Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II

We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. If |n| < 2t, we have decaying oscillation of order O(t⁻¹/²) as was proved in our previous paper. Near |n|=2t, the behavior is decaying oscillation of order O(t⁻¹/³) and the coefficient...

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Дата:	2015
Автор:	Yamane, H.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2015
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II / H. Yamane // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 6 назв. — англ.

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spelling	irk-123456789-1469962019-02-13T01:24:05Z Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II Yamane, H. We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. If \|n\| < 2t, we have decaying oscillation of order O(t⁻¹/²) as was proved in our previous paper. Near \|n\|=2t, the behavior is decaying oscillation of order O(t⁻¹/³) and the coefficient of the leading term is expressed by the Painlevé II function. In \|n\| > 2t, the solution decays more rapidly than any negative power of n. 2015 Article Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II / H. Yamane // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 6 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q55; 35Q15 DOI:10.3842/SIGMA.2015.020 http://dspace.nbuv.gov.ua/handle/123456789/146996 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 investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. If \|n\| < 2t, we have decaying oscillation of order O(t⁻¹/²) as was proved in our previous paper. Near \|n\|=2t, the behavior is decaying oscillation of order O(t⁻¹/³) and the coefficient of the leading term is expressed by the Painlevé II function. In \|n\| > 2t, the solution decays more rapidly than any negative power of n.
format	Article
author	Yamane, H.
spellingShingle	Yamane, H. Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Yamane, H.
author_sort	Yamane, H.
title	Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
title_short	Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
title_full	Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
title_fullStr	Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
title_full_unstemmed	Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II
title_sort	long-time asymptotics for the defocusing integrable discrete nonlinear schrödinger equation ii
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/146996
citation_txt	Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II / H. Yamane // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 6 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT yamaneh longtimeasymptoticsforthedefocusingintegrablediscretenonlinearschrodingerequationii
first_indexed	2025-07-11T01:07:09Z
last_indexed	2025-07-11T01:07:09Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 020, 17 pages Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II Hideshi YAMANE Department of Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japan E-mail: yamane@kwansei.ac.jp URL: http://sci-tech.ksc.kwansei.ac.jp/~yamane/ Received September 06, 2014, in final form March 03, 2015; Published online March 08, 2015 http://dx.doi.org/10.3842/SIGMA.2015.020 Abstract. We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. If \|n\| < 2t, we have decaying oscillation of order O(t−1/2) as was proved in our previous paper. Near \|n\| = 2t, the behavior is decaying oscillation of order O(t−1/3) and the coefficient of the leading term is expressed by the Painlevé II function. In \|n\| > 2t, the solution decays more rapidly than any negative power of n. Key words: discrete nonlinear Schrödinger equation; nonlinear steepest descent; Painlevé equation 2010 Mathematics Subject Classification: 35Q55; 35Q15 1 Introduction In our previous paper [6], we studied the long-time behavior of the defocusing integrable discrete nonlinear Schrödinger equation (IDNLS) i d dt Rn + (Rn+1 − 2Rn +Rn−1)− \|Rn\|2(Rn+1 +Rn−1) = 0 (1) in the region \|n\| ≤ (2− V0)t, 0 < V0 < 2. (In the present paper we refer to it as Region A.) We have proved that there exist Cj = Cj(n/t) ∈ C and pj = pj(n/t), qj = qj(n/t) ∈ R (j = 1, 2) depending only on the ratio n/t such that Rn(t) = 2∑ j=1 Cjt −1/2e−i(pjt+qj log t) +O ( t−1 log t ) as t→∞. The behavior of each term in the sum is decaying oscillation of order t−1/2. Here Cj and qj are defined in terms of the reflection coefficient r = r(z) ([1], [6]) corresponding to the initial potential {Rn(0)}. In the present paper, we study (1) in other regions, namely one including the rays n = ±2t and another with \|n\| > 2t. Painlevé asymptotics has been observed in the cases of the MKdV equation ([2]) and the Toda lattice ([3]). The proofs are based on the nonlinear steepest descent method. Unlike the saddle point case, one has to deal with a phase function of degree 3. Following these results, especially [2], we obtain the long-time asymptotics of (1) in Region B, i.e. near n = ±2t. Roughly speaking, up to a time shift t 7→ t− t0, our result is as follows (Theorem 1). Consider a curve defined by t2/3 2− n/t (6− n/t)1/3 = a real constant. (2) mailto:yamane@kwansei.ac.jp http://sci-tech.ksc.kwansei.ac.jp/~yamane/ http://dx.doi.org/10.3842/SIGMA.2015.020 2 H. Yamane It approaches n/t = 2 with an error of O(t−2/3) as t→∞. The behavior of Rn(t) on it is of the form Rn(t) = const ei(−4t+πn)/2t−1/3 +O ( t−2/3 ) . The constant in the above expression is written in terms of the Painlevé II function with pa- rameters determined by the reflection coefficient corresponding to {Rn(0)}. A similar result was obtained in [5] at least formally. Notice that an analogous phenomenon can be found in a different context [4]. In the result of [2] about the MKdV equation, no oscillatory factor appears together with the Painlevé function. Remark 1. The equation (1) is invariant under the reflection n 7→ −n. In the later sections, we assume n > 0 without loss of generality. In Section 2 we state our main results. Sections 3–6 are devoted to the study of the region 2t −Mt1/3 < n < 2t. In Section 7 we study 2t ≤ n < 2t + M ′t1/3. In Section 8 we investigate n > 2t. 2 Main results Let r(z) be the reflection coefficient determined by the initial potential {Rn(0)}. See [6] for the precise definition. We assume that {Rn(0)} decreases rapidly in the sense that for any s > 0 there exists a constant Cs > 0 such that \|Rn(0)\| ≤ Cs/(1 + \|n\|)s.1 Then r(z) is smooth on C : \|z\| = 1. Let Region B2 be defined by 2t−Mt1/3 < n < 2t+M ′t1/3, (3) where M and M ′ are arbitrary positive constants. The solution to an initial value problem for (1) has the following asymptotic behavior there: Theorem 1. Let t0 be such that π−1(arg r(e−πi/4) − 2t0) − 1/2 is an integer. Set t′ = t − t0, p′ = i(−4t′ + πn)/4, α′ = [12t′/(6t′ − n)]1/3, q′ = −2−4/331/3(6t′ − n)−1/3(2t′ − n). Then we have Rn(t) = e2p′−πi/4α′ (3t′)1/3 u ( 4q′ 31/3 ) +O ( t′−2/3 ) . Here u is a solution of the Painlevé II equation u′′(s) − su(s) − 2u3(s) = 0 and is specified in (20). Let Region C be defined by n > (2 + V0)t, where V0 is an arbitrary positive constant. Theorem 2. Let j be an arbitrary positive integer. Then in Region C, we have Rn(t) = O(n−j). More precisely, there exists a constant C = C(j, V0) > 0 such that \|Rn(t)\| ≤ Cn−j holds. The solution decays exponentially if r(z) is analytic on the circle \|z\| = 1: there exists a positive constant ρ = ρ(V0) with 0 < ρ < 1 and a positive constant C = C(V0) such that \|Rn(t)\| ≤ Cρn holds. Remark 2. A sufficient condition for the analyticity of r(z) is that {Rn(0)} is finitely sup- ported. 1It is equivalent to saying that ∑ s (1 + \|n\|)s\|Rn(0)\| converges for any s, see [6]. 2We only consider the case n > 0, see Remark 1 above. Integrable Discrete Nonlinear Schrödinger Equation 3 Figure 1. Real part of ϕ. 3 Decomposition and reduction First we consider the long-time asymptotics in the ‘left-hand half’ of Region B defined in (3), namely 2t−Mt1/3 < n < 2t. (4) Notice that a curve like (2) is in this kind of region for M suitably chosen. Set ϕ = ϕ(z) = ϕ(z;n, t) = 2−1it ( z − z−1 )2 − n log z, ψ = ψ(z) = ϕ(z)/(it). Choice of the branch of the logarithm is irrelevant because ϕ always appears in the form e±ϕ. We formulate a Riemann–Hilbert problem (RHP): m+(z) = m−(z)v(z) on C : \|z\| = 1, (5) m(z)→ I as z →∞, (6) v(z) = e−ϕ adσ3 [ 1− \|r(z)\|2 −r̄(z) r(z) 1 ] . (7) Here m+ and m− are the boundary values from the outside and inside of C respectively of the unknown matrix-valued analytic function m(z) = m(z;n, t) in \|z\| 6= 1. Namely C is endowed with clockwise orientation (a convention adopted by [1]). We employ the usual notation σ3 = diag(1,−1), aadσ3Q = aσ3Qa−σ3 (a: a scalar, Q: a 2× 2 matrix). In formulating other RHPs in the remaining part of the present article, we will always assume the normalization condition (6), which we often neglect to mention. The sign of the real part of ϕ is as in Fig. 1. The function ϕ(z) has four saddle points. They are S1 = e−πi/4A, S2 = e−πi/4Ā, S3 = −S1, S4 = −S2, where A = 2−1( √ 2 + n/t− i √ 2− n/t). One can reconstruct {Rn(t)} from m(z) by Rn(t) = − lim z→0 1 z m(z)21 = − d dz m(z)21 ∣∣∣∣ z=0 . (8) We will frequently use the factorization v = v(z) = e−ϕ adσ3 {[ 1 −r̄(z) 0 1 ] [ 1 0 r(z) 1 ]} and its outcomes. 4 H. Yamane For ψ = ψ(z) = ϕ(z)/(it), we have ψ′(z) = z − z−3 − n it z−1, ψ′′(z) = 1 + 3z−4 + n it z−2, ψ′′′(z) = −12z−5 − 2n it z−3. Third-order approximation of ψ will be necessary, since we will deal with coalescence of saddle points. We do not need the ‘∆-conjugation’ as in [6, § 4], where a function called ρ was introduced. Here we decompose r̄ and r on arc(S2S3) ∪ arc(S4S1) by using Taylor’s theorem and Fourier analysis3. Set ϑ = θ − π/4, z = eiθ and ϑ0 = π/2 + argA = π/2 − arctan √ (2t− n)/(2t+ n). (The definitions of ϑ and ϑ0 are different from those in [6].) Then arc(S2S3) corresponds to −ϑ0 ≤ ϑ ≤ ϑ0. We regard the function r̄ on arc(S2S3) as a function in ϑ and denote it by r̄(ϑ) by abuse of notation. We have r̄(ϑ) = He ( ϑ2 ) + ϑHo ( ϑ2 ) , −ϑ0 ≤ ϑ ≤ ϑ0, for smooth functions He and Ho. By Taylor’s theorem, they are expressed as follows: He ( ϑ2 ) = µe 0 + · · ·+ µe k ( ϑ2 − ϑ2 0 )k + 1 k! ∫ ϑ2 ϑ2 0 H(k+1) e (γ) ( ϑ2 − γ )k dγ, Ho ( ϑ2 ) = µo 0 + · · ·+ µo k ( ϑ2 − ϑ2 0 )k + 1 k! ∫ ϑ2 ϑ2 0 H(k+1) o (γ) ( ϑ2 − γ )k dγ. Here k = 4q + 1 and q can be any positive integer. We set R(ϑ) = Rk(ϑ) = k∑ i=0 µe i ( ϑ2 − ϑ2 0 )i + ϑ k∑ i=0 µo i ( ϑ2 − ϑ2 0 )i , α(z) = (z − S2)q(z − S3)q, h(ϑ) = r̄(ϑ)−R(ϑ) and, by abuse of notation, α(ϑ) = α ( ei(ϑ+π/4) ) = [ ei(ϑ+π/4) − ei(−ϑ0+π/4) ]q[ ei(ϑ+π/4) − ei(ϑ0+π/4) ]q . Notice that we have R(±ϑ0) = r̄(±ϑ0). The function R extends analytically from arc(S2S3) to a complex neighborhood. By abuse of notation, R(z) denotes the analytic function thus obtained, so that R(ei(ϑ+π/4)) = R(ϑ) and R(Sj) = r̄(Sj). On arc(S2S3) we have dψ/dϑ = −2 cos 2ϑ − n/t. Since [−ϑ0, ϑ0] 3 ϑ 7→ ψ ∈ R is strictly decreasing, we can consider its inverse ϑ = ϑ(ψ), ψ(ϑ0) ≤ ψ ≤ ψ(−ϑ0). We set (h/α)(ψ) = { h(ϑ(ψ))/α(ϑ(ψ)) if ψ(ϑ0) ≤ ψ ≤ ψ(−ϑ0), 0 otherwise. Then (h/α)(ψ) is well-defined for ψ ∈ R. It can be shown that h/α ∈ Hp(−∞ < ψ < ∞), where p can be any positive integer if we choose a sufficiently large value of k. Its norm is uniformly bounded with respect to (n, t). This argument is a ‘curved’ version of [2, equa- tion (1.33)]. Notice that dψ/dϑ = −2 cos 2ϑ− n/t has a zero of order two at ϑ = ±π/2 if n/t = 2. It may worsen the estimate of the Sobolev norm (cf. [2, equation (1.33)]) of h/α as a function of ψ in 3We sometimes denote arc(SjSk) by SjSk. Integrable Discrete Nonlinear Schrödinger Equation 5 Figure 2. Σ(1). Figure 3. Σ(2). contract to ϑ, especially in the case of Section 7, since it involves d/dψ = (dψ/dϑ)−1d/dϑ. This kind of difficulty is overcome by choosing a sufficiently large value of k. Set (ĥ/α)(s) = ∫ ∞ −∞ e−isψ(h/α)(ψ) dψ√ 2π , hI(ϑ) = α(ϑ) ∫ ∞ t eisψ(ϑ)(ĥ/α)(s) ds√ 2π , hII(ϑ) = α(ϑ) ∫ t −∞ eisψ(ϑ)(ĥ/α)(s) ds√ 2π , then we have h(ϑ) = hI(ϑ) +hII(ϑ) and r̄(ϑ) = R(ϑ) +hI(ϑ) +hII(ϑ) on \|ϑ\| ≤ ϑ0. We can apply the same process to r. We have r̄(z) = r̄ = hI + hII +R, r(z) = r = h̄I + h̄II + R̄, r̄(Sj) = R(Sj), r(Sj) = R̄(Sj). (9) The decomposition on arc(S4S1) immediately follows by symmetry. Notice that hII, R, h̄II and R̄ can be analytically continued to certain open sets. We still employ the same notation for the extended functions. For example, h̄II = h̄II(z) is analytic, although the bar may seem a little strange. We introduce a new contour Σ(1) as in Fig. 2. It is a variation of Σ in [6, Fig. 2]. The part L(1) is bent so that it stays away from the circle as n → 2t (except near the saddle points.) Some open sets, not necessarily connected, are defined in Fig. 2. Notice that Σ(1) remains finite even as n→ 2t. We introduce a new unknown matrix m(1) by setting m(1) =  m in Ω (1) 1 ∪ Ω (1) 4 , me−ϕ adσ3 [ 1 0 −h̄II 1 ] in Ω (1) 2 , me−ϕ adσ3 [ 1 −hII 0 1 ] in Ω (1) 3 . 6 H. Yamane We define a new jump matrix v(1) by v(1) = v =  e−ϕ adσ3 {[ 1 −r̄ 0 1 ][ 1 0 r 1 ]} on S1S2 ∪ S3S4, e−ϕ adσ3 [ 1 −hII 0 1 ] on L(1), e−ϕ adσ3 {[ 1 −hI −R 0 1 ][ 1 0 h̄I + R̄ 1 ]} on S2S3 ∪ S4S1, e−ϕ adσ3 [ 1 0 h̄II 1 ] on L′(1). Then we have m (1) + = m (1) − v(1) on Σ(1), m(1) → I as z →∞. By (8), we have Rn(t) = − d dz m(1)(z)21 ∣∣∣∣ z=0 . (10) We need some estimates. First, in the same way as [2, equation (1.36)] and [6, equation (43)], \|e−2ϕhI\| ≤ C/t(3q+1)/2, \|e2ϕh̄I\| ≤ C/t(3q+1)/2 holds for some C > 0 on S2S3 ∪ S4S1. The estimates of \|e−2ϕhII\| on L(1) and of \|e2ϕh̄II\| on L′(1) must be handled with greater care. We have [6, § 4] ψ′′(Sj) = (−1)j2S−2 j (2 + n/t)1/2(2− n/t)1/2. It can be infinitely small and does not lead to a reasonably good estimate. We would rather rely on ψ′′′(Sj). The following lemma replaces [6, equation (44)] in our context. Lemma 1. Let L(1)(Sj) (resp. L′(1)(Sj)) be the segment ⊂ L(1) (resp. ⊂ L′(1)) emanating from Sj. Let d be the distance from Sj to z ∈ L(1)(Sj) (resp. to z ∈ L′(1)(Sj)). Then there exists a positive constant C ′ such that Re iψ(z) ≥ C ′d3, z ∈ L(1)(Sj), Re iψ(z) ≤ −C ′d3, z ∈ L′(1)(Sj). (11) Proof. First we assume j = 2. In view of Fig. 1, L(1) is in the region Re(iψ) = Re(t−1ϕ) > 0. Since ψ′(S2) = 0, we have iψ(z) = iψ(S2) + iψ′′(S2) 2 (z − S2)2 + iψ′′′(S2) 6 (z − S2)3 + higher order terms (12) and iψ(S2) is purely imaginary. It holds that ψ′′(S2) = 2S−2 2 (2 + n/t)1/2(2− n/t)1/2, ψ′′′(S2) = −12S−5 2 − 2n it S−3 2 . The segment L(1)(S2) is tangent to the steepest ascent path of ϕ = itψ, hence also of iψ. Assume z ∈ L(1)(S2). If n/t ≈ 2, then S2 is close to T1 = e−πi/4 and z − S2 is close to id. We have iψ′′(S2)(z − S2)2 ≈ 2(2 + n/t)1/2(2− n/t)1/2d2, (13) Integrable Discrete Nonlinear Schrödinger Equation 7 iψ′′′(S2)(z − S2)3 ≈ ( 12eπi/4 + 2n t e−3πi/4 ) d3. (14) The right-hand side of (13) is positive. In estimating Re iψ from below, we can neglect the term of degree 2 in (12). On the other hand, the quantity in the parentheses on the right-hand side of (14) has a positive real part (close to 4 √ 2). Hence we get the first inequality of (11). By symmetry, (11) also holds on L(1)(S4). In a similar way, we can show that the second inequality of (11) holds on L′(1)(S2). Notice that z − S2 ≈ d on it. The case j = 2 is now finished. By symmetry, we get (11) for j = 4. Since S1 ≈ S2, the estimates on L(1)(S1) and L′(1)(S1) are similar to those on L′(1)(S2) and L(1)(S2) respectively. Notice that L and L′ are exchanged. We get (11) for j = 1. The case j = 3 follows by symmetry. � Assume z ∈ L(1)(Sj). We have \|α(z)\| ≤ const dq. By modifying the argument of [6, equa- tion (45)], we obtain \|e−2ϕhII\| ≤ const dqe−2C′td3 ≤ const t−q/3 sup τ>0 τ q/3e−2C′τ ≤ const t−q/3. (15) This kind of estimate obviously holds on any compact subset of {Reϕ > 0}. Hence we get the following lemma. Lemma 2. \|e−2ϕhII\| ≤ const t−q/3 on L(1), \|e2ϕh̄II\| ≤ const t−q/3 on L′(1). The contribution to Rn(t) by hII and h̄II on L(1)∪L′(1), as well as by hI and h̄I on S3S2∪S1S4, are of order t−l as t→∞, where l > 0 is arbitrarily large. It is justified by choosing sufficiently large q. We are left with an RHP over Σ(2) = C, the union of four arcs oriented clockwise. See Fig. 3. We follow [2, equations (5.9) and (5.10)]. The new jump matrix v(2) = v(2)(z) is given by v(2) =  e−ϕ adσ3 {[ 1 −r̄ 0 1 ][ 1 0 r 1 ]} on S1S2 ∪ S3S4, e−ϕ adσ3 {[ 1 −R 0 1 ][ 1 0 R̄ 1 ]} on S2S3 ∪ S4S1. Here SjSk denotes the minor arc joining Sj and Sk. Let m(2) be the solution to the RHP corresponding to v(2). Then by (10), for any l > 0, Rn(t) = − d dz m(2)(z)21 ∣∣∣∣ z=0 +O ( t−l ) . See Section 4 for a more precise (routine) argument based on the Beals–Coifman formula. Let Σ(3) be the contour in Fig. 4. The parts inside and outside the circle are denoted by L(3) and L′(3) respectively. The latter consists of four half-lines. Following [2, equations (5.13)– (5.15)], we set v(3) = v(2) =  e−ϕ adσ3 {[ 1 −r̄ 0 1 ][ 1 0 r 1 ]} on S1S2 ∪ S4S3, e−ϕ adσ3 [ 1 −R 0 1 ] on L(3), e−ϕ adσ3 [ 1 0 R̄ 1 ] on L′(3). 8 H. Yamane Figure 4. Σ(3). Figure 5. Σ(4). Figure 6. Σ(5). Notice that v(3) = v(3)(z)→ I as L′(3) 3 z →∞. The new unknown function m(3)(z) is defined in the usual way: m(3)(z)→ I. It implies that m(3)(z) = m(2)(z) near z = 0. By using the method of [6, § 7.2, § 9] (originally of [2, § 2, § 3]), we can replace Σ(3) by the bounded contour Σ(4) in Fig. 5 up to an error of order O(t−1), hence without changing the leading part in the asymptotics. We can assume that the lengths of the ‘branches’ emanating from the saddle points are independent of n and t. The new jump matrix v(4) equals v(3) on Σ(4) and is the identity matrix elsewhere. Owing to the technique of [2, Proposition 3.66] and [6, Proposition 9.2], the contribution from the two connected components of Σ(4) can be separated out, with an error of O(t−1). Notice that the two terms arising from the two components are actually the same because r(−z) = −r(z) for z ∈ C (cf. [6, Proposition 12.4]). It is enough to investigate the lower part (containing T1 = e−πi/4), which is referred to as Σ (4) lower. 4 Scaling and rotation We have ϕ(T1) = i 4 (−4t+ πn), ϕ′(T1) = (2t− n)eπi/4, ϕ′′(T1) = i(−2t+ n), ϕ′′′(T1) = (−12t+ 2n)e−πi/4, ϕ(z) = 3∑ k=0 ϕ(k)(T1) k! (z − T1)k + ϕ4(z), ϕ4(z) = O ( (z − T1)4 ) near z = T1. Let ε > 0 be such that Σ (4) lower is within the circle \|z − T1\| = ε/2. Integrable Discrete Nonlinear Schrödinger Equation 9 Now we define an operator sc by z 7→ sc(z) = t−1/3e−3πi/4z + T1, T1 = T2 = e−πi/4. Set σj = sc−1(Sj) = t1/3e3πi/4(Sj−T1). We have Sj−T1 = O (√ 2− n/t ) = O(t−1/3), the latter equality being a consequence of (4). It follows that σj is bounded in spite of the magnifying factor t1/3. There is a constant M̃ > 0 such that \|Reσj \| < M̃ . Let µj be such that Reµj = (−1)j−1M̃ and that \| sc(µj)\| = 1, = sc(µj) < 0. We modify sc−1 ( Σ (4) lower ) without moving the endpoints to get the contour Σ(5) in Fig. 6. The arc µ1µ2 is a part of a circle of radius t1/3 and looks like a segment of length 2M̃ if t is large. The lengths of L(5) and L′(5) are of order t1/3 and their directions approach ±π/4 or ±3π/4 as t→∞. We choose Σ(5) so that sc ( Σ(5) ) is within the circle \|z − T1\| = ε. We want to approximate ϕ(sc(z)) by a cubic polynomial which is related to the Painlevé II function (up to a constant term). We introduce φ = φ(z) = i 4 (−4t+ πn) + i(−2t+ n)t−1/3z + i(6t− n)t−1 3 z3. Then we have ϕ(sc(z)) = φ(z) + 2−1(2t−n)t−2/3z2 +ϕ4(sc(z)). The following proposition is an analogue of [6, Proposition 10.1]. Proposition 1. Fix a constant γ with 0 < γ < 1 < (6t− n)t−1/3. Then on L(5), we have∣∣e−2ϕ(sc(z))R(sc(z))− e−2φ(z)r̄(T1) ∣∣ ≤ Ct−1/3 ∣∣e−iγz3∣∣,∣∣ sc(z)−2e−2ϕ(sc(z))R(sc(z))− T−2 1 e−2φ(z)r̄(T1) ∣∣ ≤ Ct−1/3 ∣∣e−iγz3∣∣ for some constant C > 0. Proof. We show only the latter inequality; the former is easier. We have eiγz 3[ sc(z)−2e−2ϕ(sc(z))R(sc(z))− T−2 1 e−2φ(z)r̄(T1) ] = e−iγz 3[ sc(z)−2ER(sc(z))− T−2 1 e−2φ(z)+2iγz3 r̄(T1) ] , where E = exp(−2ϕ(sc(z)) + 2iγz3). Each factor is uniformly bounded. Notice that sc(z) remains in the ε-neighborhood of T1. Set f(w) = w−2. For any fixed z, sc(z)−2 = f(sc(z)) and R(sc(z)) tend to T−2 1 and r̄(T1) respectively as t→∞. This convergence is uniform on L(5) in the following sense:∣∣e−iγz3[ sc(z)−2 − T−2 1 ]∣∣ ≤ ∣∣e−iγz3∣∣∣∣t−1/3e−3πi/4z ∣∣ sup \|w−T1\|≤ε \|f ′(w)\| ≤ const t−1/3,∣∣e−iγz3[ R(sc(z))− r̄(T1) ]∣∣ ≤ ∣∣e−iγz3∣∣∣∣t−1/3e−3πi/4z ∣∣ sup \|w−T1\|≤ε \|R′(w)\| ≤ const t−1/3. We have used the fact that e−iγz 3 z is bounded on either branch of L(5). Since e−iγz 3 zj (j = 2, 4) is bounded and 2t− n = O(t1/3), we have∣∣e−iγz3( E − e−2φ(z)+2iγz3)∣∣ ≤ ∣∣e−iγz3∣∣ sup 0≤s≤1 ∣∣∣∣ dds exp ( −2φ(z) + 2iγz3 + s [ (2t− n)t−2/3z2 − 2ϕ4(sc(z)) ])∣∣∣∣ ≤ C ∣∣e−iγz3∣∣[(2t− n)t−2/3\|z\|2 + 2\|t−1/3z\|4 ] ≤ Ct−1/3. Combining the three estimates above, we can derive the desired inequality. � 10 H. Yamane Figure 7. Σ(6). The factorization problem on Σ (4) lower is equivalent, up to the change of variables z 7→ sc(z), to one on Σ(5), where the jump matrix v(5) = v(5)(z) is v(5)(z) =  e−ϕ(sc(z)) adσ3 {[ 1 −r̄(sc(z)) 0 1 ][ 1 0 r(sc(z)) 1 ]} on σ2σ1, e−ϕ(sc(z)) adσ3 {[ 1 −R(sc(z)) 0 1 ][ 1 0 R̄(sc(z)) 1 ]} on σ1µ1 ∪ µ2σ2, e−ϕ(sc(z)) adσ3 [ 1 −R(sc(z)) 0 1 ] on L(5), e−ϕ(sc(z)) adσ3 [ 1 0 R̄(sc(z)) 1 ] on L′(5). Notice that v(5) is smooth across σ1 and σ2 to any desired order (choose k sufficiently large). Let Σ(6) be the contour in Fig. 7 obtained by extending L(5) and L′(5) infinitely. Then we can regard v(5)(z) as a jump matrix on Σ(6): set v(5) = I on Σ(6) \ Σ(5). Because of Proposition 1 (and its variants about L′(5) and µ1µ2), the jump matrix v(5)(z) is approximated by v(6)(z) up to an error of order O(t−1/3), where v(6)(z) =  e−φ(z) adσ3 {[ 1 −r̄(T1) 0 1 ][ 1 0 r(T1) 1 ]} on µ1µ2, e−φ(z) adσ3 [ 1 −r(T1) 0 1 ] on L(6), e−φ(z) adσ3 [ 1 0 r(T1) 1 ] on L′(6). We rescale by the factor α = [12t/(6t − n)]1/3 > 0, which satisfies α3t−1(6t − n)/3 = 4 and tends to 31/3 as t→∞. We have φ(αz) = i 4 (−4t+ πn) + 4i { z3 + αt−1/3 4 (−2t+ n)z } . Set p = i(−4t+ πn)/4, q = αt−1/3(−2t+ n)/4 = 2−4/331/3(6t− n)−1/3(−2t+ n), then we have φ(αz) = p+ 4i(z3 + qz), p ∈ iR. We have normalized the coefficient of z3. The term 4i(z3 + qz) will play an important role in Section 6. Integrable Discrete Nonlinear Schrödinger Equation 11 Figure 8. Σ(7) = Σ(8). The jump matrix v(7)(z) = v(6)(αz) on Σ(7) = α−1Σ(6) is given by v(7)(z) =  e−[p+4i(z3+qz)] adσ3 {[ 1 −r̄(T1) 0 1 ][ 1 0 r(T1) 1 ]} on ( α−1µ1 )( α−1µ2 ) , e−[p+4i(z3+qz)] adσ3 [ 1 −r̄(T1) 0 1 ] on L(7), e−[p+4i(z3+qz)] adσ3 [ 1 0 r(T1) 1 ] on L′(7), where (α−1µ1)(α−1µ2) is the arc in Σ(7). We have an RHP m (7) + (z) = m (7) − (z)v(7)(z) on Σ(7). We want to remove p in v(7)(z). (Notice that p contributes to the oscillatory factor in Theorem 1.) Set m(8)(z) = ep adσ3m(7)(z). Then m(8)(z) is the solution to m (8) + (z) = m (8) − (z)v(8)(z) on Σ(8) = Σ(7), m(8)(z)→ I as z →∞, where v(8)(z) = ep adσ3v(7)(z). We have v(8)(z) =  e−[4i(z3+qz)] adσ3 {[ 1 −r̄(T1) 0 1 ][ 1 0 r(T1) 1 ]} on ( α−1µ1 )( α−1µ2 ) , e−[4i(z3+qz)] adσ3 [ 1 −r̄(T1) 0 1 ] on L(8) = L(7), e−[4i(z3+qz)] adσ3 [ 1 0 r(T1) 1 ] on L′(8) = L′(7). We have explained steps of reduction in terms of contours and jump matrices. It should be supplemented with reconstruction formulas (up to some errors) involving integrals. Recall that each v(j) has a factorization of the form v(j) = (I + w (j) − )(I + w (j) + ), where the diagonal components of w (j) ± is zero. We have (I + w (j) − )−1 = I − w(j) − . Let( C (j) ± f ) (z) = ∫ Σ(j) f(ζ) ζ − z± dζ 2πi = lim y→z y∈{±-side ofΣ(j)} ∫ Σ(j) f(ζ) ζ − y dζ 2πi , z ∈ Σ(j), be the Cauchy operators on Σ(j). Define Cw(j) : L2(Σ(j))→ L2(Σ(j)) by Cw(j)f = C (j) + ( fw (j) − ) + C (j) − ( fw (j) + ) 12 H. Yamane for a 2×2 matrix-valued function f (cf. [2, § 2], [6, § 7]). The Cauchy differential form is invariant under an affine change of variables: z = az′+b and ζ = aζ ′+b imply (ζ−z)−1dζ = (ζ ′−z′)−1dζ ′. The operator Cw(j) commutes with an affine change of variables in the sense that (Cw(j)(z)f(•))(az′ + b) = (Cw(j)(az′+b)f(a •+b))(z′). We have m(j)(z) = I + ∫ Σ(j) ( (1− Cw(j))−1I ) (ζ)w(j)(ζ) ζ − z dζ 2πi , where w(j)(ζ) = w (j) + (ζ) + w (j) − (ζ). By (10), we obtain Rn(t) = − ∫ Σ(1) z−2 [( (1− Cw(1))−1I ) w(1) ] 21 (z) dz 2πi . Repeated replacement of contours and integrands leads to (cf. [6]) Rn(t) = − ∫ Σ(4) z−2 [( (1− Cw(4))−1I ) w(4) ] 21 (z) dz 2πi +O ( t−1 ) = −2 ∫ Σ (4) lower z−2 [( (1− Cw(4))−1I ) w(4) ] 21 (z) dz 2πi +O ( t−1 ) . By repeated affine changes of variables and Proposition 1, we get Rn(t) = −2e−3πi/4 t1/3 ∫ Σ(5) sc(z′)−2 [( (1− Cw(5))−1I ) w(5) ] 21 (z′) dz′ 2πi +O ( t−1 ) = −2e−3πi/4 t1/3 ∫ Σ(6) T−2 1 [( (1− Cw(6))−1I ) w(6) ] 21 (z′) dz′ 2πi +O ( t−2/3 ) = −2e−πi/4α t1/3 ∫ Σ(7) [( (1− Cw(7))−1I ) w(7) ] 21 (z) dz 2πi +O ( t−2/3 ) , (16) where α = (12t)1/3(6t− n)−1/3 > 0. We have used the fact that sc(z′)− T1 = O(t−1/3) and the second resolvent identity. See [6, Remark 7.4]. Let us calculate the integral in (16). As z →∞, z [ σ3,m (j)(z) ] 21 → 2 [∫ Σ(j) ( (1− Cw(j))−1I ) w(j) ] 21 (ζ) dζ 2πi . (17) On the other hand, we have [σ3,m (8)(z)] = ep adσ3 [σ3,m (7)(z)]. These two formulas imply[∫ Σ(8) ( (1− Cw(8))−1I ) w(8) ] 21 (ζ) dζ 2πi = [ ep adσ3 ∫ Σ(7) ( (1− Cw(7))−1I ) w(7) ] 21 (ζ) dζ 2πi = e−2p [∫ Σ(7) ( (1− Cw(7))−1I ) w(7) ] 21 (ζ) dζ 2πi . (18) By using (16), (17) and (18), we obtain Rn(t) = −2αe2p−πi/4 t1/3 ∫ Σ(8) [( (1− Cw(8))−1I ) w(8) ] 21 (z) dz 2πi +O ( t−2/3 ) = αe2p−πi/4 t1/3 lim z→∞ { −z [ σ3,m (8)(z) ] 21 } +O ( t−2/3 ) . (19) Integrable Discrete Nonlinear Schrödinger Equation 13 Figure 9. Σ(9). Figure 10. Σ(10). 5 Time shift If r(Tj) is purely imaginary, it is easy to apply the argument of [2, p. 359] to our case. Otherwise, we perform the following reduction. As is proved in [1], the time evolution of the reflection coefficient is given by r(T1, t) = r(T1) exp ( it(T1 − T̄1)2 ) = r(T1) exp(−2it), r(T1) = r(T1, 0). Therefore r(T1, t0) is purely imaginary for some t0. The condition to be satisfied is arg r(T1)− 2t0 − π/2 ∈ πZ. Notice that (3) is preserved if t is replaced by t− t0. 6 Painlevé function We assume that r(T1) is purely imaginary. See the previous section for justification. Augment Σ(8) → Σ(9) (cf. [2, Fig. 5.5]) as in Fig. 9. The contour Σ(9) contains four pairs of parallel half-lines. Define the new unknown function m(9)(z) by m(9)(z) =  m(8)(z), z ∈ Ω (9) 1 ∪ Ω (9) 3 ∪ Ω (9) 5 ∪ Ω (9) 7 , m(8)(z)e−{4i(z 3+qz)} adσ3 [ 1 0 −r(T1) 1 ] , z ∈ Ω (9) 2 ∪ Ω (9) 4 , m(8)(z)e−{4i(z 3+qz)} adσ3 [ 1 −r̄(T1) 0 1 ] , z ∈ Ω (9) 6 ∪ Ω (9) 8 . Direct computation shows that m(9)(z) has no jump across Σ (9) j , j = 1, 4, 5, 8, 9, 10. Its jump is given by J23 across Σ (9) 2 ∪ Σ (9) 3 and by J67 across Σ (9) 6 ∪ Σ (9) 7 , where J23 = e−{4i(z 3+qz)} adσ3 [ 1 0 r(T1) 1 ] , J67 = e−{4i(z 3+qz)} adσ3 [ 1 −r̄(T1) 0 1 ] . Thus the RHP is reduced to one along Σ (9) 2 ∪ Σ (9) 3 ∪ Σ (9) 6 ∪ Σ (9) 7 . It is not exactly a cross, but a simple deformation enables us to replace it by Σ(10) as in Fig. 10, the counterpart of the contour in [2, Fig. 5.6]. In the upper and lower halves, the jump matrix coincides with J23 and J67 respectively. We apply the argument in [2, pp. 357–360]. In particular, we employ the 14 H. Yamane parameters p, q, r (roman font) in it. See Appendix for explanation. We use \|r(Tj)\| < 1 and r(Tj) + r̄(Tj) = 0, the latter being true if the time variable t is replaced by t − t0 for some t0 (see the previous section). We employ the notation explained in Appendix. We set p = r(T1), q = −r(T1) = r(T1), r = (p + q)/(1 − pq) = 0 and consider the solution u(s; r(T1),−r(T1), 0) to the Painlevé II equation u′′ − su− 2u3 = 0. Since 4i ( z3 + qz ) = 4i 3 ( 31/3z )3 + i 4q 31/3 ( 31/3z ) , we have lim z→∞ ( −z [ σ3,m (8)(z) ] 21 ) = 1 31/3 u ( 4q 31/3 ; r(T1),−r(T1), 0 ) . (20) We combine (20) with (19). The result is Rn(t) = e2p−πi/4α (3t)1/3 u ( 4q 31/3 ; r(T1),−r(T1), 0 ) +O ( t−2/3 ) . Theorem 1 holds at least in the region (4). 7 Asymptotics in the remaining part of Region B We consider the long-time asymptotics in the region 2t ≤ n < 2t+Mt1/3, (21) where M ′ is an arbitrary positive constant. It is the ‘right-hand half’ of the Region B defined by (3). If 2t = n, then the function ϕ(z) has no saddle points. Indeed, Sj and Sj+1 (j = 1, 3) coalesce. If 2t < n, then ϕ(z) has four saddle points on the line Re z + Imz = 0. Set A = 2−1( √ 2 + n/t + √ −2 + n/t), A′ = 2−1( √ 2 + n/t − √ −2 + n/t), then the four saddle points are ±e−πi/4A and ±e−πi/4A′. Notice that A > 1, AA′ = 1, 0 < A′ < 1. For z = reiθ (here r is not the reflection coefficient), we have Reϕ = −1 2 t(r 2 − r−2) sin 2θ − n log r. It vanishes for any θ if r = 1. If r 6= 1, the equation Reϕ = 0 is equivalent to saying that sin 2θ = −2n t log r r2 − r−2 . The function log r/(r2−r−2) can be continuously extended to 0 < r <∞. It is strictly increasing in 0 < r < 1 and is strictly decreasing in r > 1. It attains its maximum 1/4 at r = 1. We can calculate the number of solutions θ (modulo 2π) for each fixed value of r. Figs. 1, 11 and 12 show the curve Reϕ = 0 in the cases n < 2t, n = 2t and 2t < n respectively. Set ψ0 = ψ0(z) = 2−1(z−z−1)2+2i log z. It is nothing but what ψ is if n = 2t. We employ it as the Fourier variable in the region (21), not only on the ray n = 2t. Then we get a decomposition like (9) on arc(T2T3) and on arc(T4T1), where T1 = T2 = e−πi/4 and T3 = T4 = e3πi/4. In the formulas below, hI, hII etc. denote the terms obtained by this decomposition. Set ϕ0 = itψ0. We have Reϕ > Reϕ0 in \|z\| < 1, and Reϕ < Reϕ0 in \|z\| > 1. We introduce a new contour Σ(11) as in Fig. 13. Notice that L(11) and L′(11) are in {Reϕ0 > 0, \|z\| < 1} and {Reϕ0 < 0, \|z\| > 1} respectively. In the same way as (15), we can derive estimates of \|e−2ϕ0hII\| on L(11). It is good enough even in the case n > 2t, because we have \|e−2ϕhII\| ≤ \|e−2ϕ0hII\| on L(11). By this observation, we can perform a simplified version of the argument in the preceding section. We conclude that Theorem 1 holds in the whole region (3). Integrable Discrete Nonlinear Schrödinger Equation 15 Figure 11. n = 2t. Figure 12. n > 2t. Figure 13. Σ(11). Figure 14. Σ(12). 8 Region C We consider the case 2t < n→∞. The four saddle points of ϕ are not on the circle C : \|z\| = 1. Two of them are inside and the other two are outside. For z = reiθ, we have Re [ 2ϕ n ] = − t n ( r2 − r−2 ) sin 2θ − 2 log r. Set f(r) = n−1t(r2 − r−2) − 2 log r. If r > 1, then Re[2ϕ/n] ≤ f(r) and if r < 1, then Re[−2ϕ/n] ≤ −f(r). Notice that f(1) = 0, f ′(1) = 2(2t−n)/n < 0. If r > 1 is sufficiently close to 1, then we have Re[2ϕ/n] < 0. On the other hand, if r < 1 is sufficiently close to 1, then we have Re[−2ϕ/n] < 0. We introduce a contour as in Fig. 14 consisting of three concentric circles L(12), L′(12) and C : \|z\| = 1. Their radii are sufficiently close. There exists a positive number p = p(V0) < 1 such that \|e−2ϕ\| ≤ pn on L(12) and \|e2ϕ\| ≤ pn on L′(12). Since r(z) is smooth on \|z\| = 1, its complex conjugate can be written in terms of a Fourier series: r̄(z) = ∞∑ k=−∞ ake ikθ = ∞∑ k=−∞ akz k. 16 H. Yamane For any α ∈ N, there exists a constant Aα > 0 such that \|ak\| ≤ Aα/\|k\|α+1 holds for any k ∈ Z. If r(z) is analytic, then a contour deformation leads to ak = (2πi)−1 ∫ \|z\|=1±ε z −k−1r̄(z)dz. So ak is exponentially decreasing: \|ak\| ≤ const(1± ε)k. Set hI(z) = ∑ k<−n akz k, hII(z) = ∑ k≥−n akz k, h̄I(z) = ∑ k<−n ākz −k and h̄II(z) = ∑ k≥−n ākz −k. We have r(z) = h̄I(z) + h̄II(z). We employ z−k rather than z̄k with analytic continuation in mind. Indeed, hII and h̄II can be analytically continued up to L(12) and L′(12) respectively. It is easy to see that hI and h̄I decay faster than any negative power as n → ∞ on the circle, since they would have fewer terms. On the other hand, we can show that e−2ϕhII and e2ϕh̄II decay exponentially on L(12) and L′(12) respectively. We define a new jump matrix v(12) by v(12) =  e−ϕ adσ3 [ 1 −hII 0 1 ] on L(12), e−ϕ adσ3 {[ 1 −hI 0 1 ][ 1 0 h̄I 1 ]} on \|z\| = 1, e−ϕ adσ3 [ 1 0 h̄II 1 ] on L′(12). The factorization problem (5)–(7) is equivalent to the one involving v(12). We can show that v(12) tends to the identity matrix as n → ∞. The error is smaller than any negative power of n. Indeed, we have exponential decay on L(12) and L′(12) due to ϕ. The decay on the circle \|z\| = 1 is not so good in general. If r(z) is analytic, however, hI and h̄I decay exponentially as n→∞. This completes the proof of Theorem 2. A Parametrization of the Painlevé functions For readers’ convenience, we collect some useful facts employed in [2]. Let p, q and r be constants satisfying the constraint r = p + q + pqr. We define six matri- ces Si by S1 = [ 1 0 p 1 ] , S2 = [ 1 r 0 1 ] , S3 = [ 1 0 q 1 ] , S4 = [ 1 −p 0 1 ] , S5 = [ 1 0 −r 1 ] , S6 = [ 1 −q 0 1 ] . We introduce the contour Σ(13) (the intersection is the origin and all the rays are oriented outward) and the regions Ω (13) i in Fig. 15. Then we consider the Riemann–Hilbert problem Ψi+1(s, z) = Ψi(s, z)Sj on Σ (13) i (1 ≤ i ≤ 6), where Ψi (Ψ7 = Ψ1) is holomorphic in Ω (13) i . It has a unique solution with the asymptotics Ψ(s, z) = ( I + (Ŷi)1 z + (Ŷi)2 z2 + · · · ) e−([4i/3]z3+isz)σ3 as z →∞ in Ω (13) i . The function u defined by u = u(s; p, q, r) = − lim z→∞ z [ σ3, Ŷi(z) ] 21 , where the limit is taken with respect to z ∈ Ω (13) i for any i ∈ {1, . . . , 6}, satisfies the Painlevé II equation u′′(s)− su(s)− 2u3(s) = 0. Integrable Discrete Nonlinear Schrödinger Equation 17 Figure 15. Σ(13). Acknowledgments This work was partially supported by JSPS KAKENHI Grant Number 26400127. Parts of this work were done during the author’s stay at Wuhan University. He wishes to thank Xiaofang Zhou for helpful comments and hospitality. References [1] Ablowitz M.J., Prinari B., Trubatch A.D., Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Note Series, Vol. 302, Cambridge University Press, Cambridge, 2004. [2] Deift P., Zhou X., A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295–368, math.AP/9201261. [3] Kamvissis S., On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity, Comm. Math. Phys. 153 (1993), 479–519. [4] Kitaev A.V., Caustics in 1 + 1 integrable systems, J. Math. Phys. 35 (1994), 2934–2954. [5] Novokshenov V.Yu., Asymptotic behavior as t → ∞ of the solution of the Cauchy problem for a nonlinear differential-difference Schrödinger equation, Differ. Equ. 21 (1985), 1288–1298. [6] Yamane H., Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, J. Math. Soc. Japan 66 (2014), 765–803, arXiv:1112.0919. http://dx.doi.org/10.2307/2946540 http://arxiv.org/abs/math.AP/9201261 http://dx.doi.org/10.1007/BF02096951 http://dx.doi.org/10.1063/1.530495 http://dx.doi.org/10.2969/jmsj/06630765 http://arxiv.org/abs/1112.0919 1 Introduction 2 Main results 3 Decomposition and reduction 4 Scaling and rotation 5 Time shift 6 Painlevé function 7 Asymptotics in the remaining part of Region B 8 Region C A Parametrization of the Painlevé functions References

Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II

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