but there is another jump (2.8) coming from the arc γc,j , which is the part of γm,j that connects one of the endpoints with the point of intersection. Thus, we can assume that there are always L arcs γc,j in (2.8), with the understanding that only proper complementary arcs have to satisfy the sign requirements from Section 2.2.2. Remark 2.6. Assuming that the contours constituting M and C, as well as all the constants `, ηj , $j , j = 1, 2, . . . , L, are known, the conditions 1-5 form an RHP for the function g(z). If a branch of ln z in (2.6) is fixed, that RHP would have a unique solution (constructed below), provided that the constants can be chosen so that the first requirement (2.6) is satisfied. We start the construction of g(z) by removing the jump on the unbounded contour γ0. Let λ2j , λ2j+1 be the beginning and the end points of the (oriented) main arc γm,j , j = 0, . . . , L. We define the radical R(z) = √√√√2L+1∏ j=0 (z − λj), (2.10) Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 11 where the branchcuts of R(z) are γm,j , j = 0, 1, . . . , L, and the branch of R is defined by limz→∞ R(z) zL+1 = 1. The hyperelliptic Riemann surface R(t) is the Riemann surface of R(z). Let ω̃ = P̃ (ζ)dζ R(ζ) be a normalized meromorphic differential of the third kind with two simple poles at ∞± with residues ±1 respectively (i.e., P̃ (z) is monic of degree L). It is well known [14] that the above conditions define P̃ (z) uniquely. The definition of “normalized” means that∫ γm,j P̃ (ζ)dζ R+(ζ) = 0 for all j = 1, . . . , L. (2.11) Let u(z) = ∫ z λ0 P̃ (ζ) R(ζ) dζ, (2.12) where λ0 is the beginning of the main arc γm,0. It will be also convenient to choose λ0 as the (finite) endpoint of the contour γ0. Then u(z) is analytic in C \ (M ∪ γ0), has analytic limiting values on M ∪ γ0 (except, possibly, the branchpoints) and satisfies u(λ0) = 0, u+(z)− u−(z) = 2iπ, z ∈ γ0, u+(z) + u−(z) = 0, z ∈ γm,0, and u(z)− ln z is analytic at z =∞. We will be looking for g(z) in the form g(z) = u(z) + v(z). Then v(z) must be analytic in C̄ except for the main arcs M and bounded complementary arcs ∪Lj=1γc,j , where, according to (2.9), (2.7), it satisfies jump conditions v+(z) + v−(z) = f(z)− [u+(z) + u−(z)] + `+ 2πi$j , v+(z)− v−(z) = 2πiηj on the corresponding contours. By Sokhotski–Plemelj formula, v(z) = R(z) 2πi [∫ M f(ζ)− [u+(ζ) + u−(ζ)] + ` (ζ − z)R(ζ)+ dζ + L∑ j=1 (∫ γm,j 2πi$jdζ (ζ − z)R(ζ)+ + ∫ γc,j 2πiηjdζ (ζ − z)R(ζ) )] . The solution g(z) to the RHP from Remark 2.6 is then given by g(z) = u(z) + R(z) 2πi [∫ M f(ζ)− [u+(ζ) + u−(ζ)] + ` (ζ − z)R(ζ)+ dζ + L∑ j=1 (∫ γm,j 2πi$jdζ (ζ − z)R(ζ)+ + ∫ γc,j 2πiηjdζ (ζ − z)R(ζ) )] = u(z) + R(z) 4πi [∮ M̂ f(ζ)− 2u(ζ) + ` (ζ − z)R(ζ) dζ + L∑ j=1 (∮ γ̂m,j 2πi$jdζ (ζ − z)R(ζ) + ∮ γ̂c,j 2πiηjdζ (ζ − z)R(ζ) )] , (2.13) where γ̂m,j , γ̂c,j is a negatively oriented loop around γm,j , γc,j respectively, with the latter traversing both sheets of the Riemann surface R(t), j = 1, . . . , L, and M̂ denotes a negatively oriented loop around M that (because of the u(z) term) passes through λ0. Here z is chosen outside all the loops. 12 M. Bertola and A. Tovbis Remark 2.7. Note that the jumps of g are analytic (in z) functions; thus, the interior of any main or complementary arc can be smoothly deformed without affecting the value of g(z), as long as the deformed arc does not pass through z. Moreover, the limiting values g±(z) are analytic functions on the contour Ω with the exception of the branchpoints of the Riemann surface R(t) (endpoints of the main and complementary arcs). Remark 2.8. Similarly to g, the function h = 2g − f − ` (2.14) can be considered as the (unique) solution of the scalar RHP with the jumps h+(z)− h−(z) = 4πiηj , z ∈ γc,j , h+(z) + h−(z) = 4πi$j , z ∈ γm,j , h+(z)− h−(z) = 4iπ, z ∈ γ0, (2.15) across all the main and all the bounded complementary arcs and with the asymptotic behavior h(z) = −f(z)− `+ 2 ln z +O ( z−1 ) , z →∞. (2.16) It follows immediately from (2.15) that <h(z) is continuous across the complementary arcs C. Also, (2.13) implies that h(z) = R(z) 2πi [∮ M̂ f(ζ) + `− 2u(ζ) (ζ − z)R(ζ) dζ + L∑ j=1 (∮ γ̂m,j 2πi$jdζ (ζ − z)R(ζ) + ∮ γ̂c,j 2πiηjdζ (ζ − z)R(ζ) )] , (2.17) where z is inside the loop M̂ but outside all other loops. Notice that ` can be removed from (2.17) since ∮ M̂ dζ (ζ−z)R(ζ) = 0 when z is inside M̂. 2.2.2 Inequality (sign) requirements (or sign distribution requirements) for h and the modulation equation The following sign distribution requirements must be satisfied: 1) in the interior of every bounded complementary arc γc,j ⊂ C, j = 1, . . . , L, as well as in the interior of every unbounded complementary arc, we have <h(z) < 0; 2) if z0 is an interior point of a main arc γm,j ⊂ M, j = 0, 1, . . . , L then on both sides γm,j in close proximity of z0 we have <h(z) > 0. Remark 2.9. The second requirement together with (2.15) imply that <h(z) ≡ 0 on M. Thus, the jump conditions (2.15) imply that <h(z) is continuous everywhere in C. Remark 2.10. We shall call the above mentioned case regular, with the same connotation as in [10]. Violation of the above mentioned strict inequalities is allowed in exceptional (non- regular) cases at no more than a finite number of interior points of the corresponding main and/or complementary arcs Ω = C∪M. Let z∗ ∈ Ω denote one of such points. Then <h(z∗) = 0. Indeed, this is true when z∗ ∈ M, see Remark 2.9; otherwise, when z∗ ∈ C, this is implied by the continuity of <h(z). Moreover, the above requirements imply that there are at least four zero level curves of <h(z) = 0, emanating from z = z∗ (pinching). Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 13 2.3 Modulation equations The sign distribution requirements of Section 2.2.2 imply that <h(z) should attain both positive and negative values in a vicinity of any branchpoint λ, where a main and a complementary arcs meet. On the other hand, if an endpoint λ of a main arc is an interior point of another main arc, then there are at least three zero level curves of <h(z) emanating from λ. Combined with the requirement g± ∈ L2 loc(Ω), we see that <h(z) = O(z − λk) 3 2 , z → λk, (2.18) at every branchpoint λk, k = 0, 1, . . . , 2L + 1. In other words, <h(z) → 0 at least at the order (z − λk) 3 2 (or faster) as z → λk. Consider equation (2.17) in the light of the latter requirement. Moving z to a vicinity of a branchpoint λk, k = 0, . . . , 2L + 1, one has to cross some loops γ̂m,j , γ̂c,l, which yield terms 2πiωj and ±2πiηl respectively, j, l = 1, . . . , L. These terms do not affect <h(z). It now follows from (2.17) that <h(z) = O(z − λk) 1 2 , k = 0, 1, . . . , 2L+ 1. Then, the conditions (2.18) can be equivalently restated as∮ M̂ f(ζ) + ` (ζ − λk)R(ζ)+ dζ + L∑ j=0 ∮ γ̂m,j 2πi$jdζ (ζ − λk)R(ζ)+ + L∑ j=1 ∮ γ̂c,j 2πiηjdζ (ζ − λk)R(ζ) + ∫ γ̃0 2πidζ (ζ − λk)R(ζ) = 0, where k = 0, 1, . . . , 2L+ 1 and γ̃0 is a positively oriented loop around γ0. Assuming that all the constants `, ηj , ωj , j = 1, 2, . . . , L, are known (see Section 3.2), the system (2.3) is a system of 2L+ 2 equations (2.18) for the 2L+ 2 unknown branchpoints λk. It is known as the system of modulation equations that governs the location of the endpoints λk provided the genus L of the corresponding hyperelliptic Riemann surface R(t) is given. The modulation equations (2.18) guarantee that there are at least three zero level curves of <h emanating from each branch-point λk, k = 0, 1, . . . , 2L + 1. They do not guarantee, however, that the sign requirements for h are satisfied. The latter requires a particular choice of the genus L of R(t) depending on t. The modulation equations (2.18) imply that solutions of the RHPs for g and for h commute with differentiation, since the limiting values of g±, h± and g′±, h′± along Ω belong to L2 loc(Ω). Thus, the modulation equations (2.18) can be equivalently stated as h′(z) = O (√ z − λk ) , z → λk, k = 0, 1, . . . , 2L+ 1 (note that h′(z) is not continuous on the main arcs). The function h′ can be obtained as the solution of the RHP for h′(z) with the following jump conditions and the asymptotics at infinity h′+(z) + h′−(z) = 0, z ∈ γm,j , j = 0, 1, . . . , L, h′(z) = −f ′(z) + 2 z +O ( z−2 ) , z →∞. (2.19) In view of (2.14), the RHP for g′(z) (the jump conditions and asymptotics at infinity) is given by g′+(z) + g′−(z) = f ′(z), z ∈ γm,j , j = 0, 1, . . . , L, g′(z) = 1 z +O ( z−2 ) , z →∞. 14 M. Bertola and A. Tovbis Then solution to (2.3) is given by g′(z) = R(z) 4πi ∮ M̂ f ′(ζ) (ζ − z)R(ζ) dζ, (2.20) where z is outside the contour of integration. Using the residue theorem, we see that h′(z) = R(z) 2πi ∮ M̂ f ′(ζ) (ζ − z)R(ζ) dζ = R(z)M(z), (2.21) where z is now inside the contour M̂. Equation (2.21) shows that M(z) is a polynomial of degree deg f ′ − L − 1. If R(z) (that is, the Riemann surface R(t)) is given, then M(z) is defined by the asymptotics in (2.19). This argument, in particular, shows that for a quartic polynomial f the genus L of R(t) cannot exceed 2. In fact, our Riemann surface R(t) is the nodal curve y2(z) = h′2(z) = M2(z) 2L+1∏ j=0 (z − λj) = P (z), (2.22) where P (z) is a polynomial and degP = 2 deg f − 2. In case of a general potential f(z;~t), the genus L is bounded by L ≤ deg f − 2. (2.23) The modulation equations now can be again recast as the requirements that: i) the mero- morphic differential h′(z)dz on the hyperelliptic surface R(t) has the asymptotics as required in (2.19), and ii) the Riemann surface R(t) satisfies the Boutroux condition for all t ∈ C∗, that is, all the periods of h′(z)dz are purely imaginary (for all t ∈ C). The corresponding equations can be written as 1 4πi ∮ M̂ ζkf ′(ζ)dζ R(ζ) = −δk,L+1, < ∮ γ̂m,j h′(ζ)dζ = 0, < ∮ γ̂c,j h′(ζ)dζ = 0, (2.24) where k = 0, 1, . . . , L+ 1, j = 1, . . . , L and δk,l is the Kronecker delta. The first L+ 2 equations in (2.24), also known as moment conditions, express the fact that the large z asymptotics of g′(z) is given by (2.3) (requirement i)). The remaining equations (2.24), also known as integral conditions [25], express the requirement ii). Equivalence of modulation equations (2.18) and (2.24) follows from the fact that the solution of the RHP for g, h commutes with the differentiation and from the existence and uniqueness of the solutions of the RHPs for g, h, g′, h′. 2.4 Schematic conclusion of the steepest descent analysis In this subsection we briefly describe the construction of the leading order asymptotics in the case of a positive genus. The corresponding construction for the genus zero case was given in [4]. For convenience of the reader, we repeat here some common parts (for any genus). Fix some t ∈ C∗ \ {t0, t2}. Let ~λ = (λ0, . . . , λ2L+1) be a solution to the modulation equations with some L ∈ N∪{0}. Deform the contour Ω in such a way that λk ∈ Ω for all k = 0, . . . , 2L+1 and partition it into a collection of the main and complementary arcs as discussed in Section 2.2. If the g-function (2.13) satisfies all the requirements of Section 2.2 (equalities and inequalities), then we say that t belongs to the genus L region of C and proceed with the construction of the leading order asymptotics of the recurrence coefficients, as outlined below. The choice of the correct genus for a given t ∈ C∗ \ {t0, t2} will be discussed in Section 4.3. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 15 The remaining steps in the steepest descent analysis involve inserting additional contours, the lenses, which enclose each main arc and lie entirely within the −<h < 0 region, the so- called sea. (On the panels in Fig. 2, these regions (the sea) are shown in blue (darker) color.) One then re-defines T (z) from the RHP (2.5) within the regions between the main arc and its corresponding lens by using the factorization( a d 0 a−1 ) = ( 1 0 a−1d−1 1 )( 0 d −d−1 0 )( 1 0 ad−1 1 ) (2.25) of the jump matrices of T so that T+(z) = T−(z) [ 1 0 ν−1 m e−Nh− 1 ] [ 0 νm −ν−1 m 0 ] [ 1 0 ν−1 m e−Nh+ 1 ] , where νm is the (constant!) value of ν(z) on a given main arc γm under consideration. (For simplicity, we use here notation γm instead of γm,j for a given particular main arc.) Therefore, defining T̂ (z) as T outside of the lenses and by T̂ (z) := T (z) [ 1 0 ∓ν−1 m e−Nh 1 ] in the regions within the lenses and adjacent to the± sides of γm one obtains a new RHP for T̂ (z). In this RHP, the jump matrices on the lenses and on the complementary arcs turn out to be exponentially close to the identity in any Lp, (1 ≤ p < +∞) as N → ∞ due to the inequality requirements of Section 2.2.2, see, for example, [2] for details. (Because of the asymptotics (2.16), where f(z) is a polynomial, we can always assume that supz∈Ω<h(z) < 0 for all sufficiently large z ∈ Ω; that establishes the above statement for any unbounded complementary arc.) The problem corresponding to the remaining jumps can be solved exactly. The fact that the neglected jump matrices do not approach the identity matrix 1 in L∞ is addressed by the construction of appropriate local solutions of the RHP called parametrices [2, 10]. The type of local RHP depends on the behavior of h(z) near the endpoints. For regular values of t (away from the breaking curves) we use the standard Airy parametrix at any branchpoint λj where a main arc meets a complementary arc. This Airy parametrix should be appropriately modified [2] if at least three main arcs come together at λj . Finally, if t is a regular point on the breaking curve and if λ is one of the interior points of M ∪ C, where the sign requirement of Section 2.2.2 is violated, the local parametrix near λj is of the same type as in [3]. Consider, for simplicity, a regular point t. In the final steps of the approximation we fix sufficiently small disks Dj , around the endpoints λj , j = 0, . . . , L+ 1, and define a suitable approximate solution Φ(z) := { Φext(z) for z outside ∪jDj , Φext(z)Pj(z) inside Dj , such that the error matrix E(z) := T̂ (z)Φ−1(z) solves a small-norm Riemann–Hilbert problem (as N →∞) and thus can be – in principle – completely solved in Neumann series. Here by Pj(z) we denote the parametrices near the endpoints λj respectively, j = 0, . . . , L + 1. The RHP for Φext = Ψ0 (“model solution” or “exterior parametrix”) will be discussed below, see (2.26). Once we have achieved a suitable approximation for T̂ (z), the recurrence coefficients for the orthogonal polynomials can and will be recovered via Proposition 2.3: hn = −2iπeN` (T1)12 , αn = (T1)12 (T1)21 , βn = (T2)12 (T1)12 − (T1)22 . 16 M. Bertola and A. Tovbis Here we used (2.4) and the fact that T̂ (z) near ∞ equals T (z) (since we are in the exterior region) and has expansion T̂ (z) = T (z) = 1 + T1 z + T2 z2 + · · · , z →∞. The matrix entry T1,2 can be obtained from the corresponding expansion of Φext(z) near infinity, to within the error determined by the error matrix E = T̂Φ−1. In the case of a regular t, the standard error analysis (which we do not report here) shows that E introduces an error of order O(N−1) that is uniform on compact subsets of regular t. The higher genus case (i.e., more than one main arc) with real potentials and on the real line was first fully treated in [10] and was later extended to the complex plane and complex potentials in [2]. Consider first the case of interlacing main and complementary arcs, i.e., the region inside the two Schwarz-symmetrical curves connecting the points t0 = − 1 12 and t2 = 1 4 on Fig. 2 (we, obviously, exclude the genus zero region bounded by contours connecting the points t0 = − 1 12 and t = 0). Remark 2.11. Let ρj denote the traffic on the main arc γm,j , j = 0, 1, . . . , L. Without any loss of generality, see Remark 2.2, we assume that ρ0 = 1. Alternatively, we can replace ρj by ρj/ρ0 in the formulae below. Let σ1 = [ 0 1 1 0 ] , σ2 = [ 0 −i i 0 ] , σ3 = [ 1 0 0 −1 ] denote the Pauli matrices. Following the standard arguments, we obtain the following RHP (model problem) Ψ0(z) is analytic in C \ Ω̂, Ψ0+(z) = Ψ0−(z)e(2πi$jN+ln ρj)σ3iσ2 on γm,j , j = 0, 1, . . . , L, Ψ0+(z) = Ψ0−(z)e−2πiηjNσ3 on γc,j , j = 1, . . . , L, Ψ0(z) = 1 +O(z−1) as z →∞, Ψ0(z) = O ( (z − λk)− 1 4 ) , z → λk, k = 0, 1, . . . , 2L+ 1 (2.26) for the“model solution” or “exterior parametrix” matrix Ψ0(z). Here Ω̂ is the collection of all main and all bounded complementary arcs, the real constants $j , ηj are given by (2.15) and the constants ρj ∈ C∗ are equal to the traffic on the oriented main arc γm,j , obtained by deformation of the original contours into Ω. In the case with whiskers, that is, when three main arcs come together, some jumps γc,j are created according to Remark 2.5. In this case, the diagonal jump matrix on γc,j in (2.26) will have additional ln ρ terms in the exponential. 2.5 Solution of the model problem To solve the model RHP (2.26) in terms of Riemann theta-functions we follow the approach of [12], see also [25] (for an alternative approach in the case of real main arcs see [19]). First, we use the transformation Ψ̃0(z) = e−Ng̃(∞)σ3Ψ0(z)eNg̃(z)σ3 in order to remove the jumps e−2πiηjNσ3 of Ψ0(z) on the complementary arcs. The function g̃ is analytic in C\Ω̂ and have constant jumps on Ω̂. If V denotes the jump matrix for Ψ0(z), then the jump matrix Ṽ for Ψ̃0(z) is Ṽ = e−Ng̃−(z)σ3V eNg̃+(z)σ3 . (2.27) To remove the jumps on γc,j , we require that g̃+ − g̃− = 2πiηj on γc,j and g̃+ + g̃− = 2πiκj on γm,j , j = 1, . . . , L, Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 17 where the complex constants κj are defined by the requirement that g̃ is analytic at z = ∞. Then Ψ̃0(z) solves the RHP Ψ̃0(z) is analytic in C \ Ω̂, Ψ̃0+(z) = Ψ̃0−(z)e(2πiN$̃j+ln ρj)σ3iσ2 on γm,j , j = 0, 1, . . . , L, Ψ̃0(z) = 1 +O ( z−1 ) as z →∞, Ψ̃0(z) = O(z − λk)− 1 4 , z → λk, k = 0, 1, . . . , 2L+ 1, (2.28) where $̃j = $j + κj , j = 1, . . . , L, and $̃0 = 0. The solution to the RHP (2.28) is known [12, 25]. To present it, we need to introduce the Riemann theta function associated with the hyperelliptic Riemann surface R(t). Let Aj = γ̂m,j , j = 1, . . . , L be A-cycles of R, whereas the (standard) set of B-cycles Bj consists of the properly oriented loops passing through the branchcuts γm,0 and γm,j . The basis ~ω = (ω1, . . . , ωL) of normalized holomorphic differentials, dual to the A-cycles, is defined by∫ γ̂m,j ωk = δjk, j, k = 1, . . . , L, (2.29) where δjk denotes the Kronecker symbol. Each of these differentials ωj has the form ωj = Pj(z) R(z) dz, where Pj(z) is a polynomial of degree less than L. The Riemann period matrix is defined τ = (τkj) = (∫ Bj ωk ) , k, j = 1, . . . , L. (2.30) τ is known to be symmetric and have positive definite imaginary part. The Riemann theta function is defined as Θ(s) = ∑ l∈Z2N e2πi(l,s)+πi(l,τ l), s ∈ C2N . It satisfies (a) Θ(s) = Θ(−s), (b) Θ(s+ ej) = Θ(s), (c) Θ(s+ τj) = e−2πisj−πiτjjΘ(s), (2.31) where ej is the j-th column of 1L×L and τj = τej . The set Λ = ZL + τZL is called the period lattice and J = CL mod Λ the Jacobian. Define the Abel map by u(z) = ∫ z λ0 ~ω, and the vector of functions M(z, d) ≡ (M1,M2) = ( Θ(u(z)− Ŵ + d) Θ(u(z) + d) , Θ(−u(z)− Ŵ + d) Θ(−u(z) + d) ) , (2.32) where Ŵ = ( Ŵ1, . . . , ŴL ) = ( N$̃1 + ln ρ1 2πi , . . . , N$̃L + ln ρL 2πi )t (2.33) 18 M. Bertola and A. Tovbis and d ∈ CL is a vector that will be chosen appropriately later on. Note that u+ + u− = −τj on γm,j , j = 0, 1, . . . , L, where τ0 = 0, and u+ = u− − L∑ k=j ek on γc,j , j = 1, . . . , L, (2.34) and u(λ0) = 0, u(λ2j+1) = −1 2 τj + L∑ k=j+1 ek  , j = 0, 1, . . . , L, u(λ2j) = −1 2 τj + L∑ k=j ek  , j = 1, . . . , L. (2.35) Although u(z) is multivalued, M(z, d) is single-valued and meromorphic on C \M. Moreover, according to (2.34) and the third equation in (2.31), M satisfies M+ =M− ( 0 e2πiŴj e−2πiŴj 0 ) on γm,j , j = 0, 1, . . . , L, where Ŵ0 = 0. Theorem 2.12 ([14, p. 308]). Let f ∈ CL be arbitrary, and denote by u(p) the Abel map (extended to the whole Riemann surface R). The (multi-valued) function Θ(u(z) − f) on the Riemann surface either vanishes identically or vanishes at L points p1, . . . , pL (counted with multiplicity). In the latter case we have f = L∑ j=1 u(pj) +K, (2.36) where K = L∑ j=1 u(λ2j). The sign of K in (2.36) is irrelevant since 2K ∈ Λ. An immediate consequence of Theorem 2.12 is the following statement. Corollary 2.13. The function Θ vanishes at e ∈ Λ if and only if there exist L − 1 points q1, . . . , qL−1 on R such that e = L−1∑ j=1 u(qj) +K. Definition 2.14. The Theta divisor is the locus e ∈ J such that Θ(e) = 0. It will be denoted by the symbol (Θ). Let r(z) =  L∏ j=0 z − λ2j+1 z − λ2j  1 4 , (2.37) Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 19 where the branch cuts of r(z) are along the main arcs M and r(z) is normalized by lim z→∞ r(z) = 1. One can also construct the solution to the RHP (2.28) in the form of L(z) = 1 2 ( (r(z) + r−1(z))M1(z, d) −i(r(z)− r−1(z))M2(z, d) i(r(z)− r−1(z))M1(z,−d) (r(z) + r−1(z))M2(z,−d) ) , (2.38) where the vector d ∈ CL is to be defined and M(z, d), r(z) were given by (2.32), (2.37) re- spectively. Indeed, using the facts that (r ± r−1)+ = i(r ∓ r−1)−, it is straightforward to check that L(z) satisfies the jump conditions listed in (2.28). The vector d ∈ CL is found by requiring the analyticity of L(z) in C\M; to this end, consider the equation r4(z)− 1 = 0 ≡ L∏ j=0 (z − λ2j+1)− L∏ j=0 (z − λ2j) R(z) = 0. Note that r4(z)− 1 is a meromorphic function on the Riemann surface R with a zero z0 =∞+ (infinity on the main sheet) and 2L (counted with multiplicities) zeroes z1, . . . , zL counted twice (for each of the two sheets of R). Zeroes z1, . . . , zL are finite provided L∑ j=0 λ2j 6= L∑ j=0 λ2j+1. With a slight abuse of notation, we denote by z1, . . . , zL zeroes of r2(z) − 1 = 0 other than z0 that can be located on either sheet of R. It is straightforward to check that if z is a zero of r2(z)− 1 = 0 then ẑ is a zero of r2(z) + 1 = 0, where ẑ denotes the projection of z on the other sheet of the hyperelliptic surface R. If we now choose d = − L∑ j=1 ∫ ẑj λ2j ~ω, then, according to Theorem 2.12, the set of all zeroes of Θ(u(z) + d) in R consists of ẑj , j = 1, . . . , L. Indeed, Θ(u(z) + d) = Θ u(z) +K − L∑ j=1 u(λ2j)− L∑ j=1 ∫ ẑj λ2j ~ω  = Θ u(z) +K − L∑ j=1 u(ẑj)  . Since Θ(u(zj)− d) = Θ(−u(ẑj)− d) = Θ(u(ẑj) + d) = 0 we obtain that the set of all zeroes of Θ(−u(z) + d) in R consists of zj , j = 1, . . . , L. Since all the zeroes zj , j = 1, . . . , L, of Θ(u(z)− d) are also zeroes of r(z)− 1 r(z) and all the zeroes ẑj , j = 1, . . . , L of Θ(u(z) + d) are also zeroes of r(z) + 1 r(z) , we conclude that L(z) does not have poles on R. In particular, L(z) is analytic in C\M. It obviously satisfies the boundary requirements of (2.28) at the branchpoints. Lemma 2.15. d = −u(∞+) mod Λ, (2.39) where Λ is the period lattice. 20 M. Bertola and A. Tovbis Proof. By construction, r2(z) − 1 is a meromorphic function on R with simple poles at λ2j , j = 0, 1, . . . , L, and simple zeroes at zj , j = 1, . . . , L, and at infinity on the main sheet∞ =∞+. Let D denote the divisor of r2(z) − 1. Then according to Abel’s theorem (see [14]), U(D) = 0 mod Λ. Thus, u(∞) + L∑ j=1 u(zj) = L∑ j=0 u(λ2j) mod Λ. (2.40) Then (2.39) follows from (2.40), L∑ j=1 u(zj) = L∑ j=0 u(λ2j) + d and u(λ0) = 0. � With Lemma 2.15, we obtain the following lemma. Lemma 2.16. The matrix L(z) with d = u(∞+) and Ŵ as in (2.33) is analytic in C \M and satisf ies the same jump conditions as Ψ̃0(z) from the RHP (2.28). Since L(z) satisfies the same jump conditions as Ψ̃0(z), the determinant detL(z) must be analytic in C. On the other hand, according to (2.38) and (2.37) detL(∞) =M1(∞, d)M2(∞,−d) = Θ ( u(∞)− Ŵ + d ) Θ ( u(∞) + Ŵ + d ) Θ2(u(∞) + d) = Θ2 ( Ŵ ) Θ2(0) . (2.41) L(z) is bounded at z =∞, so that detL(z) ≡ Θ2 ( Ŵ ) Θ2(0) . Thus, we obtain Ψ̃0(z) = L−1(∞)L(z). Consequently we have proven the following analog of Theorem 2.17. Theorem 2.17. The RHP (2.28)admits a solution Ψ̃0 if and only if Θ ( Ŵ ) 6= 0. Corollary 2.18. If Ŵ = u(λ2j) mod Λ for some j = 1, . . . , L, that is N ~̃$ + ~ln ρ 2πi = 1 2 τj + L∑ k=j ek  mod Λ, (2.42) for some j = 1, . . . , L, where ~̃$ = ($̃1, . . . , $̃L), ~ln ρ = (ln ρ1, . . . , ln ρL), then the RHP (2.28) does not have a solution. Condition (2.42) is necessary and sufficient in the case L = 1. Proof. The corollary follows from Corollary 2.13 and (2.41), (2.35) and (2.33). � Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 21 Theorem 2.19. Assume that the h function h(z; t), defined by the RHP (2.15), (2.16) with L+1 main arcs, satisfies the sign requirements (strict inequalities) in some compact region T ⊂ C∗. Then the recurrence coefficients αn = αn(t) and the pseudonorms hn = hn(t) satisfy αn = 1 16  L∑ j=0 (λ2j − λ2j+1) 2 Θ ( 2u∞ − Ŵ ) Θ ( 2u∞ + Ŵ ) Θ2(0) Θ2 ( Ŵ ) Θ2(2u∞) +O ( N−1 ) , (2.43) hn = −π 2  L∑ j=0 (λ2j − λ2j+1)  Θ ( 2u∞ + Ŵ ) Θ(0) Θ ( Ŵ ) Θ(2u∞) eN` +O ( N−1 ) , (2.44) in the limit N → ∞ uniformly in t ∈ T provided that Θ ( Ŵ ) is separated from zero. Here u∞ = u(∞+) and `, Ŵ are given by (3.19) and (2.33), (3.18) respectively. Proof. Note that r(z)− r−1(z) = 1 2z L∑ j=0 (λ2j − λ2j+1) +O ( z−2 ) (2.45) as z →∞. According to (2.45) and (2.41), the O(z−1) (residue) term in the expansion of Ψ̃0 at infinity has (1, 2) and (2, 1) terms given by −i L∑ j=0 (λ2j − λ2j+1) · M2(∞, d) 4M1(∞, d) and i L∑ j=0 (λ2j − λ2j+1) · M1(∞,−d) 4M2(∞,−d) . (2.46) Then, combining (2.3), (2.4), (2.27), we obtain αn = 1 16  L∑ j=0 (λ2j − λ2j+1) 2 Θ ( u∞ − Ŵ − d ) Θ ( u∞ + Ŵ − d ) Θ2(u∞ + d) Θ ( u∞ − Ŵ + d ) Θ ( u∞ + Ŵ + d ) Θ2(u∞ − d) . We now use (2.3), Lemma 2.16 and properties (2.31) of theta functions to obtain (2.43), (2.44). The error estimates follow from assumptions of the theorem. � Remark 2.20. Theorem 2.19 is valid for general polynomial potentials f(z;~t). 3 Formal calculation of the g-function and h-function In this section, following [4], we calculate explicitly g(z; t) and h(z; t) in the genus zero case. We will also derive the determinantal expressions for g and h in the higher genera regions for general polynomial potentials f = f(z;~t). 3.1 Explicit form of g and h functions in the genus zero region In the genus zero case, M consists of a single main arc γm with the endpoints λ0,1 = λ0,1(t). Applying the Sokhotski–Plemelj formula for the RHP (2.3) and using the analyticity of f ′, we obtain g′(z) = R(z) 4πi ∮ γ̂m f ′(ζ) (ζ − z)R(ζ)+ dζ, R(z) = √ (z − λ1)(z − λ0), (3.1) 22 M. Bertola and A. Tovbis where the branchcut of R(z) is γm and the branch of R is defined by lim z→∞ R(z) z = 1; the contour γ̂m encircles the contour γm and has counterclockwise orientation, and; z is outside γ̂m. Note that if z is inside γ̂m, the right hand side of (3.1) is equal to g′(z)− 1 2f ′(z) = 1 2h ′(z). The modulation equations (2.24) in the case L = 0 reduce to the following moment conditions:∮ γ̂m f ′(ζ) R(ζ)+ dζ = 0 and ∮ γ̂m ζf ′(ζ) R(ζ)+ dζ = −4iπ. (3.2) We use the moment conditions (3.2) to define the location of the endpoints λ0,1. Since f is a polynomial, the equations (3.2) can be solved using the residue theorem on equations (3.2), setting λ0 = a− b, λ1 = a+ b. We obtain the system a+ ta ( a2 + 3 2 b2 ) = 0 and a2 + 1 2 b2 + t ( a4 + 3a2b2 + 3 8 b4 ) = 2. (3.3) There are two possibilities: a = 0 or a 6= 0. The Generic configuration of the traffic, see Section 1, considered in the paper, forces the first (symmetric) case, see [4]. In this case we obtain solutions to the system (3.3) as a = 0, b2 = − 2 3t ( 1∓ √ 1 + 12t ) , (3.4) λ0,1 = ∓b = ∓ √ − 2 3t ( 1− √ 1 + 12t ) . (3.5) The choice of the negative sign in (3.4) comes from the requirement that b is bounded as t→ 0. Observe that for t > t0 = − 1 12 the values ±b coincide with the branch-points, derived in [13]. The second pair of roots b = ± √ − 2 3t(1 + √ 1 + 12t) are sliding along the real axis from ±∞ to ±b0 = ± √ 8 as real t varies from 0− to t0, and are sliding along the imaginary axis from ±i∞ to 0 as real t varies from 0+ to +∞. At the point t = t0 = − 1 12 the two pairs of roots (3.4) coincide, creating five zero level curves of <h(z) emanating from the endpoints ±b0 (see [4, 13]). Once the values of branch-points (endpoints) λ0,1 = λ0,1(t) are determined, one can calculate explicitly g(z) = g(z; t) and h(z) = h(z; t), where h(z) = 2g(z)− f(z)− `. We are first looking for h′(z) = −(k + tz2) √ z2 − b2, where the endpoints ±b of γm are given by (3.4). The asymptotics in (2.19) implies k = 1 + 1 2 tb 2, so that h′(z) = − [ tz2 + 1 + tb2 2 ] ( z2 − b2 ) 1 2 = − [ tz2 + √ 1 + 12t 3 + 2 3 ] ( z2 − b2 ) 1 2 . (3.6) Since the branch-cut of the radical is [−b, b] we conclude that h′(z) is an odd function. Direct calculations yield h(z) = 2 ln z + √ z2 − b2 b − z 8 ( 2tz2 + tb2 + 4 )( z2 − b2 ) 1 2 . (3.7) It is clear that h(b) = 0. Combined with (3.6), that implies h(z) = O(z − b) 3 2 (3.8) (or a higher power of (z − b)). It is easy to check that at t0 = − 1 12 the order O(z − b) 3 2 in (3.8) can be replaced O(z − b) 5 2 . There is the oriented branch-cut of h(z) along the ray (−∞,−b), where h+(z) − h−(z) = 4πi. Because of this jump, the function h(z) does not have O(z + b) 3 2 behavior near z = −b; however, <h(z) does have O(z + b) 3 2 behavior near z = −b. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 23 Remark 3.1. One can verify directly that h(z) satisfies the following RHP: 1) h(z) is analytic (in z) in C \ {γm ∪ (−∞, λ0)} and h(z) = −1 4 tz4 − 1 2 z2 + 2 ln z − `+O ( z−1 ) as z →∞, where ` = ln b2 4 − b2 8 − 1 2 ; 2) h(z) satisfies the jump condition h+ + h− = 0 on γm and h+ − h− = 4πi on (−∞, λ0). This RHP is the genus zero case (L = 0) of the general RHP for h from Remark 2.8. Remark 3.2. As it was mentioned above, the solution to the scalar RHP for h commutes with differentiation in z; on the same basis, it commutes with differentiation in t as well. Thus, we obtain the following RHP for ht: 1) ht(z) is analytic (in z) in C̄ \ γm and ht(z) = −1 4 z4 − `t +O ( z−1 ) as z →∞, (3.9) where `t = 3 32 b4 = 2 + 12t− 2 √ 1 + 12t 24t2 ; 2) ht(z) satisfies the jump condition ht+ + ht− = 0 on γm. This RHP has the unique solution ht(z) = −z 8 ( 2z2 + b2 )√ z2 − b2 (3.10) that can be verified directly. The explicit formula of ht is useful in controlling the sign of <h(z; t) when t is varying. 3.2 The g-function and h-function and their properties in the genus L case For the rest of this section we will be considering the case of a general polynomial potential f(z) = f(z,~t) = n∑ j=1 tj j zj (3.11) with ~t = (t1, . . . , tn) being a vector of (complex) coefficients of f . Then the genus L of the corresponding Riemann surface R = R(~t) satisfies (2.23). 24 M. Bertola and A. Tovbis In the case of genus L > 0, we start construction of g, h by defining the constants ηj , $j ∈ R, j = 1, 2, . . . , L, and ` ∈ C. Using (2.6), (2.13) (the second expression) and the expansion 1 ζ−z = −1 z ∞∑ k=0 ζk zk , we obtain 1 2πi ∮ M̂ ζk[f(ζ)− 2u(ζ)] R(ζ) dζ + ` 2πi ∮ M̂ ζkdζ R(ζ) + L∑ j=1 $j ∮ γ̂m,j ζkdζ R(ζ) + L∑ j=1 ηj ∮ γ̂c,j ζkdζ R(ζ) = −2δk,Lu∗, (3.12) k = 0, 1, . . . , L, where u∗ = lim z→∞ u(z)− ln z and δk,L is the Kronecker delta. It is clear that the second integral in (3.12) is zero for all k < L, so that the first L equations in (3.12) is a system of linear equations for 2L real unknowns ηj , $j . Then from the last equation we obtain ` = 2u∗ + 1 2πi ∮ M̂ ζL[f(ζ)− 2u(ζ)] R(ζ) dζ + L∑ j=1 $j ∮ γ̂m,j ζLdζ R(ζ) + L∑ j=1 ηj ∮ γ̂c,j ζLdζ R(ζ) . (3.13) Taking complex conjugates of the first L equations (3.12), we obtain − 1 2πi ∮ M̂ ζk(f(ζ)− 2u(z)) R(ζ) dζ + L∑ j=1 $j ∮ γ̂m,j ζkdζ R(ζ) + L∑ j=1 ηj ∮ γ̂c,j ζkdζ R(ζ) = 0, (3.14) k = 0, 1, . . . , L− 1. The first L equations from (3.12) combined with (3.14) form a system of linear equations D ( ~$ ~η ) = 1 2πi (∮ M̂ (f(ζ)− 2u(z)) R(ζ) ~ς(ζ)dζ, − ∮ M̂ (f(ζ)− 2u(z)) R(ζ) ~ς(ζ)dζ )t , (3.15) where ~$, ~η are vectors of constants $1, . . ., $L and η1, . . ., ηL respectively, ~ς(ζ)=(1, ζ, . . ., ζL−1)t and Dt =  ∮ γ̂m,1 dζ R(ζ) · · · ∮ γ̂m,1 ζL−1dζ R(ζ) ∮ γ̂m,1 dζ R(ζ) · · · ∮ γ̂m,1 ζL−1dζ R(ζ) · · · · · · · · · · · · · · · · · ·∮ γ̂m,L dζ R(ζ) · · · ∮ γ̂m,L ζL−1dζ R(ζ) ∮ γ̂m,L dζ R(ζ) · · · ∮ γ̂m,L ζL−1dζ R(ζ)∮ γ̂c,1 dζ R(ζ) · · · ∮ γ̂c,1 ζL−1dζ R(ζ) ∮ γ̂c,1 dζ R(ζ) · · · ∮ γ̂c,1 ζL−1dζ R(ζ) · · · · · · · · · · · · · · · · · ·∮ γ̂c,L dζ R(ζ) · · · ∮ γ̂c,L ζL−1dζ R(ζ) ∮ γ̂c,L dζ R(ζ) · · · ∮ γ̂c,L ζL−1dζ R(ζ)  . (3.16) Here ∗t denotes the transposition. Note that the contours γ̂m,j can be taken as α-cycles of the Riemann surface R = R(~t). Then the contours γ̂c,1, γ̂c,1 ∪ γ̂c,2, . . . , ∪Lj=1γ̂c,j with the opposite orientation form the β-cycles. Let us denote D = ( A B Ā B̄ ) , Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 25 where A and B are the corresponding L×L blocks of D. Then, taking proper linear combinations of the rows of A we can replace the integrands ζj R(ζ) with the normalized holomorphic differentials pj(ζ) R(ζ) of the Riemann surface R, where pj(ζ) are polynomials of degree not exceeding L−1. Thus the matrix A is nonsingular and D = ( A 0 0 Ā )( 1 A−1B 1 Ā−1B̄ ) , (3.17) so that Dt = ( 1 1 θ θ̄ )( At 0 0 Āt ) = ( 1 0 0 −H )( 1 1 τ τ̄ )( At 0 0 Āt ) , where θt = A−1B, H is a lower triangular matrix with all entries on and below the main diagonal being one. Then θ = −Hτ , where τ is the Riemann matrix of periods. As it is well known, =τ is a positive definite matrix. Thus \|D\| = \|Dt\| = (2i)L\|A\|2\|=τ \|, where \|D\| denotes detD, and we have proved the following lemma. Lemma 3.3 ([24]). If all the branchpoints λ0, λ1, . . . , λ2L+1 are distinct then (−i)L\|D\| > 0. Lemma 3.3 shows that there exists a unique set of real constants ~$, ~η satisfying the first L equations (3.12). Of course, this system of equations could be solved with all ~η = 0 if we allow ~$ to have complex components. Denoting the corresponding complex vector by ~̃$, we obtain ~̃$ = −A −1 2πi ∮ M̂ [f(ζ)− 2u(ζ)] R(ζ) ~ς(ζ)dζ = − 1 2πi ∮ M̂ [f(ζ)− 2u(ζ)]~ω = − 1 2πi ∮ M̂ f(ζ)~ω + 2u∞. (3.18) According to (2.11), we can also reduce (3.13) to ` = 2u∗ + 1 2πi ∮ M̂ [f(ζ)− 2u(ζ)]P̃ (ζ) R(ζ) dζ = 2u∗ + 1 2πi ∮ M̂ f(ζ)P̃ (ζ) R(ζ) dζ − 2π2, (3.19) since, according to (2.12),∮ M̂ u(ζ)P̃ (ζ) R(ζ) dζ = ∮ M̂ u(ζ)u′(ζ)dζ = 1 2 u2(z) ∣∣∣∣z=λ0e−2πi z=λ0 = −2π2. In view of Lemma 3.3, the system (3.15) has a unique solution given by( ~$ ~η ) = D−1 ( ~r ~̄r ) , where the vector ~r denotes the first L entries in the right hand side of (3.15). Moreover, this solution is real since the system (3.15) can be written as( <A <B =A =B )( ~$ ~η ) = ( <~r =~r ) . Using (3.15) and (3.17), we calculate ~η = ( = ( A−1B ))−1= ( A−1~r ) , ~$ = <(A−1~r)−< ( A−1B )( = ( A−1B ))−1= ( A−1~r ) . (3.20) 26 M. Bertola and A. Tovbis Alternatively, one can say that the vector −(~$t, ~ηt) coincides with the last row of the matrix K(z) diag((Dt)−1, 1) Now, according to (3.12), the expression (2.17) for h(z) can be represented as h(z) = R(z) \|D\| K(z), (3.21) where K(z) = 1 2πi ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∮ γ̂m,1 dζ R(ζ) · · · ∮ γ̂m,1 ζL−1dζ R(ζ) ∮ γ̂m,1 dζ R(ζ) · · · ∮ γ̂m,1 ζL−1dζ R(ζ) ∮ γ̂m,1 dζ (ζ−z)R(ζ) · · · · · · · · · · · · · · · · · · · · ·∮ γ̂m,L dζ R(ζ) · · · ∮ γ̂m,L ζL−1dζ R(ζ) ∮ γ̂m,L dζ R(ζ) · · · ∮ γ̂m,L ζL−1dζ R(ζ) ∮ γ̂m,L dζ (ζ−z)R(ζ)∮ γ̂c,1 dζ R(ζ) · · · ∮ γ̂c,1 ζL−1dζ R(ζ) ∮ γ̂c,1 dζ R(ζ) · · · ∮ γ̂c,1 ζL−1dζ R(ζ) ∮ γ̂c,1 dζ (ζ−z)R(ζ) · · · · · · · · · · · · · · · · · · · · ·∮ γ̂c,L dζ R(ζ) · · · ∮ γ̂c,L ζL−1dζ R(ζ) ∮ γ̂c,L dζ R(ζ) · · · ∮ γ̂c,L ζL−1dζ R(ζ) ∮ γ̂c,L dζ (ζ−z)R(ζ)∮ M̂ f̃(ζ)dζ R(ζ) · · · ∮ M̂ ζL−1f̃(ζ))dζ R(ζ) ∮ M̂ f̃(ζ)dζ R(ζ) · · · ∮ M̂ ζL−1f̃(ζ))dζ R(ζ) ∮ M̂ (f̃(ζ)+`))dζ (ζ−z)R(ζ) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ (3.22) with f̃(ζ) = f(ζ) − 2u(z), and where z is inside the loop M̂ but outside all other loops. In- deed, multiplying the first L rows of K(z) by 2πi$1, 2πi$2, . . . , 2πi$L respectively, the next L rows of K(z) by 2πiη1, 2πiη2, . . . , 2πiηL, and adding them to the last row, we obtain (3.21). Equation (3.21) gives 2g(z) when z is outside the loop M̂. It is clear that moving z inside the loops γ̂m,l, γ̂c,k would generate residue terms 2πi$l and ±2πiηk (depending on the direction z crosses the oriented loop γ̂c,k) in the right hand side of (3.21). Combining this fact with (2.3), we obtain a new form of modulation equations K(λj) = 0, j = 0, 1, . . . , 2L+ 1. (3.23) Theorem 3.4. Let λ denote an arbitrary branchpoint λj in the set of 2L+2 distinct branchpoints j = 0, 1, . . . , 2L+ 1. Then the following statements are equivalent: 1) K(λ) = 0; 2) ∂(~$,~η)t ∂λ = 0; 3) ∂h(z) ∂λ ≡ 0 for all z ∈ C; 4) ∂g(z) ∂λ ≡ 0 for all z ∈ C. Proof. Combining (2.17) with the identity ∂ ∂λ [ R(z) (ζ − z)R(ζ) ] = − R(z) 2(z − λ)(ζ − λ)R(ζ) , we obtain ∂h(z) ∂λ = − R(z)K(λ) 2(z − λ)\|D\| + R(z) 4πi (∮ M̂ ( ∂` ∂λ − 2∂u(ζ) ∂λ ) dζ (ζ − z)R(ζ) + L∑ j=1 [ ∂$j ∂λ ∮ γ̂m,j 2πidζ (ζ − z)R(ζ)+ + ∂ηj ∂λ ∮ γ̂c,j 2πidζ (ζ − z)R(ζ) ]) , (3.24) where z is inside M̂. According to (2.12), ∂u(z) ∂λ = 1 2 ∫ z λ0 P̃ (ζ)dζ (ζ − λ)R(ζ) + ∫ z λ0 P̃ ′(ζ)dζ R(ζ) (3.25) Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 27 provided that λ 6= λ0. If λ = λ0, the integral in (2.12) seems to have singularity at z = λ0, but it is compensated by the extra term − P̃ (ζ) R(ζ) ∣∣∣ ζ→λ0 obtained when differentiating by λ0. In any case, ∂u(ζ) ∂λ is analytic outside the closed contour M̂ and at ζ =∞. Thus, the first integral in (3.24) vanishes for every λ. According to (2.16), ∂h(z) ∂λ = O(1). (3.26) Then, K(λ) = 0 implies the system of equations L∑ j=1 [ ∂$j ∂λ ∮ γ̂m,j 2πiζkdζ R(ζ)+ + ∂ηj ∂λ ∮ γ̂c,j 2πiζkdζ R(ζ) ] = 0, k = 0, 1, . . . , L− 1. (3.27) Considering (3.27) together with L complex conjugated equations, we obtain the system (3.15) with zero right hand side. Then, according to Lemma 3.3, (~$,~η)t ∂λ = 0. Hence, we proved that 1) implies 2). Similarly, (3.26) combined with (~$,~η)t ∂λ = 0 imply 3), that is, 2) implies 3). Let us assume 3). Then, according to (2.14), (2.6), ∂` ∂λ = 0 and so ∂g(z) ∂λ ≡ 0 for all z ∈ C. Thus, 3) implies 4). Let us now assume 4). Then, differentiating in λ the jump conditions (2.7)–(2.9) for g(z), we obtain ∂` ∂λ = 0 (` is the jump on γm,0) and ∂(~$,~η)t ∂λ = 0 (the jumps on the remaining main and complementary arcs). Now 1) follows from (3.24), (3.26). � Remark 3.5. According to (3.10), in the case of the quartic potential f(z; t), the function h(z; t) is analytic in t in the genus zero region when z is on the Riemann surface R(t) away from the branchpoints. But, according to (3.21), (3.22), in general, in the higher genera regions the function is only smooth in t since the determinant K depends on both t and t̄. Fix some j ∈ {0, 1, . . . , 2L + 1}. The equation K(λj) = 0 defines a co-dimension one mani- fold Λj in the C2L+2 space of all branchpoints λj . Then Theorem 3.4 implies the following corollary. Corollary 3.6. Let potential f = f(z;~t) be given by (3.11). If ~λ ∈ Λ, where Λ = ∩2L+1 j=0 Λj, then d dtj h(z;~t) ≡ ∂ ∂tj h(z;~t), d dtj g(z;~t) ≡ ∂ ∂tj g(z;~t), d dtj $k(~t) = ∂ ∂tj $k(~t), d dtj ηk(~t) = ∂ ∂tj ηk(~t). Similar formulae are valid for the derivatives in t̄j. According to Corollary 3.6 and (3.11), modulation equation from (3.23) can be written in the so-called hodograph form as n∑ m=1 tmKtm(λj) +K0(λj) = 0, j = 0, 1, . . . , 2L+ 1, where Ktm(z), K0(z) can be obtained from K(z) by replacing f̃ in (3.22) by ζm m and by −2u(ζ) respectively. Let us now rewrite the equations (3.20) in terms of the data of the Riemann surfaceR = R(~t). If ~ω denotes the vector of normalized holomorphic (first kind) differentials and τ of R, then 28 M. Bertola and A. Tovbis A−1~r = 1 2πi ∮ M(f − 2u)~ω, A−1B = τ t(−Ht), so that (3.20) become ~η = J ( =τ t )−1= [ 1 2πi ∮ M (f − 2u)~ω ] , ~$ = < [ 1 2πi ∮ M (f − 2u)~ω ] −< ( τ t ( −Ht )) ~η, (3.28) where J = −(Ht)−1 is the L× L Jordan block with the eigenvalue −1. Since f is linear in the entries tm of ~t, Corollary 3.6 implies another set of hodograph equations n∑ m=1 tm∂m~η + ~η0 = 0, n∑ m=1 tm∂m~$ + ~$0 = 0, (3.29) where ∂m = ∂ ∂tm , ∂m~η = J ( =τ t )−1= res ζ=∞ ζm~ω m , ∂m~$ = < res ζ=∞ ζm~ω m + <τ tHt∂m~η, m = 1, . . . , L, and ~η0, ~$0 can be obtain from (3.28) by setting f = 0, ~η = ~η0, ~$ = ~$0 there. The following set of conservation equations = res ζ=∞ [ ∂l ζm m − ∂m ζ l l ] ~ω = = ( ∂mτ t ) Ht∂l~η −= ( ∂lτ t ) Ht∂m~η, < res ζ=∞ [ ∂l ζm m − ∂m ζ l l ] ~ω = < ( ∂mτ t ) Ht∂l~η −< ( ∂lτ t ) Ht∂m~η or res ζ=∞ [ ∂l ζm m − ∂m ζ l l ] ~ω = ∂lτ t ( =τ t )−1= res ζ=∞ ζm~ω m − ∂mτ t ( =τ t )−1= res ζ=∞ ζ l~ω l , m, l = 1, . . . , L, follows from (3.29). Lemma 3.7. If λ = λj, j = 0, 1, . . . , 2L+ 1 and the modulation equations (3.23) hold, then ∂K(z) ∂λ = [ ∂ ln \|D\| ∂λ + 1 2(z − λ) ] K(z). (3.30) In particular, for any j,m = 0, 1, . . . , 2L+ 1 ∂ ∂λm K(λj) = 0 if j 6= m and ∂ ∂λj K(λj) = 1 2 K ′(z)\|z=λj . (3.31) Proof. Formula (3.30) is a direct consequence of Theorem 3.4 and (3.21), whereas (3.31) follows from the analyticity of K(z) at z = λj (3.22) and (3.23). The analyticity of K(z) at z = λ0 follows from the fact that u(z) R(z) does not have a jump across γm,0 and, thus, R(z) 2πi ∮ M̂ u(ζ)dζ (ζ − z)R(ζ) = u(z) + R(z) 2πi ∮ M̃ u(ζ)dζ (ζ − z)R(ζ) . Here M̃ is a small deformation of M̂ near λ0, such that M̃ crosses γm,0 and does not contain a neighborhood of λ0. � Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 29 Corollary 3.8. For any j = 0, 1, . . . , 2L+1 and any m = 1, . . . , L the motion of λj with respect to tm is given by ∂m(λj) = −2∂mK(λj) K ′(λj) . Moreover, the system of branchpoints satisfies the so-called Whitham equations ∂mK(λj)∂lλj = ∂lK(λj)∂mλj , (3.32) for any l = 1, . . . , L. The proof follows directly from Lemma 3.7. Equations (3.32) are, in fact, an instance of the well-known Whitham equations for the evolution of branchpoints of the Riemann surface R(~t), see, for example, [23]. 4 Existence of genus zero g and h functions, breaking curves and continuation principle The L = 0 g-function constructed in Section 3 for our quartic potential f(z; t) satisfies the equality requirements (jump conditions) of Section 2.2.1, but so far there is no information about the sign requirements of Section 2.2.2. We first discuss the sign requirements in the genus zero region (they were established in [13] for t ∈ (−1/12, 0) and later in [4] for any t) Then we introduce the notion of the breaking curve, derive equations for the breaking curve and find their exact locations. Finally, we show that the sign requirements of Section 2.2.2 are satisfied in the region of genera one and two, that is, at any regular point of C∗ = C \ {0}. 4.1 Sign requirements in the genus zero region The explicit expression for h, given by (3.7), satisfies the RHP in Remark 3.1 and, thus, g = 1 2(h + f + `) satisfies the equality requirements of Section 2.2.1. It will be convenient to make some deformations, see Remark 2.7, of the contour Ω that are similar to that of [13]. Consider first t ∈ (− 1 12 , 0), that is, arg t = −π. The contour Ω can first be deformed into the union of the segment [−b, b] and four rays connecting the endpoints ±b with infinity, where b = b(t) is given by (3.5), see Fig. 3, left. The two rays emanating from b have directions ±π 4 and the two rays emanating from −b have directions ±3π 4 . The rays in the left half-plane are oriented towards z = −b whereas the rays in the right half-plane are oriented towards infinity; the segment [−b, b] has the standard (left to right) orientation. As we consider Generic traffic configuration, see (1.4), (1.5), the traffic on [−b, b] equals 1, while all the traffics %j on the four rays are different from zero. The following Lemma 4.1 [4] shows that the segment [−b, b] represents the main arc, whereas the rays can be deformed into unbounded complementary arcs. Lemma 4.1. If t ∈ (− 1 12 , 0), the four rays of the contour Ω can be deformed so that the func- tion <h(z), where h(z) is given by (3.7), satisfies the sign requirements of Section 2.2.2 along the contour Ω. Lemma 4.1 implies the existence of the genus zero h function (and, thus, the existence of the genus zero g-function) for all t ∈ (− 1 12 , 0). The following arguments shows that the genus zero region extends from (− 1 12 , 0) into a region in C∗. First observe that if a point t ∈ C∗ = C \ {0} is a regular point of genus L, see Remark 2.10, that is, the branchpoints in ~λ(t) are distinct and the sign requirements of Section 2.2.2 for h(z; t) are satisfied in every interior point of M, C, then there is a neighborhood U of t consisting of regular point of genus L. The proof of this well known fact is given in the lemma below (see also [1, 25] etc.). 30 M. Bertola and A. Tovbis Lemma 4.2. The set of regular points is open. This statement is true for a general polynomial potential f = f(z;~t), see (3.11). Proof. Writing modulation equations (3.23) as F (~λ;~t) = 0, we obtain from Lemma 3.7 that the Jacobian matrix ∂F ∂~λ = 1 2 diag ( K ′(λ0), . . . ,K ′(λ2L+1) ) . (4.1) If ~t0 is a regular point then all the branchpoints are finite and distinct. In this case the determi- nant K(z) has simple zeroes at every branchpoint λj , see (2.18) and (3.21), so that the Jacobian matrix (4.1) is invertible. By the implicit function theorem, we obtain a unique continuation of the branchpoints ~λ = ~λ(~t) into a neighborhood of the original parameter vector ~t0. Since ~λ(~t) is a continuous function of ~t in this neighborhood, then, according to (3.21), so is h(z;~t). Thus, if this neighborhood is sufficiently small, then the sign requirements of Section 2.2.2 for h will be preserved, and we complete the proof. � 4.2 Breaking curves. Regular and critical breaking points. Symmetries Let the h-function h = h(z; t) satisfy the sign requirements of Section 2.2.2 except, possibly, at finitely many points. Define z0 as a saddle point of h(z; t) if h′(z; t) has zero at z0 of order at least one, that is, h′(z; t) = O(z−z0) as z → z0. A point t ∈ C∗ is called a breaking (non-regular) point, if there is a saddle point z0 of h(z; t) that is either on a main arc or on a complementary arc that cannot be deformed away from z0. In the latter case, <h(z0, t) = 0 and we say that the complementary arc is pinched at z0. So, if tb is a breaking point, then there exists a saddle point z0 ∈ Ω satisfying the equations h′(z0; tb) = 0 and <h(z0; tb) = 0. (4.2) (In the case z0 is a branchpoint, the first equation (4.2) should be replaced by the condition: h′(z; t) is of order at least O(z − z0) 3 2 .) A breaking point tb ∈ C∗ is called critical breaking point if one of the following applies: 1) a saddle point z0 of h(z; t) coincides with a branchpoint; in this case we have h′(z; tb) = O(z − z0)m with m ≥ 3 2 ; 2) there are at least two (counted with multiplicity) saddle points of h(z; t) on Ω that, in the case when a saddle point is on C, cannot be deformed away. For the case Ω = R considered in [10], critical points correspond to special cases of irregular potentials. A breaking point tb that is not a critical point is called a regular breaking point. In this case the saddle point z0, corresponding to tb, is simple and does not coincide with any branchpoint. It is called a double point. This name reflects the fact that a double point can be considered as a double branchpoint, that can move apart and form an extra main or complementary arc of Ω in a higher genus region, that is, an extra branchcut of R. Considering (4.2) as a system G(u, v, tj) = 0, j = 1, 2 of three real equations with four real variables z = u+ iv and t = t1 + it2, we calculate the Jacobian det ( ∂G ∂(u, v, tj) ) = ij−1\|h′′(z; t)\|<htj (z; t) ∣∣∣∣ (z,t)=(z0,tb) , (4.3) where we can choose j to be either 1 or 2. If tb is a regular breaking point and if ht(z0; tb) 6= 0, then, by the implicit function theorem, there is a smooth curve passing through tb ∈ C that consists of breaking points. Such curves are called breaking curves. Thus, we have proved the following lemma. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 31 Lemma 4.3. Any regular breaking point tb belongs to a locally smooth branch of the breaking curve provided that at least one of the derivatives htj (z0; tb), j = 1, 2, is not zero. Lemma 4.3 can be extended for a critical breaking point tb corresponding to a situation where there are several double points not coinciding with the branchpoints. In this case each double point, according to Lemma 4.3, will define its own local piece of a breaking curve, passing through tb. Thus, generically, there would be several breaking curves intersecting each other at tb. However, under certain symmetries, all these breaking curves could coincide and form just one breaking curve that is smooth at tb. Examples of such symmetries include the case of the semiclassical focusing NLS [25], where the Schwarz symmetry of h(z) implies that the saddle points of h(z) on Ω appears in complex conjugated pairs. As it will be shown below, they also include the case of Generic traffic configuration. In particular, we will prove that h′(z; t) is an odd function (in z) and, thus, the saddle points ±z0 6= 0 appear simultaneously on the contour Ω at a critical point tb. We will also show that at least one of the derivatives htj (z0; tb) 6= 0 if tb is a breaking point and z0 is a corresponding double point, so that the implicit function theorem guarantees existence of a smooth local breaking curve for each double point ±z0. Because of the symmetry, these (local) breaking curves coincide with each other. Therefore, in the case of symmetric h′ we extend the notion of a regular breaking point tb to the cases when there is a pair of symmetric double points on Ω that generate the same smooth breaking curve passing through tb. Lemma 4.4. Under the assumption of Generic traffic configuration, the branchpoints ~λ are symmetrical for any t ∈ C∗, that is, λ is a branchpoint if and only if −λ is a branchpoint. Proof. Let %0 = %2 and %1 = %3 in (1.1). Then, since the weight is even, the orthogonal polynomials πn(z) are also symmetrical, and so, the curves of accumulation of zeroes will also be even. It is well known that these curves coincide with the main arcs (see for example [2]). But, within the Generic traffic configuration, the RHP contour Ω does not depend on the traffics ~%. Thus, in the Generic case with any admissible traffics, the branchpoints are symmetrical. � Corollary 4.5. If the branchpoints ~λ = ~λ(t) are symmetrical, the function h′(z; t) is odd and the functions htj (z; t) are even, j = 1, 2, t = t1 + it2. Proof. If the branchpoints are symmetrical, the corresponding radical R(z), see (2.10), is even for odd L and odd for even L. It follows then that the solution g′(z; t) to the scalar RHP (2.3), given by (2.20), is odd, and so, h′(z; t) is odd, regardless of the genus. Because of the modulation equations, solution of the RHP (2.15), (2.16) for h commutes with the operator d dtj . Writing solution to the differentiated RHP for htj in the same form as (2.17), we can show, in a similar way to h′, that ht is even. � Now, according to (4.2), a breaking point tb is critical if either h′(z; tb) R(z) ∣∣∣∣ z=±z0 = 0 or h′(z0; tb) = h′′(z0; tb) = 0, (4.4) where z0 is a branchpoint in the former case and an interior point on a main or a complementary arc in the latter case. In the genus zero case, according to (3.5), the branchpoints are ±b. Using equation (3.6) for h′, we easily calculate that the first equation (4.4) yields t = t0 = − 1 12 , whereas the second set of equations yields z0 = 0, tb2 2 = −1 and, thus, t = t2 = 1 4 . As we will show below, there are no other critical points for Generic traffic configurations (the critical point t1 = 1 15 occurs for different traffic configurations, see [4]). It follows then that the arcs of breaking curves begin and end at a critical point (t0 = − 1 12 , t2 = 1 4) or at t = 0 or at infinity. They separate either regions of different genera or regions 32 M. Bertola and A. Tovbis of the same genus but with different topology of main arcs. In Section 6 we shall exclude the possibility of a breaking curve forming a smooth closed curve (loop) in the complex t-plane while not passing through any critical point or through t = 0. 4.3 Continuation principle for Boutroux deformations In Section 3.2 we have derived the explicit formula (3.21), (3.22) for the h function in the genus L region, where L = 1, 2, . . . . For quartic exponential weight considered here (L ≤ 2) there is no guarantee yet that the function h, given by (3.21), (3.22), satisfies the sign requirements of Section 2.2.2. In Section 4.1, we showed that the sign requirements are satisfied in the genus zero region. To prove that the correct signs of <h persist beyond the genus zero region, we use the continuation principle for Boutroux deformations. Let ϑ be a smooth bounded curve in the external parameters space (complex t-plane in our case). According to (2.24), the deformation of the vector of (all the) branchpoints ~λ(t) along ϑ, governed by the modulation equations, preserves the Boutroux condition (Boutroux deformation). The idea of the continuation principle is that moving along ϑ, we can show that the sign conditions for <h(z; t) will be satisfied for all t along the curve, provided that: • the sign conditions for <h(z; t) were satisfied at the beginning point of ϑ; • the curve ϑ does not contain any critical point or any singular point (t = 0), and; • every time a breaking curve is crossed (at a regular breaking point), the genus is properly adjusted. Let t be a regular point in a genus L region, that is, all the branchpoints in ~λ(t) are distinct and <h(z; t) satisfies the strict inequalities from Section 2.2.2 at every point of the contour Ω = Ω(t) except the branchpoints. Take another regular point t∗ in the same open connected component G of the genus L region, that is, t, t∗ ∈ G, and let ν ⊂ G be a simple smooth contour, connecting t and t∗. Using Lemma 4.2, we can continue ~λ(t) from t to t∗ along ν preserving the condition λj(t) 6= λk(t), j 6= k. Therefore, we can also continue h(z; t) from t to t∗ along ν. The only possibility for the inequalities for h(z; t) to fail at some point t ∈ ν is if t is a breaking point, which is excluded by construction of ν. Thus, we proved the existence of h-function satisfying the strict inequalities at any point of G. In view of the above results, it remains only to prove that the strict inequalities can be preserved after a transversal crossing (along ϑ) of the breaking curve at a regular breaking point tb. Since L ≤ 2, at any regular breaking point tb, there can be either one or two symmetrical double points z0 = 0 or ±z0 respectively. Because of the symmetry of hz(z; t), it is sufficient to consider only one double point z0 in any case. At a regular breaking point tb the topology of zero level curves of <h(z; t) undergoes a change according to one of the following four scenarios: • two endpoints of a main arc collide and the arc becomes a double point z0; • two endpoints of neighboring main arcs collide and the corresponding bounded comple- mentary arc turns into a double point z0; • a complementary arc is pinched by zero level curves of <h(z; t) and a new double point z0 appears; • a main arc collides with (another) zero level curve of <h(z; t) and a new double point z0 appears. The first two scenarios lead to decrease of the genus whereas the last two lead to its increase. Let h(n)(z; t) denote the h function (that means that all the strict inequalities of Section 2.2.2 are satisfied) in the genus n region. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 33 Theorem 4.6. Let tb be a regular breaking point on a breaking curve, separating regions of genus n and m, where n,m ∈ N∪{0}. Then h (m) z (z; tb) ≡ h(n) z (z; tb) and h (m) zt (z; tb) ≡ h(n) zt (z; tb) for all z ∈ C. For the RHP, describing the NLS evolution, and with \|m − n\| = 2, the first identity of the theorem was proved in [25, Theorem 3.1], and the second identity – in [24, Theorem 3.1]. In the context of orthogonal polynomials, this theorem was proved in [1]. Finally, in the most general case which include both regular breaking points and the critical points, the theorem follows from the continuity of Boutroux-normalized meromorphic differentials, proven in [5]. Corollary 4.7. In the conditions of Theorem 4.6, h(m)(z; tb) ≡ h(n)(z; tb) and h (m) t (z; tb) ≡ h (n) t (z; tb) for all z ∈ C. We can now return to the continuation principle for Boutroux deformations. Let t(s), s ∈ R, be a parametrization of the smooth curve ϑ that is transversal to the breaking curve at a regular breaking point tb = t(0). Then, by Lemma 6.1, ht(z0, tb) 6= 0, where z(s) is the corresponding saddle point and z0 = z(0). Since <[ht(z0; tb)∆t] = 0 when ∆t is tangential to the breaking curve at tb, we conclude that <dhds = <[ht dt ds ] 6= 0 at z0, tb. Let us assume, for example, that <dhds (z0; tb) < 0, and that the sign conditions for h(z, t) = h(L)(z, t) hold when t = t(s) with s < 0. Then, as s→ 0−, either a main arc γmj collapsed into the double point z = z0, or a main arc collides with another zero level curve of <h(z; t) at z0. We now need to prove the sign requirements on the other side of the breaking curve in a small vicinity of z0, that is, when t = t(s) with a small s > 0. Consider first the case of a collapsing simple (without whiskers) main arc γmj . This case can be illustrated by Fig. 3 as transition from the second to the first (counting from the left) panel, caused by collapse of the two symmetrical small main arcs (on the right and on the left). We will now consider only a vicinity of z0 and disregard the symmetrical to z0 double point. If s < 0 is sufficiently close to zero then γmj is the only main arc in a vicinity of z0. Moreover, z0 is a saddle point of <h(L)(z, tb) with <h(L)(z0, tb) = 0. Let us desingularize the hyperelliptic Riemann surface R(tb) = R(L)(tb) of the genus L by removing a pair of branchpoints that col- lided into z = z0. The genus of the desingularized Riemann surface R(L−1)(tb) is L − 1. The corresponding h(L−1)(z; tb) ≡ h(L)(z; tb) satisfies the strict sign inequalities everywhere except z = z0. However, since h (L−1) t (z; tb) ≡ h (L) t (z; tb) and <dh(L) ds (z0; tb) < 0, we conclude that for any small s > 0 the function <h(L−1)(z; t(s)) < 0 in some deleted neighborhood of z0. Then the strict sign inequalities for h(L−1)(z; t(s)), where s > 0 and small, will be valid everywhere in this neighborhood. So, we proved that h(z, t(s)) = h(L)(z, t(s)), s < 0, can be continued by h(z, t(s)) = h(L−1)(z, t(s)), s > 0, across the breaking curve while preserving the sign inequali- ties. Consider the case when γmj is not a simple main arc, that is, γmj is a disappearing “whisker”, see the center top panel in Fig. 2 and the other panel right below it (see also transition from the third to the second panel (counting from the left) panel on Fig. 3). In this case we do not change the genus after crossing the breaking curve, so that h(z, t(s)) = h(L)(z, t(s)) for all s close to zero. By repeating the previous arguments we see that a new complementary arc is formed near z = z0, so that strict inequalities are valid everywhere for small s > 0. Here we would like to mention one more issue: at the breaking point tb solution of the modulation equations F (~λ, t) = 0 does not produce a set of distinct branchpoints; in fact, two of them collided at z = z0. Under this circumstances we cannot use Theorem 3.3 from [25] to prove existence and continuity of the branchpoints ~λ(t(s)) for s > 0, since the Jacobian ∂F ∂~λ = 0 at t = tb. However, after a proper rescaling of t − tb, it was shown that ~λ(t(s)) can be continuously extended for s > 0 in [25, Theorem 6.4]. This fact also follows from Remark 5.9. 34 M. Bertola and A. Tovbis The remaining case of a main arc colliding with another zero level curve can be considered similarly. Finally, the remaining two cases of different topological change at the breaking curve correspond to <dhds (z0; tb) > 0. They can be considered in the similar way by just reversing the orientation of the curve t(s), i.e., by replacing s 7→ −s. Thus, we have essentially proved the following continuation principle: if t(s), 0 ≤ s ≤ 1, is a smooth curve in the complex t-plane that does not contain critical points, and if h(z, t(0)) satisfies all the conditions (equations and inequalities) for h-functions, then h(z, t(0)) can be continuously (in s) deformed into h(z, t(1)), so that for any s ∈ [0, 1], the function h(z, t(s)) satisfies all the conditions for h-functions. Since the curve t(s) and the breaking curve are smooth and their intersections are transversal, there can be no more than a finite values of s ∈ [0, 1] for which the inequalities are not strict; for these values of s, t(s) are regular breaking points. The next step in the proof of the sign requirements for h(z; t) for any t ∈ C∗ \ {t0, t2} is to show that for all s ∈ [0, 1] the set of branchpoints λ(t(s)) is bounded. (Note that the dimension of the vector λ(t(s)) may vary with s). This will be done in Section 5. 5 Boundedness of the branchpoints Continuation principle allows us to prove that if the g-function (or h-function) exists for some t = t̂b, then it can be continued to a given nonsingular t = t̂1 along any simple, compact, piece-wise smooth contour t(s) in C, where s ∈ [0, 1] and t(0) = t̂b, t1 = t̂(1), provided that t(s) does not contain any critical points. However, the arguments of Section 4.3, as well as general theorem about continuation principle from [1], does not exclude the possibility that some of the branchpoints λj(t(s)) may approach infinity as s → s∗ for some s∗ ∈ (0, 1). In this section we prove the boundedness of the branchpoints and nodes of a nodal hyperelliptic curve R under the Boutroux evolution, which implies boundedness of the branchpoints for general polynomial potentials. We start our analysis by considering polynomials P (z) of the form P (z) = M2(z)S(z), M(z) := t2 m∏ j=1 (z − µj), S(z) := R2(z) = 2L+1∏ j=0 (z − λj), (5.1) where all λj are distinct. We are motivated by the fact that (5.1) coincides with (2.22), where y(z) = √ P (z). Let us assume that y(z) = f ′(z) + T z +O ( z−2 ) , (5.2) where T ∈ R is a residue of y(z) at infinity and the polynomial potential f has the form f(z) = f(z,~t) = m+L+2∑ j=1 tj j zj (5.3) with ~t = (t1, . . . , tm+L+2) being a vector of (complex) coefficients of f and t = tm+L+2. (The constant term in f(z) does not play any significant role and hence is conventionally set to zero.) Note that y2(z) = P (z) defines a nodal hyperelliptic curve R of the (geometric) genus L. We further assume that the meromorphic differential y(z)dz satisfies the Boutroux condition: < ∮ γ y(z)dz = 0 (5.4) along any loop γ on R. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 35 Remark 5.1. A simple counting of the parameters shows that the m+L+2 complex coefficients from ~t, the T ∈ R and the 2L+ 1 real constraints (5.4) account for all the m+ 2L+ 3 complex parameters (~λ, ~µ and t) defining a polynomial P (z) of the form (5.1). Remark 5.2. It is clear that y(z) satisfies the jump conditions (2.19), where γm,j are the branchcuts of R, and the asymptotics of y at infinity is given by (5.2). Then the Boutroux condition implies that h(z) = ∫ z λ0 y(u)du satisfies the jump conditions (2.15) with some real constants ηj , ωj . Note that the jump across γ0 in (2.15) has to be replaced by 2πiT . Here γc,j is a collection of complementary loops on R and γ0 is a contour connecting λ0 and ∞. The asymptotics of h(z) at infinity is given by h(z) = f(z) + `+ T ln z +O ( z−1 ) , z →∞, where ` ∈ C and f is given by (5.3). Then various form of modulation equations, obtained in Section 2, as well as determinantal formulae for h, obtained in Section 4, are valid for h(z) =∫ z λ0 y(u)du. In particular, condition (5.4) together with T ∈ R is, of course, equivalent to the integral conditions of (2.24). Then, as was shown in Section 2.3, conditions (5.4), (5.2) define the branchpoints ~λ = (λ1, . . . , λm+L+2), the nodes ~µ = (µ1, . . . , µm) and T ∈ R. Our goal is to study finite deformations of the parameters ~t (we consider T = const) that preserve the Boutroux condition. Such deformations will be called Boutroux deformations. Let ~t(s) be a smooth finite curve in Cm+L+2, where s ∈ [0, 1] is the arclength parameter. We assume the residue T to be constant. Consider a Boutroux deformation P (z, s) := P (z,~t(s)) of the polynomial P (z) = P (z;~t). The simple roots ~λ(s) and the double roots ~µ(s) now depend continuously on s (as long as they are distinct). As we move along ~t the genus of the nodal curve R = R(s) can change as some simple roots may become double roots and vice versa. We want to prove that the roots ~λ(s), ~µ(s) are bounded provided that ~t(s) is separated from the hyperplane t = tm+L+2 = 0. The smooth deformation contour ~t(s) can be then interpreted as a parametrized integral curve of the corresponding vector field ~v(~t), that is, ~̇t(s) = ~v(~t(s)) and ‖~v(~t)‖ = 1. Thus, we need to show how an arbitrary unitary vector field on the ~t induces an infinitesimal deformation of the coefficients of P (z), namely, a vector field on the space of polynomials P (z) of the form (5.1) that is tangent to the manifold of constraints (5.4). 5.1 A general theorem on ODE’s for polynomials Let P (z, s) = p0(s)zn + p1(s)zn−1 + · · ·+ pn−1(s)z + pn(s), (5.5) where the coefficients ~p(s) = (p1(s), . . . , pn(s)) of P satisfy an autonomous system of differential equations of the form ~̇p(s) = ~g(~p(s)), s ∈ [0, 1], (5.6) where ~g(~p) = (g1(~p), . . . , gn(~p)). To prove that the roots of the polynomial P are bounded (in s), it would be sufficient to show that p0(s) is separated from zero and all the coefficients ~p(s) are bounded on [0, 1]. Theorem 5.3. Let ~g(~p) in (5.6) be continuous and suppose that there exists K > 0 such that the functions gj(~p) satisfy the bound \|gj(~p)\| ≤ Kρj , j = 1, . . . , n, (5.7) 36 M. Bertola and A. Tovbis where ρ = ρ(~p) := max `≥1 \|p`\| 1 ` . (5.8) Then for any solution of the system (5.6) the function ρ(s) := ρ(~p(s)) remains bounded for all s ∈ [0, 1]. To prove Theorem 5.3, we need to establish the following statements from nonsmooth analysis. Lemma 5.4. Let ~p(s) ∈ C1[0, 1]. Then ρ(s) , defined by (5.8), admits Dini’s derivative D+ρ(s) := lim sup s̃→s+ ρ(s̃)− ρ(s) s̃− s , s ∈ [0, 1]. (5.9) Moreover, D+ρ(s) = max 1≤j≤n { d ds \|pj(s)\| 1 j : ρ(s) = \|pj(s)\| 1 j } . (5.10) In particular, the derivative is finite at all points where ρ(s) > 0. Proof. Denote fj(s) := \|pj(s)\| 1 j . Let s be such that ρ(s) > 0. Fix a sequence sk → s from above and let jk be such that fjk(sk) = ρ(sk). Since the set of indices is finite, there is a subsequence of jk that eventually becomes constant, say j0. Then, along any such subsequence {jk`}, fj0(s) = lim ` fjk` (s) = lim ` fjk` (sk`) = lim ` ρ(sk`) = ρ(s). Then D+ρ(s) ≥ lim ` ρ(sk`)− ρ(s) sk` − s = lim ` fjk` (sk`)− ρ(s) sk` − s = lim ` fj0(sk`)− fj0(s) sk` − s = f ′j0(s), (5.11) where we used the fact that the subsequence jk` is eventually constant with limiting value j0. Since f ′j0(s) belongs to the set in (5.10) we prove the inequality. Now let us prove (≤). Let the limsup in (5.9) is attained along the sequence {sk}, introduced above. We have D+ρ(s) = lim k ρ(sk)− ρ(s) sk − s = lim k fjk(sk)− ρ(s) sk − s ≤ lim k fjk(sk)− fjk(s) sk − s . The limit set of the sequence jk necessarily belongs to the set {j̃ : fj̃(s) = ρ(s)} and hence lim k fjk(sk)− fjk(s) sk − s ∈ { d ds fj(s) : fj(s) = ρ(s) } . Thus the inequality ≤ holds. The latter equation and (5.11) prove (5.10). � Next we state the following well known fact that will be used in the proof of Theorem 5.3. Proposition 5.5 ([17]). If D+M(s) exists everywhere (finite) in the interval [a, b] then M(b)−M(a) ≤ ∫ b a D+M(t)δt where ∫ b a denotes the upper Riemann integral. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 37 Proof of Theorem 5.3. Since (` ≥ 1), we have d ds \|p`(s)\| 1 ` = 1 ` \|p`(s)\| 1 ` −1 d ds \|p`(s)\|≤\|p`(s)\| 1 ` −1 d ds \|p`(s)\| ≤ \|p`(s)\| 1 ` −1 \|ṗ`(s)\| = \|p`(s)\| 1 ` −1 \|g`(~p(s))\| ≤Kρ`\|p`(s)\| 1 ` −1, (5.12) where the assumption of the theorem was used in the latter inequality. By formula (5.10) we have that D+ρ(s) is the maximum of all derivatives amongst the functions f` = \|p`(s)\| 1 ` that realize the maximum at s. Thus in (5.12) d ds \|p`(s)\| 1 ` ≤ Kρ`ρ1−` = Kρ and, taking into the account (5.10), we obtain D+ρ(s) ≤ Kρ(s). We would like to use Gronwall’s theorem to complete the proof. In order to use this theorem, we introduce F (s) = max{1, ρ(s)}. Since ρ(s) is the maximum of locally Lipshitz functions fj , j = 1, . . . , n, (when ρ(s) > 1) then F (s) is locally Lipshitz as well. It is straightforward to verify that D+F (s) =  0, ρ(s) < 1, D+ρ(s), ρ(s) > 1, max{0, D+ρ(s)}, ρ(s) = 1. Then, if ρ(s) > 1 then D+F (s) = D+ρ(s) ≤ Kρ(s) ≤ KF (s); if ρ(s) = 1 then also D+F (s) = max{0, D+ρ(s)} ≤ max{0,K} = KF (s). If ρ(s) < 1 then D+F (s) = 0 ≤ Kρ(s) ≤ KF (s). Thus in all cases D+F (s) ≤ KF (s). Integrating this inequality from 0 to s and using (5.5), we obtain F (s)− F (0) ≤ ∫ s 0 D+F (t)dt ≤ ∫ s 0 KF (t)dt = ∫ s 0 KF (t)dt, where in the last equality we have replaced the upper Riemann integral by a regular Riemann integral because M(t) is continuous. Now Gronwall’s theorem yields F (s) ≤ F (0)eKs for any s ∈ [0, 1]. The theorem is proved. � Corollary 5.6. If p0(s) remains separated from 0 and the remaining coefficients pj(s), j = 1, . . . , n, of (5.5) satisfy the assumptions of Theorem 5.3, on [0, 1] then the roots of P (z; s) are uniformly bounded on [0, 1]. Proof. According to Rouché’s theorem, the roots of the polynomial (5.5) are within a circle of the radius N max n≥`≥1 ∣∣∣∣p`(s)p0(s) ∣∣∣∣ 1` ≤ Nρ(s) min{\|p0(s)\|, 1} , (5.13) where the constant N depends only on the order of the polynomial P (z; s). Now the corollary follows from Theorem 5.3. � 38 M. Bertola and A. Tovbis 5.2 Properties of the polynomial (5.1) In this section we prove that the ODEs (5.6) for the coefficients pj(s), j = 1, . . . , n of the polynomial P (z, s) = t2 m∏ j=1 (z − µj(s))2 2L+1∏ j=0 (z − λj(s)), (5.14) see (5.1), where n = 2m+ 2L+ 2, satisfy the conditions (5.7). Under the additional assumption that p0(s) = t2(s) is separated from zero on [0, 1], it would then follow from Corollary 5.6 that the roots µj(s), λj(s) of (5.14) are uniformly bounded on [0, 1]. Let ∂j = ∂ ∂tj , j = 0, 1, . . . , n. Then Ṗ (z, s) = ∇~tP · ~v(~t) and, taking into account ~̇t(s) = ~v(~t) and (5.6), we obtain g`(~p) = ṗ`(s) = ∇~tp` · ~v(~t). Since ‖~v‖ = 1, it is clear that if there exist some Kj,` > 0 such that \|∂jp`\| < Kj,`ρ ` for all j = 1, . . . ,m+ L+ 2, ` = 1, . . . , n, (5.15) then the condition (5.7) in Theorem 5.3 will hold for all `. Therefore, we need to prove (5.15) for some fixed j and all ` = 1, . . . , n. According to (5.1), we have yj = Pj 2 √ P = Q R , (5.16) where the subindex j is used to denote ∂ ∂tj , so that Pj = 2MQ. (5.17) Expressions (5.16), (5.17) together with (5.3), (5.2) imply that Q is a polynomial of degree L+j, and the condition (5.4) in differentiated form implies that the differential ytdz = Q(z)dz R(z) (5.16) is a second kind differential on the Riemann surface of R(z) uniquely determined by the condition that Q R = zj +O(z−2) for \|z\| → ∞ and by the requirement that it is an imaginary normalized, namely, all integrals on closed paths are purely imaginary. Then the results of [5] apply and we have the following theorem. Theorem 5.7 ([5]). The polynomial Q(z;~λ) is continuous in ~λ. Lemma 5.8. The polynomial Q(z) = Q(z;~λ) satisfy the homogeneity Q(z;~λ) = νL+jQ ( ν−1z; ν−1~λ ) , ν ∈ R+. Proof. The Boutroux condition (5.4) guarantees the validity of Corollary 3.6. Using this corol- lary, (3.21), (3.22) and the fact that y(z) = h′(z), see (2.22), we obtain yj(z) = h′j(z) = 1 R(z) [( R2(z) )′ Kj(z) +R2(z)K ′j(z) 2D ] , so that, according to (5.16) Q(z;~λ) = ( R2(z;~λ) )′ Kj(z;~λ) +R2(z;~λ)K ′j(z; ~λ) 2D(~λ) , (5.18) Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 39 where ()′ denotes d dz . Taking into account (3.16), a straightforward calculation shows that D ( ~λ ν ) = ν(L+1)LD(~λ), R ( z ν ; ~λ ν ) = ν−(L+1)R(z;~λ),∫ ζkdζ R(ζ, ~λ ν ) = νL−k ∫ ζkdζ R(ζ, ~λ) . (5.19) Note that, according to Corollary 3.6, Kj(z) = 1 2πij ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∮ γ̂m,1 dζ R(ζ) · · · ∮ γ̂m,1 ζL−1dζ R(ζ) ∮ γ̂m,1 dζ R(ζ) · · · ∮ γ̂m,1 ζL−1dζ R(ζ) ∮ γ̂m,1 dζ (ζ−z)R(ζ) · · · · · · · · · · · · · · · · · · · · ·∮ γ̂m,L dζ R(ζ) · · · ∮ γ̂m,L ζL−1dζ R(ζ) ∮ γ̂m,L dζ R(ζ) · · · ∮ γ̂m,L ζL−1dζ R(ζ) ∮ γ̂m,L dζ (ζ−z)R(ζ)∮ γ̂c,1 dζ R(ζ) · · · ∮ γ̂c,1 ζL−1dζ R(ζ) ∮ γ̂c,1 dζ R(ζ) · · · ∮ γ̂c,1 ζL−1dζ R(ζ) ∮ γ̂c,1 dζ (ζ−z)R(ζ) · · · · · · · · · · · · · · · · · · · · ·∮ γ̂c,L dζ R(ζ) · · · ∮ γ̂c,L ζL−1dζ R(ζ) ∮ γ̂c,L dζ R(ζ) · · · ∮ γ̂c,L ζL−1dζ R(ζ) ∮ γ̂c,L dζ (ζ−z)R(ζ)∮ M̂ ζjdζ R(ζ) · · · ∮ M̂ ζj+L−1dζ R(ζ) ∮ M̂ ζjdζ R(ζ) · · · ∮ M̂ ζj+L−1dζ R(ζ) ∮ M̂ ζjdζ (ζ−z)R(ζ) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . (5.20) In light of (5.19) and (5.20), we obtain Kj ( z ν ; ~λ ν ) = ν(L+1)2−jKj(z, ~λ), K ′j ( z ν ; ~α ν ) = ν(L+1)2−j+1K ′j(z, ~λ). (5.21) Now the statement of the lemma follows from (5.18), (5.19) and (5.21). � Let ~α = (~λ, ~µ). According to (5.17), (see (5.1) for the definition of M) ∂jP (z; ~α) = 2M(z; ~µ)Q(z;~λ) = G(z; ~α). (5.22) Then, as an immediate consequence of Lemma 5.8, we have G(z; ~α) = νm+L+jG ( ν−1z; ν−1~α ) , ν ∈ R+. Equation (5.22) implies (p`)t = G`, where G (z; ~α) = L+m+j∑ k=0 Gk(~α)zL+m+j−k. (5.23) Thus, G` ( ν−1~α ) = ν−`G` (~α) . (5.24) Remark 5.9. According to Theorem 5.7, the polynomial Q(z;~λ) is continuous even when any number of branchpoints in ~λ coincide. Therefore G`(~α) defined in (5.22) is continuous with respect to ~α. That also mean that G`(~α) is continuous in the symmetric polynomials p` of the entries of vector ~α. Thus, the right hand side of equation (5.22), considered as a nonlinear autonomous system of ODEs, is continuous in the p`’s. This right hand side is Lipshitz when the branchpoints in ~λ are distinct, so the t-evolution of ~α is defined uniquely. On the discriminant locus (where two or more αj ’s coincide) the right hand side of equation (5.22) is still continuous but not Lipshitz-continuous; in fact, the uniqueness of solutions of (5.22) fails there (but the existence still holds). This non-uniqueness of the solutions for initial data with coinciding α’s (i.e., on the breaking curves) is important because solutions of the continuation equations (5.22) of different genera coincide at those points and hence it allows us to select the appropriate solution that also satisfies the sign inequality requirements. 40 M. Bertola and A. Tovbis Lemma 5.10. Conditions (5.15) are satisfied. Proof. We prove this statement for an arbitrary fixed j = 1, . . . ,m + L + 2. According to (5.22), (5.23), ∂jp` = G`(~α) for any ` = 0, 1, . . . , n. Let ν = ρ, where ρ is defined by (5.8). Then, all the roots of the polynomial P , that is, all the components of ~α = (~λ, ~µ), are within the a circle K̂ρ, where K̂ = N min{\|p0\|,1} , see (5.13). Since G`(~α) is a continuous function of ~α, it is bounded on the set maxk \|αk\| ≤ K̂ by some Kj` > 0. Then, according to (5.24), \|∂jp`\| = \|G`(~α)\| ≤ Kj,`ρ `, so that conditions (5.15) are satisfied. � Thus, we have proved the following theorem. Theorem 5.11. Suppose that the polynomial P (z)P = P (z;~t), given by (5.1), (5.2), is such that the Boutroux condition (5.4) is satisfied and tM+L+2 6= 0. Let ~t(s), s ∈ [0, 1], be a smooth deformation of ~t, with ~t = ~t(0) preserving the Boutroux condition (5.4) and such that tM+L+2(s) remains separated from zero. Then the roots ~λ(s), ~µ(s) are uniformly bounded for s ∈ [0, 1]. As an immediate consequence of Theorem 5.11 we obtain the following two corollaries. Corollary 5.12. Let t(s), s ∈ [0, 1], be simple, compact, separated from zero piece-wise smooth contour, connecting points t(0) = t̂0 and t1 = t̂(1) of C. If f(z, t) is given by (1.1) and T = 2, then the roots of the polynomial P (z, t(s)), given by (5.1), (5.2), are uniformly bounded on s ∈ [0, 1], provided that the Boutroux condition (5.4) holds for all s ∈ [0, 1]. Corollary 5.13. For any t ∈ C∗ there exists the g function g(z; t), satisfying all the sign requirements of Section 2.2.2. The corresponding hyperelliptic surface R(t) has genus L = 0, 1, 2. The function h is given by (3.7) for L = 0 and by (3.21) L > 0. The explicit expression for h with L = 1 is given in (6.11). 6 Breaking curves for quartic potential f(z; t) In this section, we prove existence of the breaking curve and confirm their topology, shown on Fig. 2. We start with the following lemma. Lemma 6.1. If tb is a regular breaking point and ±z0 are the corresponding saddle points on Ω then ht(±z0; tb) 6= 0. Proof. Since the maximal genus for a quartic exponential potential cannot exceed two, see equation (2.23), the genus can be only zero or one along any breaking curve. Comparing (3.6) with (3.10), we see that in the case of genus zero, the conditions ht(z; t) = 0 and hz(z; t) = 0 while z 6= ±b (6.1) imply that t = 1 4 , which is the critical breaking point t2. In the remaining case of genus one with symmetrical branchpoints, the functions hz, ht are given by hz(z; t) = −tz √( z2 − λ2 1 )( z2 − λ2 2 ) , ht(z; t) = −1 4 ( z2 +K )√( z2 − λ2 1 )( z2 − λ2 2 ) , (6.2) Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 41 where ±λ1, ±λ2 are the branchpoints and K is a constant in z. The hz expression follows from (2.22). To derive the second expression, we observe that, according to Corollary 3.6, ht(z; t) = R(z) \|D\| Kt(z). In the case of genus one with symmetrical branchpoints, R(z) is an even function. Since the residue at ζ = ∞ of any even function is zero, the last row of the determinant Kt (see (5.20)) contains all zero entries except of the diagonal (the last) entry. Thus, ht(z) = −R(z) 8πi ∮ M̂ ζ4dζ (ζ−z)R(ζ) , which implies (6.2). Using the large z asymptotics (2.19) and (3.9) for hz and ht respectively, we obtain λ2 1 + λ2 2 = −2 t and K = −1 t . Then, conditions (6.1), where z 6= ±b is replaced by z 6= ±λj , j = 1, 2, are not compatible. Thus, we have shown that for the Generic traffic configuration zeroes of ht cannot coincide with any regular breaking point. � Let tb be an arbitrary regular breaking point. Then, according to Lemmas 4.5 and 6.1, the breaking curve passing through tb is a locally smooth curve. Thus, we obtain the following corollary. Corollary 6.2. Breaking curves are smooth simple curves consisting of regular breaking points (except, possibly, the endpoints). They do not intersect each other except, possibly, at the critical points t0 = − 1 12 , t2 = 1 4 , at the singularity t = 0 and at infinity. They can originate and end only at the above mentioned points. 6.1 Existence of breaking curves The continuation principle shows that the existence of the g-function (or h-function) on the interval (− 1 12 , 0) implies the existence of the g function for any t ∈ C∗. However, it does not describe the breaking curves or even prove their existence. The collection of the breaking for the Generic traffic configuration, the so-called “asymptotic phase portrait”, shown on Fig. 2, was calculated numerically in [4]. Let us briefly characterize the breaking curves on Fig. 2. The curves connecting t0 = − 1 12 and t = 0 are the boundaries of the genus zero region. They are described by (4.2) with h given by (3.7), where b = b(t) is given by (3.4) with the choice of the negative sign. The next breaking curves, connecting t0 with t2 = 1 4 are given by the same equations, except that now b = b(t) is given by (3.4) with the choice of the positive sign. Two different values of b(t) given by (3.4) coincide at the critical point t0. Moreover, with the positive choice of sign in (3.4) both equations (4.2) with h given by (3.7) are satisfied at t = 1 4 and z = 0. To derive equations of the genus zero breaking curves we note that, according to (3.6), the two saddle points ±z0 are defined by z2 0 = −1 t − b2 2 . (6.3) Thus, the values of h, ht, see (3.7), (3.10) respectively, at either of the two saddle points ±z0 = ±z0(t) are h(z0; t) = ln 2z2 0 − b2 + 2z0 √ z2 0 − b2 b2 − z0 4 √ z2 0 − b2, ht(z0; t) = z0 4t √ z2 0 − b2. (6.4) Introducing u = √ 1 + 12t = 2 √ 3 √ t− t0, where arg u = 0 when t ∈ (− 1 12 , 0), and using (3.4) and (6.3), we obtain h(z0(t); t) = ln 1± 2u+ √ 3 √ u2 ± 2u 1∓ u − √ u2 ± 2u 4 √ 3t , ht(z0(t); t) = √ u2 ± 2u 4 √ 3t2 . (6.5) 42 M. Bertola and A. Tovbis Thus, the equations of genus zero breaking curves are ln ∣∣∣∣∣1± 2 √ 1 + 12t+ √ 3 √ 1 + 12t± 2 √ 1 + 12t 1∓ √ 1 + 12t ∣∣∣∣∣ = < √ 1 + 12t± 2 √ 1 + 12t 4 √ 3t . (6.6) Let us now consider the local structure of the breaking curves near t = t0. It would be convenient to introduce a new “hyperbolic” variable p by cosh p = z0 b . Then sinh p = √ z2 0 b2 − 1, and, according to (6.4), equation <h(z0; t) = 0 for genus zero breaking curves becomes < [ 2p− b2 sinh 2p 8 ] = 0. Direct calculations yields sinh2 p = ±3u 2(1∓ u) , b2 = 8 1± u = 8 2 cosh2 p+ 1 4 cosh2 p− 1 = 8 cosh 2p+ 2 2 cosh 2p+ 1 , (6.7) which transforms (6.1) into <[2p] = <sinh 2p(cosh 2p+ 2) 2 cosh 2p+ 1 . (6.8) The right hand side of (6.8) has Taylor expansion 2p− 8 45p 5 + · · · at p = 0. Thus, for small p, equation (6.8) becomes <p5 ≈ 0. Taking into account (6.7), we obtain that the breaking curves (6.6) with upper signs approach t0 = − 1 12 at angles ±2π 5 , and the breaking curves (6.6) with lower signs approach t0 = − 1 12 at angles ±4π 5 . In order to prove the transition from genus 0 to genus 2 along the breaking curve (6.6) with the upper sign, it is sufficient to prove that < [ ht(z0(t); t) dt ds ] > 0, (6.9) where dt ds is the direction of the crossing from the genus 0 to genus 2 region at a regular breaking point t = tb. Indeed, the positive sign of the inequality reflects the fact that a complementary arc is pinched at a double point and, thus, a new main arc is born. Let us, for simplicity, take t = tb sufficiently close to t0 = − 1 12 . Then \| arg(tb − t0)\| ≤ π and we know tb 6∈ R. So, u is small and <√u > 0,=√u 6= 0. Then the logarithmic term in (6.6), and, thus, the right hand side of (6.6) are positive. But, according to (6.5), <ht(z0(t); t) = < [ t−1 √ u2±2u 4 √ 3t ] < 0 since arg t ≈ π. We cross the breaking curve moving along the line =t = =tb in the negative direction, so dt ds < 0. Thus, we have proved inequality (6.9) for t close to t0 = − 1 12 . Since, according to Lemma 6.1, ht(z0(t); t) 6= 0 on the breaking curve, the inequality (6.9) will hold on the breaking curve until we reach the singularity t = 0. Hence we have proved the Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 43 transition from genus zero to genus two across the breaking curve (6.6) with the upper choice of the sign. The second pair of breaking curves, emanating from t0 = − 1 12 , is given by (6.6) with the lower sign. They end at t2 = 1 4 . They separate two regions of genus 2 with different topology of the main arcs, see Fig. 2, two middle panels at the bottom and the panel in the middle (lower half plane). As we see, bounded complementary arcs shrink to points as we cross this breaking curve, so that the inequality (6.9) should hold in this case as well. The proof is identical to the previous arguments once we notice that <√−u > 0. We have finally proved the following lemma. Lemma 6.3. Four breaking curves, emanating from t0 = − 1 12 , separate regions of genera zero and two, or two regions of genus two with different topology of main arcs, as shown on Fig. 2. They are given by equation (6.6). The genus one breaking curves consist of the ray <z > 1 4 of R as well as two bounded curves, symmetrical with respect to R, that connect t2 = 1 4 with t = 0. Let us first consider this case. Expanding the expression (6.2) for h′ at z = ∞ and taking into the account the asymptotics (2.19) of h′, we obtain h′(z) = −tz3 − z + 2 z +O(z2) = −tz3 [ 1− λ2 1 + λ2 2 2z2 − ( λ2 1 − λ2 2 )2 8z4 +O ( z−6 )] , which yields λ2 1 + λ2 2 = −2 t , λ2 1 − λ2 2 = 4√ t or λ2 1 = 2√ t − 1 t , λ2 2 = − 2√ t − 1 t . (6.10) It follows that the branchpoints are situated on the imaginary axis when t ∈ (0, 1 4) and on the cross (±λ1 ∈ R, ±λ2 ∈ iR) when t > 1 4 , see Fig. 2, the panel in lower left corner and the panels in the middle on the right side). To obtain the equations of the genus one breaking curve we need to compute h. Substitut- ing (6.10) into (6.2), we obtain h′(z; t) = −tz √( z2 + 1 t )2 − 4 t , so that h(z; t) = ln tz2 + 1 + √ (tz2 + 1)2 − 4t −2 √ t − tz2 + 1 4t √( tz2 + 1 )2 − 4t, (6.11) where we integrated from the branchpoint −λ2 (because of the Boutroux condition, <h does not depend on the choice of a particular branchpoint). Then h(0; t) = ln 1 + √ 1− 4t −2 √ t − √ 1− 4t 4t , (6.12) so that the equation of the breaking curves, connecting t2 = 1 4 with t = 0 is ln ∣∣∣∣1 + √ 1− 4t −2 √ t ∣∣∣∣ = < √ 1− 4t 4t . (6.13) Moreover, it is clear that <h(0; t) < 0 when t ∈ ( 0, 1 4 ) (6.14) 44 M. Bertola and A. Tovbis and <h(0; t) = 0 when t > 1 4 . Thus, the ray t > 1 4 is a breaking curve whereas the interval (0, 1 4) is not. As one crosses the breaking curve (6.13) from the genus two region (Fig. 2, the upper panel in the center), to the genus one region (Fig. 2, the lower left panel), the central main arc in the first panel shrinks into a double point (on the breaking curve), which disappear as one gets into the region zero region. To prove this scenario, it is sufficient to prove the inequality along the breaking curve (6.13), < [ ht(0; t) dt ds ] < 0. (6.15) As it was observed above, it is sufficient to prove (6.15) at one point of the breaking curve. Let us take this point, tb = ρeiφ, sufficiently close to t = 0. Then equation (6.13) implies cosφ ≈ −ρ ln ρ or φ→ π 2 − 0 (6.16) as ρ→ 0. So, according to the second equation (6.2) and (6.16), we have < [ht(0; tb)] = < √ 1− 4tb 4t2b ≈ cos 2φ 4ρ2 < 0. Finally, dtds > 0 when we cross the breaking curve (6.13) along the line =t = =tb with a sufficiently small =tb > 0. Thus, we have proved the following lemma. Lemma 6.4. The breaking curves connecting t2 = 1 4 with t = 0, separate regions of genera one (inside the curves) and two (outside the curves), see Fig. 2. They are given by equation (6.13). The breaking curve =t = 0, <t > 1 4 is surrounded by the genus two region. All the breaking curves, shown on Fig. 2, emanate from either critical point t0 = − 1 12 or t2 = 1 4 . The former are genus zero breaking curves whereas the latter are genus one breaking curves (except those, that connect t0 and t2, which are genus zero). These breaking curves were described in Lemmas 6.3 and 6.4. To complete the proof of the asymptotic phase portrait of Fig. 2, we still need to show that there are no more breaking curves, that is, there are no breaking curves that do not pass through a critical point, or t = 0 or ∞. According to Corollary 6.2, any such curve must be a simple smooth closed loop, ν, that does not intersect any other breaking curve. We rule out existence of such ν by observing that, according to (4.2), each breaking curve is zero level curve of <h(z0(t), t) = 0, where z0(t) is the double point. Then, it remains to show that <h(z0(t), t) is a harmonic function of t. We start with a genus one level curve, where z0(t) ≡ 0 and so h(z0(t), t) = h(0, t) is given by (6.12). This is a harmonic function with a singularity at t = 0 and the branchcut R+. Thus, if <h(0, t) = 0 along bounded ν, then ν ∩ R+ 6= ∅. But ν ∩ (1 4 ,+∞) = ∅ since (1 4 ,+∞) is a breaking curve. Thus, ν ∩ (0, 1 4) 6= ∅, which contradicts (6.14). Finally, if ν is an unbounded loop that does not intersect R+, then ν originates and ends at infinity and <h(0, t) is harmonic inside ν. Since <h(0, t) = O(t−1) as t→∞, we conclude that such loop cannot exist. Consider now a genus zero level curve given by the first equation (6.5). The function ψ(t) = <h0(z0(t), t), given by (6.4) (see also (6.5)) in the complex t plane, has singularities at {t0, t2, 0,∞}, with branchcut R\ (− 1 12 , 0). Thus, if ν is bounded, it must intersect (−∞,− 1 12) or (0, 1 4). In order to prove that this is not the case, we calculate ψ′(t) = <ht(z0(t), t) + <[h′(z0(t), t)z′0(t)] = <ht(z0(t), t), since h′(z0(t), t) = 0 for every t and z′0(t) is bounded for t 6∈ {t0, t2, 0,∞}. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 45 Let us first consider the interval (−∞,− 1 12). According to (3.7), ψ(t0) = 0. Thus, it is sufficient to prove that ψ′(t) 6= 0 on (−∞,− 1 12). Direct calculations (for t ∈ R) yield ψ′(t) = 1 8t < √ z2 0(t) [ z2 0(t)− b2 ] = < √ (1 + 12t)± 2 √ 1 + 12t 8 √ 3t2 , (6.17) where we used z2 0(t) = −2∓ √ 1 + 12t 3t , z2 0(t)− b2 = ∓ √ 1 + 12t t . When t ∈ (−∞,− 1 12), we have √ 1 + 12t ∈ iR, so that the radical is in the right half plane. Thus, ψ′(t) > 0, so that ψ(t) < 0 on this interval. Therefore, ν cannot cross t ∈ (−∞,− 1 12). This estimate also shows that genus zero breaking curves are bounded. Consider the interval t ∈ (0, 1 4), we distinguish the functions ψl(t) and ψu(t), corresponding to the lower/upper choice of signs in (6.4), (6.5) (and also (6.17)). It follows directly from (6.5) that h0(z0(t), t) with the lower choice of the signs does not have branching at t = 0 (corresponds to u = 1 on the main sheet of t). Thus, we have to study only ψ = ψu. A straightforward computation yields ψu ( 1 4 ) = ln ( 2 √ 2 3 + 1 3 ) + 2 √ 2 3 > 0 and ψ′u(t) < 0, since both z2 0(t), z2 0(t) − b2 have argument π on (0, 1 4). Thus, ν cannot cross (0, 1 4), and we completed the proof of the global asymptotic portrait from Fig. 2. 7 Asymptotics of recurrence coefficients in the higher genera regions The leading order behavior of the recurrence coefficients and of the pseudonorms in the genus zero region and in full vicinities of the critical points t0, t2 was obtained in [4, 13] and [7] respectively. In this section we use Theorem 2.18 to find the leading order behaviour of αn(t) and hn(t) in regions of genus one and two. 7.1 Genus 1 region Let us start with the genus one region O1 that contains the segment (0, 1 4). Explicit formulae for the corresponding branchpoints are given by (6.10), where, thanks to the symmetry, we use notations −λ2, −λ1, λ1, λ2 for consecutive branchpoints along the Riemann–Hilbert contour instead of λ0, λ1, λ2, λ3. The negatively oriented loop γ̂m,1 around the branchcut λ1, λ2 is the A cycle, whereas the loop −γ̂c,1 is the B cycle. The period τ and the normalized holomorphic differential ω are elliptic integrals τ = − ∫ λ1 −λ1 dζ R(ζ) 2 ∫ λ2 λ1 dζ R(ζ) , ω(z) = dz 2 ∫ λ2 λ1 dζ R(ζ)R(z) , where R(ζ) = √( ζ2 − λ2 1 )( ζ2 − λ2 2 ) . 46 M. Bertola and A. Tovbis Direct calculations yield 2 ∫ λ2 λ1 dζ R(ζ) = − 2 √ t√ 1 + 2 √ t K ( 2t 1 4√ 1 + 2 √ t ) , τ = i K (√ 1−2 √ t 1+2 √ t ) K ( 2t 1 4√ 1+2 √ t ) , (7.1) where K denotes complete elliptic integral. Using the fact that R(z) is an even function and (3.18), we calculate 2u(∞) = −τ 2 , $̃ = 2u(∞) = −τ 2 . (7.2) Let us normalize the traffics %j , see (1.1), so that the traffic on γm,0 = (−λ2,−λ1) is 1. We then denote the traffic on γm,0 = (λ1, λ2) by ρ1. Note that in the Generic case, considered in the paper, ρ1 6= 0. Lemma 7.1. For any t ∈ O1 the model RHP (2.26) has a solution (for all sufficiently large N ∈ N) if and only if ln ρ1 − iπ 2πi 6= 0 mod τ 2 Z⊕ Z, (7.3) where τ is given by (7.1). Lemma 7.1 follow from Corollary 2.18 and (7.2). Combining (7.2) with Theorem 2.19 and using the properties (2.31) of the Riemann theta function, we obtain the theorem below. The constant ` = −1 2 + 1 4t − 1 2 ln(t) + iπ can be easily obtained from (6.11). Theorem 7.2. If t ∈ O1 is such that (7.3) is satisfied then αn(t) = √ 1− 4t− 1 2t ×  ρ1e −2πi τ 2 Θ2 ( ln ρ1 2πi ) Θ2(0) Θ2 ( ln ρ1 2πi − τ 2 ) Θ2 ( τ 2 ) +O ( n−1 ) if n is odd, ρ−1 1 Θ2 ( ln ρ1 2πi − τ 2 ) Θ2(0) Θ2 ( ln ρ1 2πi ) Θ2 ( τ 2 ) +O ( n−1 ) if n is even, (7.4) hn(t) = πit− n 2 √ 1 + 2 √ t− √ 1− 2 √ t (−1)(n+1)e n 2 (1− 1 2t +iπτ) √ t ×  e− πiτ 2 Θ ( ln ρ1 2πi ) Θ(0) Θ ( ln ρ1 2πi + τ 2 ) Θ ( − τ 2 ) +O ( n−1 ) if n is odd, Θ ( ln ρ1 2πi − τ 2 ) Θ(0) Θ ( ln ρ1 2πi ) Θ ( − τ 2 ) +O ( n−1 ) if n is even. According to Theorem 7.2, the even and odd subsequences of αn(t) attain, in general, different limits. The special values of ρ1, when these limits are equal, are provided in the following corollary. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 47 Corollary 7.3. Under the same conditions as Theorem 7.2, the requirement ln ρ1 2πi = ( 1 4 + k ) τ +m, where k ∈ Z, m ∈ 1 2 Z, (7.5) implies that the even and odd subsequences of αn(t) attain a single limit αn(t) = √ 1− 4t− 1 2t e2πi(m− τ4 )Θ2(0) Θ2( τ2 ) +O ( n−1 ) . (7.6) Proof. Denote x = ln ρ1 2πi . According to (7.4), the limits of the even and odd subsequences of αn(t) coincide when e2πi(2x− τ 2 )Θ4(x) = Θ4 ( x− τ 2 ) . (7.7) Substituting (7.5) into (7.7) and using properties (2.31), we obtain the first statement. Direct calculations also lead to (7.6). � 7.2 Genus 2 solutions In this section we adjust the results of Theorem 2.19 to the case of L = 2 and symmetric branchpoints with respect to the origin. Because of the symmetry, we denote the main arc containing zero as γm,0, its endpoints (branchpoints) as ±λ0, the remaining main arcs as γm,±1 and their branchpoints as ±λ1, ±λ2 respectively. In the genus two region we distinguish two different configurations of the main arcs: the standard configuration shown on two panels in the middle of Fig. 2 and configuration with whiskers shown on the central and right upper panels, the second and third (from below) left panels, etc. These two regions are separated by two symmetrical genus zero breaking curves, connecting the critical points t0 = − 1 12 and t2 = 1 4 ; the whisker region is outside and the standard genus two region (together with genera zero and one regions) is inside these breaking curves. We start with calculating the Riemann period matrix τ . We take negatively oriented loops around γm,1, γm,−1 as A cycles A1, A2 respectively, and oriented loops connecting λ1 and λ0 and −λ1 and −λ0 (on the main sheet and going back on the second sheet) as B cycles B1, B2 respectively. The normalized holomorphic differentials ωj = Pj(z)dz R(z) , defined by (2.29), in the case of genus two and symmetric branchpoints have the form ω1 = κ1z + κ2 R(z) dz, ω2 = −κ1z + κ2 R(z) dz, (7.8) where κ1 = ( 2 ∫ γm,1 zdz R(z) )−1 , κ2 = ( 2 ∫ γm,1 dz R(z) )−1 , with R(z) = √√√√ 2∏ j=0 ( z2 − λ2 j ) . Here we used the fact that R(z) is an odd function in C \M. Further, the symmetry of B cycles implies that τ11 + τ21 = τ12 + τ22, where τj,k are the entries of the Riemann period matrix τ , 48 M. Bertola and A. Tovbis see (2.30). Since τ is symmetric, we obtain τ11 = τ22, i.e., the two by two matrix τ has equal diagonal entries. Let us now consider the vector ~̃$ from the RHP (2.28). Using (3.18) and Cauchy’s residue theorem, we obtain ~̃$ = 2u∞ − 1 2πi ∫ M̂ ( tz4 4 + z2 2 ) (±κ1z + κ2)dz R(z) = 2u∞ + 1 + t 4 2∑ j=0 λ2 j∫ γm,1 dz R(z) (1, 1)t, (7.9) where, in view of (7.8), the plus sign corresponds to $̃1 and the minus sign to $̃2. According to (2.19), (2.22), in the genus two region h′(z) = −t √√√√ 2∏ j=0 ( z2 − λ2 j ) = −tz3 √√√√ 2∏ j=0 ( 1− λ2 j z2 ) = −tz3 − z + 2 z +O ( z−2 ) so that, similarly to (6.10), we obtain 2∑ j=0 λ2 j = −2 t , ∑ j<k λ2 jλ 2 k = 1 t2 − 4 t . (7.10) The remaining third condition for the branchpoints ±λj , j = 0, 1, 2, is the Boutroux condition ∫ λ1 λ0 h′(z)dz = −t ∫ λ1 λ0 √√√√ 2∏ j=0 ( z2 − λ2 j ) ∈ R. Equations (7.9) and (7.10) yield ~̃$ = 2u∞ + 1 2 ∫ γm,1 dz R(z) (1, 1)t = 2u∞ + ~κ2, (7.11) where ~κ2 = κ2(1, 1)t. According to (1.5), the traffic on γm,0 is 1. Let ~ρ = 1 2πi(ln ρ1, ln ρ2)t, where ln ρj is the logarithmic term in the exponential of the jump matrix on γm,j in the RHP (2.26). Substituting (7.8), (7.11) into (2.43), (2.44) we obtain the following theorem. Theorem 7.4. Let T be a compact subset in the genus two region. The recurrence coefficients and the pseudonorms satisfy αn = (λ0 + λ2 − λ1)2 Θ(2(N − 1)u∞ +N~κ2 + ~ρ)Θ(2(N + 1)u∞ +N~κ2 + ~ρ)Θ2(0) 4Θ2(2Nu∞ +N~κ2 + ~ρ)Θ2(2u∞) +O ( N−1 ) , hn = π(λ0 + λ2 − λ1) Θ(2(N + 1)u∞ +N~κ2 + ~ρ)Θ(0) Θ(2Nu∞ +N~κ2 + ~ρ)Θ(2u∞) eN` +O ( N−1 ) , (7.12) as N →∞ uniformly in t ∈ T provided that Θ ( 2Nu∞ +N~κ2 + ~ρ ) is separated from 0. Remark 7.5. If we change the orientation of the contour γm,2 in the model RHP (2.26), then, according to (2.43), (2.44), we will have to replace in (7.12) the prefactors (λ0 + λ2− λ1) by λ0, the vector ~ρ by ~ρ + (0, 1)t and the real part <τ = 1 21 of the period matrix for all the theta functions in (7.12) by <τ = 1 2σ3. For justification and more details see [6, Section 3]. Orthogonal Polynomials with Complex Varying Quartic Exponential Weight 49 Acknowledgements Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). 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Math. 57 (2004), 877–985. http://dx.doi.org/10.1090/conm/578/11484 http://arxiv.org/abs/1112.5713 http://dx.doi.org/10.1016/j.physd.2016.03.009 http://arxiv.org/abs/1603.03341 http://dx.doi.org/10.1007/s00220-009-0984-0 http://arxiv.org/abs/0902.1123 http://dx.doi.org/10.1002/cpa.20024 1 Introduction and main results 2 Steepest descent analysis of the RHP (2.1) 2.1 The RHP for recurrence coefficients 2.2 General requirements on the g-function 2.2.1 Equality requirements and formula for g 2.2.2 Inequality (sign) requirements (or sign distribution requirements) for h and the modulation equation 2.3 Modulation equations 2.4 Schematic conclusion of the steepest descent analysis 2.5 Solution of the model problem 3 Formal calculation of the g-function and h-function 3.1 Explicit form of g and h functions in the genus zero region 3.2 The g-function and h-function and their properties in the genus L case 4 Existence of genus zero g and h functions, breaking curves and continuation principle 4.1 Sign requirements in the genus zero region 4.2 Breaking curves. Regular and critical breaking points. Symmetries 4.3 Continuation principle for Boutroux deformations 5 Boundedness of the branchpoints 5.1 A general theorem on ODE's for polynomials 5.2 Properties of the polynomial (5.1) 6 Breaking curves for quartic potential f(z;t) 6.1 Existence of breaking curves 7 Asymptotics of recurrence coefficients in the higher genera regions 7.1 Genus 1 region 7.2 Genus 2 solutions References

On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight

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