Three-Phase Freak Waves

In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the...

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Дата:	2015
Автори:	Smirnov, A.O., Matveenko, S.G., Semenov, S.K., Semenova, E.G.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2015
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/147009
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Цитувати:	Three-Phase Freak Waves / A.O. Smirnov, S.G. Matveenko, S.K. Semenov, E.G. Semenova // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 44 назв. — англ.

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spelling	irk-123456789-1470092019-02-14T01:26:56Z Three-Phase Freak Waves Smirnov, A.O. Matveenko, S.G. Semenov, S.K. Semenova, E.G. In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the spectral curves. 2015 Article Three-Phase Freak Waves / A.O. Smirnov, S.G. Matveenko, S.K. Semenov, E.G. Semenova // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q55; 37C55 DOI:10.3842/SIGMA.2015.032 http://dspace.nbuv.gov.ua/handle/123456789/147009 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	In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the spectral curves.
format	Article
author	Smirnov, A.O. Matveenko, S.G. Semenov, S.K. Semenova, E.G.
spellingShingle	Smirnov, A.O. Matveenko, S.G. Semenov, S.K. Semenova, E.G. Three-Phase Freak Waves Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Smirnov, A.O. Matveenko, S.G. Semenov, S.K. Semenova, E.G.
author_sort	Smirnov, A.O.
title	Three-Phase Freak Waves
title_short	Three-Phase Freak Waves
title_full	Three-Phase Freak Waves
title_fullStr	Three-Phase Freak Waves
title_full_unstemmed	Three-Phase Freak Waves
title_sort	three-phase freak waves
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147009
citation_txt	Three-Phase Freak Waves / A.O. Smirnov, S.G. Matveenko, S.K. Semenov, E.G. Semenova // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 44 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT smirnovao threephasefreakwaves AT matveenkosg threephasefreakwaves AT semenovsk threephasefreakwaves AT semenovaeg threephasefreakwaves
first_indexed	2025-07-11T01:08:27Z
last_indexed	2025-07-11T01:08:27Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 032, 14 pages Three-Phase Freak Waves Aleksandr O. SMIRNOV, Sergei G. MATVEENKO, Sergei K. SEMENOV and Elena G. SEMENOVA St.-Petersburg State University of Aerospace Instrumentation (SUAI), 67 Bolshaya Morskaya Str., St.-Petersburg, 190000, Russia E-mail: alsmir@guap.ru, MatveiS239@gmail.com, sksemenov@mail.ru, egsemenova@mail.ru Received December 05, 2014, in final form April 11, 2015; Published online April 21, 2015 http://dx.doi.org/10.3842/SIGMA.2015.032 Abstract. In the article, we describe three-phase finite-gap solutions of the focusing non- linear Schrödinger equation and Kadomtsev–Petviashvili and Hirota equations that exhibit the behavior of almost-periodic “freak waves”. We also study the dependency of the solution parameters on the spectral curves. Key words: nonlinear Schrödinger equation; Hirota equation; freak waves; theta function; reduction; covering; spectral curve 2010 Mathematics Subject Classification: 35Q55; 37C55 1 Introduction This study was motivated by the intention to demonstrate the behavior of three-phase extreme waves. Most recent scientific research shows that the simplest and most universal model for such waves is the focusing nonlinear Schrödinger equation (NLS) ipt + pxx + 2 \|p\|2 p = 0, i2 = −1, (1) Since 1968 the equation (1) has been describing distribution on the surface of the ocean of weakly nonlinear quasi-monochromatic wave packets with relatively steep fronts [44]. An application of this equation to the problems of nonlinear optics was known earlier [7]. Since the equation (1) is a model of first approximation, it appears in simulations of many weakly nonlinear phenomena. This equation has a wide range of applications ranging from plasma physics [28] to financial markets [43]. Among the properties of equation (1) there is a modulation instability that leads to the appearance of the so-called “freak waves” (in hydrodynamics known as “rogue waves”) [2]. These waves represent amplitude peaks localized in space and time. In the last 20 years, first in hydrodynamics and then in nonlinear optics, these waves have been the object of numerous theoretical and experimental studies [3]. Such attention to the problem of the “freak waves” is due to the losses at oil platforms, tankers, container ships and other large vessels caused by the “rogue waves”. There are many more precise and more complex models, which give a more exact description of the “freak waves” [3]. These models can be divided into two classes. In the first class one can solve them analyticaly while in the second class one can use numerical methods only. Analytical methods include: inverse scattering transform method; finite-gap integration method; Bäcklund transform method; Darboux transform method; Hirota method. In the present work, we use a finite-gap integration method. The works of Dubrovin, Novikov, Marchenko, Lax, McKean, van Moerbeke, Matveev, Its, Krichever [9, 10, 11, 12, 13, 21, 22, 26, 29, 31, 33, 35] give a description of this method (see also the review [32]). However, another method of constructing finite-gap solutions of integrable nonlinear equations exists [23, 24, 34, 36]. Let mailto:alsmir@guap.ru mailto:MatveiS239@gmail.com mailto:sksemenov@mail.ru mailto:egsemenova@mail.ru http://dx.doi.org/10.3842/SIGMA.2015.032 2 A.O. Smirnov, S.G. Matveenko, S.K. Semenov and E.G. Semenova us remark that first method is based on Baker–Akhiezer function but the second one is based on some Fay’s identities [14]. In our paper, we use the first method and Its and Kotlyarov’s classic formulas [18, 20] (see also [6]). Our goal here is to show the behavior of three-phase algebro-geometric solutions of NLS, KP-I and Hirota equations. Section 2 of this paper contains the basic notations and classic formulas for algebro-geometric solutions of integrable nonlinear equations under consideration. Section 3 is devoted to the periodicity of three-phase solutions of NLS, KP-I and Hirota equations. In Section 4 we consider an example of three-phase algebro-geometric solutions of KP-I and Hirota equations for different values of parameters. 2 Finite-gap multi-phase solutions of the NLS equation The nonlinear differential equations that are integrated by methods of the algebraic geome- try, can be obtained as a compatibility condition of the system of ordinary linear differential equations with a spectral parameter [6, 15, 16]. In particular, let us consider the following equations [15, 19, 38] Yx = UY, Yz = VY, Yt = WY, (2) where U = −λ ( i 0 0 −i ) + ( 0 iψ −iφ 0 ) , V = 2λU + V0, W = 4λ2U + 2λW0 + W1, λ is a spectral parameter. Using these equations and additional relations (Yx)z = (Yz)x, (Yx)t = (Yt)x one can easy obtain the so-called equations of zero curvature Uz −Vx + UV−VU = 0 and Ut −Wx + UW−WU = 0, (3) which should be valid for all values of spectral parameter λ. Respectively, it follows from equations (3) that matrixes V0, W0, W1 take the forms W0 = V0 = ( −iψφ −ψx −φx iψφ ) , W1 = ( ψxφ− ψφx 2iψ2φ− iψxx −2iψφ2 + iφxx ψφx − ψxφ ) , Also, W = 2λV + W1. Conditions (3) lead to additional system of equations (parities). The first system is the coupled nonlinear Schrödinger equation iψz + ψxx − 2ψ2φ = 0, iφz − φxx + 2ψφ2 = 0, (4) and the second system is the coupled modified Korteweg–de Vries equation ψt + ψxxx − 6ψφψx = 0, φt + φxxx − 6ψφφx = 0. (5) These two systems of the nonlinear differential equations are closely related to two other ones. Specifically, differentiating equations (4) with respect to x and substituting them in (5), one obtains the coupled modified two-dimensional nonlinear Schrödinger equation in cone coor- dinates [27] iψt + ψxz + 2i(ψφx − φψx)ψ = 0, iφt − φxz + 2i(φψx − ψφx)φ = 0, Three-Phase Freak Waves 3 Also, the functions ψ(x, t,−αt) and φ(x, t,−αt) are solutions to the coupled integrable Hirota equation (α ∈ R) iψt + ψxx − 2ψ2φ− iα(ψxxx − 6ψφψx) = 0, iφt − φxx + 2ψφ2 − iα(φxxx − 6ψφφx) = 0, (6) if ψ(x, z, t) and φ(x, z, t) are solutions of (4) and (5). Systems of the nonlinear differential equations (4), (5) are the first two integrable systems from the AKNS hierarchy [15]. One of the features of finite-gap multi-phase solutions of the integrable nonlinear equations is that fact that in some sense they are the solutions of all hierarchy. Particulary, our solutions can be used for constructing solutions of generalized non- linear Schrödinger equation [42]. By substituting φ = ±ψ into equation (4) we get a standard form of the nonlinear Schrödinger equation. Particularly, for φ = −ψ equations (4) transform to (1) [11, 18, 19] and equations (6) transform to the integrable Hirota equation [4, 8, 17, 30] iψt + ψxx + 2 \|ψ\|2 ψ − iα ( ψxxx + 6 \|ψ\|2 ψx ) = 0. (7) It is also easy to check that for any ψ and φ, that satisfy both (4) and (5) simultaneously, the function u(x, z, t) = −2ψφ is a solution of the Kadomtsev–Petviashvili-I equation (KP-I) 3uzz = (4ut + uxxx + 6uux)x. (8) In case φ = ±ψ this solution is a real function. Finite-gap solutions of systems (4), (5) are parameterized by the hyperelliptic curve Γ = {(χ, λ)} of the genus g [15, 38]: Γ : χ2 = 2g+2∏ j=1 (λ− λj), The branch points (λ = λj , j = 1, . . . , 2g+ 2) of this curve are the endpoints of the spectral arcs of continuous spectrum of Dirac operator (2). Infinitely far point of the spectrum corresponds two different points P±∞ on the curve Γ. In case φ = −ψ the curve Γ has the form Γ : χ2 = g+1∏ j=1 (λ− λj)(λ− λj) = λ2g+2 + 2g+2∑ j=1 χjλ 2g+2−j , =χj = 0, =(λj) 6= 0. (9) Following a standard procedure of constructing finite-gap solutions [6, 11, 38], for Γ let us choose a canonical basis of cycles γt = (a1, . . . , ag, b1, . . . , bg) with matrix of intersection indices C0 = ( 0 I −I 0 ) . To satisfy the condition φ = −ψ, it is necessary [6, 11] that this basis of cycles is transformed according to the rules τ̂1a = −a, τ̂1b = b +Ka, (10) where τ1 is anti-holomorphic involution, τ1 : (χ, λ)→ (χ, λ). Let us also consider normalized holomorphic differentials dUj :∮ ak dUj = δkj , k, j = 1, . . . , g, 4 A.O. Smirnov, S.G. Matveenko, S.K. Semenov and E.G. Semenova and a matrix of periods B of the curve Γ: Bkj = ∮ bk dUj , k, j = 1, . . . , g. It is well known (see, for example, [5, 11]) that the matrix B is a symmetric matrix with positively defined imaginary part. Let us introduce g-dimensional Riemann theta function with characteristics η, ζ ∈ Rg [5, 11, 14]: Θ [ ηt; ζt ] (p\|B) = ∑ m∈Zg exp { πi(m + η)tB(m + η) + 2πi(m + η)t(p + ζ) } , Θ [ 0t; 0t ] (p\|B) ≡ Θ(p\|B), where B is a matrix of periods, p ∈ Cg and summation passes over an integer g-dimensional lattice. Let us also define on Γ normalized Abelian integrals of the second kind (Ωj(P), j = 1, 2, 3) and the third kind (ω0(P)) with the following asymptotic at infinitely distant points P±∞:∮ ak dΩ1 = ∮ ak dΩ2 = ∮ ak dΩ3 = ∮ ak dω0 = 0, k = 1, . . . , g, Ω1(P) = ∓i ( λ−K1 +O ( λ−1 )) , P → P±∞, Ω2(P) = ∓i ( 2λ2 −K2 +O ( λ−1 )) , P → P±∞, Ω3(P) = ∓i ( 4λ3 −K3 +O ( λ−1 )) , P → P±∞, ω0(P) = ∓ ( lnλ− lnK0 +O ( λ−1 )) , P → P±∞, χ = ± ( λg+1 +O (λg) ) , P → P±∞. Let us denote the vectors of b-periods of Abelian integrals of the second kind Ω1(P), Ω2(P), Ω3(P) by 2πiU, 2πiV, 2πiW respectively. Theorem 1 ([6, 38]). Function Y (P, x, z, t) = ( y1(P, x, z, t) y1(τ0P, x, z, t) y2(P, x, z, t) y2(τ0P, x, z, t) ) , where τ0 is hyperelliptic involution, τ0 : (χ, λ)→ (−χ, λ), y1(P, x, z, t) = Θ(U(P) + Ux+ Vz + Wt−X)Θ(Z) Θ(U(P)−X)Θ(Ux+ Vz + Wt+ Z) × exp{Ω1(P)x+ Ω2(P)z + Ω3(P)t+ iΦ(x, z, t)}, y2(P, x, z, t) = ρ Θ(U(P) + Ux+ Vz + Wt+ ∆−X)Θ(Z−∆) Θ(U(P)−X)Θ(Ux+ Vz + Wt+ Z) × exp{Ω1(P)x+ Ω2(P)z + Ω3(P)t− iΦ(x, z, t) + ω0(P)}, is the eigenfunction of the Dirac operator (2) with functions ψ(x, z, t) = 2K0 ρ Θ(Z)Θ(Ux+ Vz + Wt+ Z−∆) Θ(Z−∆)Θ(Ux+ Vz + Wt+ Z) exp{2iΦ(x, z, t)}, φ(x, z, t) = 2ρK0 Θ(Z−∆)Θ(Ux+ Vz + Wt+ Z + ∆) Θ(Z)Θ(Ux+ Vz + Wt+ Z) exp{−2iΦ(x, z, t)}, (11) Three-Phase Freak Waves 5 for any z, t and ρ 6= 0. The functions (11) satisfy the equations (4) and (5). Here ∆ is the vector of holomorphic Abelian integrals, calculated along a path connecting P−∞ and P+ ∞ without crossing any of the basic cycles, ∆ = U(P+ ∞)− U(P−∞), Φ(x, z, t) = K1x+K2z +K3t, X = K + g∑ j=1 U(Pj), Z = U(P+ ∞)−X, K is a vector of Riemann constants [5, 11, 14, 25]; Pj, j = 1, . . . , g is a non-special divisor. If the spectral curve Γ satisfies the condition (9), then the following equalities hold \|ψ\|2 = −4K2 0 Θ(Ux+ Vz + Wt+ Z−∆)Θ(Ux+ Vz + Wt+ Z + ∆) Θ2(Ux+ Vz + Wt+ Z) , (12) =U = =V = =W = =Z = 0, K2 0 < 0. It is easy to see that the corresponding solution of KP-I equation (8) has the form u(x, z, t) = −8K2 0 Θ(Ux+ Vz + Wt+ Z−∆)Θ(Ux+ Vz + Wt+ Z + ∆) Θ2(Ux+ Vz + Wt+ Z) , (13) and that the square of amplitude of solution of Hirota equation (7) equals \|ψH \|2 (x, t) = −4K2 0 Θ(Ux+ (V − αW)t+ Z−∆)Θ(Ux+ (V − αW)t+ Z + ∆) Θ2(Ux+ (V − αW)t+ Z) . 3 Features of three-phase solutions In case g = 3, the basis of normalized holomorphic differentials is defined by the formula [6, 11]: dUk = ( ck1λ 2 + ck2λ+ ck3 )dλ χ , where C = ( At )−1 , Ajm = ∮ aj λ3−m dλ χ . It follows from equation (` is an arbitrary path on Γ)∫ τ̂ ` dω = ∫ ` τ∗dω, that Ajm = ∮ aj λ3−m dλ χ = ∮ aj τ∗1 ( λ3−m dλ χ ) = ∮ τ̂1aj λ3−m dλ χ = − ∮ aj λ3−m dλ χ = −Ajm. Therefore A = −A and C = −C. Similarly, with integrals on b-cycles, we obtain B = −B −K or <B = −1 2 K. It follows from bilinear relations of Riemann (see, for example, [5, 6, 11]) that the coordinates of the vectors U, V, W can be written as Um = −i ( dUm dξ− ∣∣∣∣ ξ−=0 − dUm dξ+ ∣∣∣∣ ξ+=0 ) , Vm = −2i ( d2Um dξ2− ∣∣∣∣ ξ−=0 − d2Um dξ2+ ∣∣∣∣ ξ+=0 ) , 6 A.O. Smirnov, S.G. Matveenko, S.K. Semenov and E.G. Semenova Wm = −2i ( d3Um dξ3− ∣∣∣∣ ξ−=0 − d3Um dξ3+ ∣∣∣∣ ξ+=0 ) , where ξ± = 1/λ are local parameters in the neighborhood of infinitely distant points P±∞. Calculating the derivatives we obtain the relations Um = −2icm1, Vm = 2iχ1cm1 − 4icm2, Wm = i ( 4χ2 − 3χ2 1 ) cm1 + 4iχ1cm2 − 8icm3, or (U,V,W) = iC −2 2χ1 4χ2 − 3χ2 1 0 −4 4χ1 0 0 −8  . (14) It follows from (14) that the vectors U, V, W are real and linearly independent. Therefore, U, V, W are the basis vectors in R3. Hence, any vector from R3 can be presented in the form of the linear combinations of these vectors. In particular, for the vectors of the periods of the three-dimensional theta-functions et1 = (1, 0, 0), et2 = (0, 1, 0), et3 = (0, 0, 1) we can write the following relations ek = XkU + ZkV + TkW. Therefore, three-phase solutions (13) of equation KP-I (8) are the periodic functions in a three- dimensional space u(x+ Xk, z + Zk, t+ Tk) = u(x, z, t). If a three-phase solution of (8) has a form of freak waves, then the maxima of its amplitude are located in nodes of a three-dimensional lattice with edges (Xk,Zk, Tk). These edges can be found by an inversion of the matrix (U,V,W):X1 X2 X3 Z1 Z2 Z3 T1 T2 T3  = (U,V,W)−1 = i 1/2 χ1/4 χ2/4− χ2 1/16 0 1/4 χ1/8 0 0 1/8 At. Therefore, for three-phases solutions of equation KP-I (8) it is possible to describe their behavior in the following way: after a time interval ∆t = Tk a surface of solution u(x, z) reproduces itself with a shift on the XOZ plane by the (Xk,Zk) vector. As the three-phase solution of the equations (1) depends on two coordinates, x and z, and the third coordinate t is considered to be a parameter, the value of amplitude of this solution depends on the distance between the nodes of the given three-dimensional lattice and a plane t = t0. Hence, in contrast to the case of the two-phase solution [39, 40, 41], where the change of initial phase Z led to trivial shift of the solution on XOZ plane, the amplitude of the three-phase solution (11) of equations (1) depends on a choice of initial phase Wt0 + Z in a slightly more complicated fashion. 4 An example of three-phase solution Let us consider a spectral curve Γ3 = {χ, λ} of genus g = 3: Γ3 : χ2 = ( (λ− λ0)4 − 2a2(λ− λ0)2 cos 2ϕ+ a4 )( (λ− λ0)4 − 2b2(λ− λ0)2 cos 2ϕ+ b4 ) , where 0 < a < b, π/4 < ϕ < π/2. Three-Phase Freak Waves 7 Figure 1. Canonical basis of cycles on Γ3. Let us choose the basis of cycles on Γ3 as it is shown on Fig. 1. It is easy to check that the anti-holomorphic involution τ1 transforms the canonical basis of cycles using the rule (10) with the matrix K = 0 1 1 1 0 1 1 1 0  . There are also three holomorphic involutions on Γ3: τ0 : (χ, λ)→ (−χ, λ), τ2 : (χ, λ)→ (χ, 2λ0 − λ), τ3 : (χ, λ)→ ( a2b2(λ− λ0)−4χ, λ0 + ab(λ− λ0)−1 ) . As a corollary, the curve Γ3 covers the following two curves: 1) Γ1 = Γ3/τ2 of genus g = 1 Γ1 : χ2 + = ( t2 − 2a2t cos 2ϕ+ a4 )( t2 − 2b2t cos 2ϕ+ b4 ) , 2) Γ2 = Γ3/(τ0τ2) of genus g = 2 Γ2 : χ2 − = t ( t2 − 2a2t cos 2ϕ+ a4 )( t2 − 2b2t cos 2ϕ+ b4 ) , where t = (λ− λ0)2, χ+ = χ, χ− = (λ− λ0)χ, and dt χ+ = 2(λ− λ0)dλ χ , tdt χ− = 2(λ− λ0)2dλ χ , dt χ− = 2dλ χ . The curves Γ1 and Γ2 are shown on Figs. 2 and 3, where t1 = b2e2iϕ, t2 = a2e2iϕ. The coverings generate the following transformations of cycles:a1a2 a3 → S a1a21 a22 + P b1b21 b22  , b1b2 b3 → Q a1a21 a22 +R b1b21 b22  , 8 A.O. Smirnov, S.G. Matveenko, S.K. Semenov and E.G. Semenova Figure 2. The curve Γ1. Figure 3. The curve Γ2. where S = −1 1 0 1 0 −1 1 0 1  , P = 0 −2 0 0 0 2 0 0 −2  , Q = −1 1 0 0 0 −1 0 0 1  , R = 0 0 0 1 1 1 1 1 −1  . Recall that these matrices should satisfy the relations StQ = QtS, RtP = P tR, StR−QtP = nI, where I is identity matrix, n = 2 is the number of covering sheets. Due to involution τ3, the curve Γ2 covers two elliptic curves Γ± (Figs. 4 and 5): Γ± : ν2± = (s± 2ab) ( s2 − 2 ( a2 + b2 ) s cos 2ϕ+ a4 + b4 + 2a2b2 cos 4ϕ ) , where s = t+ a2b2 t , ν± = t± ab t2 χ−, ds ν± = (t∓ ab)dt χ− . The coverings of Γ2 on Γ± generate the following cycles mappings( a21 a22 ) → ( 1 1 −1 1 )( a+ a− ) , ( b21 b22 ) → ( 1 1 −1 1 )( b+ b− ) , As a result, we havea1a2 a3 → −1 1 1 1 1 −1 1 −1 1 a1a+ a− + 0 −2 −2 0 −2 2 0 2 −2 b1b+ b−  , (15) b1b2 b3 → −1 1 1 0 1 −1 0 −1 1 a1a+ a− + 0 0 0 1 0 2 1 2 0 b1b+ b−  . (16) Three-Phase Freak Waves 9 Figure 4. The curve Γ+. Figure 5. The curve Γ−. From (15), (16) and relations dλ χ = 1 4ab ds ν− − 1 4ab ds ν+ , λdλ χ = 1 2 dt χ+ + λ0 4ab ds ν− − λ0 4ab ds ν+ , λ2dλ χ = λ0 dt χ+ + λ20 + ab 4ab ds ν− − λ20 − ab 4ab ds ν+ it follows that C = c1 + c3 −2λ0(c1 + c3) (λ20 − ab)c1 + (λ20 + ab)c3 c1 c2 − 2λ0c1 (λ20 − ab)c1 − λ0c2 c3 c2 − 2λ0c3 (λ20 + ab)c3 − λ0c2  , B = i(b1 + b3) ib1 − 1/2 ib3 − 1/2 ib1 − 1/2 i(b1 + b2) ib2 − 1/2 ib3 − 1/2 ib2 − 1/2 i(b2 + b3)  , where c1 = 1 2(α1 − 2β1) , c2 = 1 2α2 , c3 = 1 2(α3 − 2β3) , ib1 = α1 2(α1 − 2β1) , ib2 = β2 2α2 , ib3 = α3 2(α3 − 2β3) , α1 = 1 2 ∮ a+ ds ν+ , α2 = 1 2 ∮ a1 dt χ+ , α3 = 1 2 ∮ a− ds ν− , β1 = 1 2 ∮ b+ ds ν+ , β2 = 1 2 ∮ b1 dt χ+ , β3 = 1 2 ∮ b− ds ν− . From the structure of the matrix B and from the matrix version of Appel’s theorem [37] it follows that the Riemann theta function of curve Γ3 equals: Θ(p\|B) = f(p̃1, p̃2, p̃3) = ϑ3(p̃1\|h1)ϑ3(p̃2\|h2)ϑ3(p̃3\|h3) + ϑ4(p̃1\|h1)ϑ1(p̃2\|h2)ϑ1(p̃3\|h3) + ϑ1(p̃1\|h1)ϑ4(p̃2\|h2)ϑ1(p̃3\|h3) + ϑ1(p̃1\|h1)ϑ1(p̃2\|h2)ϑ4(p̃3\|h3), (17) where p̃j = pj + pj+1 − pj+2, pj+3 ≡ pj , hj = exp(−4bj). 10 A.O. Smirnov, S.G. Matveenko, S.K. Semenov and E.G. Semenova Figure 6. Three-phase solution of KP-I equa- tion for λ0 = 0, t = 0. Figure 7. Three-phase solution of KP-I equa- tion for λ0 = 0, t = 0.1. Figure 8. Three-phase solution of KP-I equa- tion for λ0 = 0, t = 0.2. Figure 9. Three-phase solution of KP-I equa- tion for λ0 = 0, t = 0.3. The functions ϑj(p\|h) are Jacoby elliptic theta functions [1]: ϑ1(p\|h) = 2 ∞∑ m=1 (−1)m−1h(m−1/2) 2 sin[(2m− 1)πp], ϑ2(p\|h) = 2 ∞∑ m=1 h(m−1/2) 2 cos[(2m− 1)πp], ϑ3(p\|h) = 1 + 2 ∞∑ m=1 hm 2 cos(2mπp), ϑ4(p\|h) = 1 + 2 ∞∑ m=1 (−1)mhm 2 cos(2mπp). Using the reduced form of theta function (17) and values for the vectors of periods, one obtains the following formula for a squared absolute value of the three-phase solution (12) of the focusing NLS equation (1) \|ψ\|2 = −4K2 0f(k1x+ κ1t+ δ1, k2z + δ2, k3x+ κ3t+ δ3) (18) × f(k1x+ κ1t− δ1, k2z − δ2, k3x+ κ3t− δ3)× {f(k1x+ κ1t, k2z, k3x+ κ3t)}−2, Three-Phase Freak Waves 11 Figure 10. Three-phase solution of KP-I equa- tion for λ0 = k2/(4k1), t = 0. Figure 11. Three-phase solution of KP-I equa- tion for λ0 = k2/(4k1), t = 0.1. Figure 12. Three-phase solution of KP-I equa- tion for λ0 = k2/(4k1), t = 0.2. Figure 13. Three-phase solution of KP-I equa- tion for λ0 = k2/(4k1), t = 0.3. where the function f(p̃1, p̃2, p̃3) is defined by equation (17), and k1 = −4ic1, k2 = −8ic2, k3 = −4ic3, κ1 = 4k1 ( 3λ20 − ab+ ( a2 + b2 ) cos(2ϕ) ) , κ3 = 4k3 ( 3λ20 + ab+ ( a2 + b2 ) cos(2ϕ) ) . (17), (18) imply that for λ0 = 0 the amplitude of the constructed solution of NLS equation (1) is a periodic function of z, and for λ0 = 0, ϕ = 1 2 arccos ( ±ab a2+b2 ) it is a periodic function of z and t. Recall that the three-phase solution u(x, z, t) of the KP-I equation (8) and the square of am- plitude of three-phase solution of Hirota equation (7), \|ψH(x, t)\|2, can be constructed from (18) by using relations u(x, z, t) = 2 \|ψ(x, z, t)\|2 and \|ψH(x, t)\|2 = \|ψ(x, t,−αt)\|2. The three-phase solution of KP-I equation for ab = 1, √ b/a = 1.3, ϕ = 0.3π at the different moment of time t and for λ0 = 0 is presented on the Figs. 6–9. One can see the same solution for λ0 = k2/(4k1) on the Figs. 10–13. It is easy to see all three phases of solution on Figs. 6–13. Two phases are shortwaves and the third phase is a long-wave envelope. One can see also on Figs. 6–9 that the solution for λ0 = 0 is periodic in z, and that the long-wave envelope moves to the right side. The three-phase solution of Hirota equation for ab = 1, √ b/a = 1.3, ϕ = 0.3π, α = 0.1 and for different values of λ0 is presented in Figs. 14–17. It is easy to see all three phases of solution only in Fig. 15. 12 A.O. Smirnov, S.G. Matveenko, S.K. Semenov and E.G. Semenova Figure 14. Amplitude of three-phase solution of Hirota equation for λ0 = 0. Figure 15. Amplitude of three-phase solution of Hirota equation for λ0 = 4. Figure 16. Amplitude of three-phase solution of Hirota equation for λ0 = k2/(4k1). Figure 17. Amplitude of three-phase solution of Hirota equation for λ0 = k2/(4k3). Acknowledgements Authors thank Professor V.B. Matveev for his support and the discussions that we held over this paper and quasi-rational solutions of the NLS equation. This research was conducted within the framework of the State order of the Ministry of Education and Science of Russian Federation, and partially supported by RFBR (research project 14-01-00589 a). 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Three-Phase Freak Waves

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