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arxiv: 2604.19506 · v1 · submitted 2026-04-21 · 🧮 math-ph · math.MP

Painlev\'e Asymptotics of the Focusing Nonlinear Schr\"odinger Equation with a Finite-Genus Algebro-Geometric Background

Ruihong ma , Engui Fan This is my paper

Pith reviewed 2026-05-10 01:38 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords focusing NLSfinite-genus solutionslong-time asymptoticsPainlevé transcendentparabolic cylinder functionsRiemann-Hilbert problemDeift-Zhou methodalgebro-geometric background
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The pith

The focusing NLS equation with finite-genus algebro-geometric initial data has long-time asymptotics given by the second Painlevé transcendent for odd genus and parabolic cylinder functions for even genus, with uniform error bounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives the leading-order long-time asymptotics for solutions of the focusing nonlinear Schrödinger equation when the initial data approach finite-genus algebro-geometric quasi-periodic solutions at both spatial infinities. It applies the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method to obtain explicit expressions and error bounds that hold uniformly for all real positions x as time t tends to positive infinity. The analysis separates into cases of odd and even genus for the underlying hyperelliptic Riemann surface, leading to different special functions in the asymptotics. A sympathetic reader would care because this provides a precise description of how solutions evolve from quasi-periodic backgrounds into specific transcendental behaviors at large times, extending classical results for simpler backgrounds.

Core claim

For the Cauchy problem for the focusing NLS equation with initial data satisfying q(x,0) ~ q^alg(x,0) as x → ±∞ where q^alg is a finite-genus algebro-geometric solution, the solution q(x,t) as t → +∞ has leading-order asymptotics expressed in terms of the second Painlevé transcendent when the genus is odd and in terms of parabolic cylinder functions when the genus is even, together with explicit error bounds that are uniform in x ∈ ℝ.

What carries the argument

Riemann-Hilbert problem associated with the finite-genus background, analyzed via the Deift-Zhou nonlinear steepest descent method to extract asymptotics in different ξ = x/t regions.

Load-bearing premise

The initial data must asymptotically match the finite-genus algebro-geometric solution at both positive and negative spatial infinity, allowing the Riemann-Hilbert problem to be set up and analyzed with the nonlinear steepest descent technique.

What would settle it

Numerical computation of the solution q(x,t) for a concrete finite-genus initial condition at large t, say t=100, in a region where the Painlevé or parabolic cylinder expression is predicted, failing to agree within the error bound would falsify the result.

Figures

Figures reproduced from arXiv: 2604.19506 by Engui Fan, Ruihong ma.

Figure 1
Figure 1. Figure 1: Initial configurations of stationary phase points illustrated for the cases of genus one [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In the transition region |ξ − ξ1|t 2/3 < C, two stationary phase points coalesce when C † > 3B 2 0 . the long-time asymptotics. Without loss of generality, we consider the case where κ R 3 lies to the left of Γ1 under the choice of branch configuration {Bk , Ak } n k=0 as shown in the first figure in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The branch cut coalesces at infinity precisely when complex critical points remain [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The branch cut coalesces with real stationary phase point on the real axis, yielding [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The coalescence of two stationary phase points [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The jump contour is given by Lj = S4 k=1 L k j . vϖϖ = ϖv +2v 2u. (3.43) Thus, asymptotics in this region are expressible via Painlevé II transcendents in ϖ. In view of the structural properties exhibited by the preceding expansion, we define appropriate scaled spectral variables. These are designed to ensure correspondence between the coefficients of exponential terms and those appearing in the Painlevé I… view at source ↗
Figure 7
Figure 7. Figure 7: The infinite cut coalesces with the real stationary phase point as [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The infinite cut coalesces with the real stationary phase point as [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Complex-complex coalescence precedes infinity branch coalescence for the finite [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The infinity branch coalescence occurs prior to the complex-complex coalescence [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Distribution scenarios of genus 6 and stationary phase points [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The standard (left) and appropriate (right) choice of canonical homology base [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The distribution of stationary phase points for [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The contour Y j κ C 1 in the Dϵ(E00) of κ C j and the region {S 1 κs0−1 } for j = 1,2,3,4 where θE00 (z) = θ(z) − θ(E00) = R z E00 p s −E00 f (s)d s, and both f (s) and R z E00 f (s)d s are analytic and nonzero at s = E00. In order to relate N E00 (z) to the Airy solution N Ai(z) of Appendix C, we make a local change of variables for z → E00 and introduce the new variable ζ = ζ(z) = µ 3i t 2 (θ(z)−θ(E00))… view at source ↗
Figure 15
Figure 15. Figure 15: The contour in the ϵ neighborhood Dϵ(E00) of E00 and the sets {S 1 κ2 } phase function and the coordinate transformation.Define the local coordinate ζ(z) which maps the neighborhood of z = E01 to ζ = 0 by ζ(z) = µ 3i t 2 (θ(z)−θ(E01)) ¶2/3 = µ 3t 2 ¶2/3 (z −E01)ψE01 (z), (4.48) ψE01 (z) ≡ ψE01 (z,ξ) is analytic in Dϵ(E01) with ψE01 (E01) ̸= 0. Observe that these jump conditions coincide with the standard … view at source ↗
Figure 16
Figure 16. Figure 16: The contour and region in the ϵ neighborhood Dϵ(κ R 1 ) of κ R 1 We want to eliminate the jumps across Y 6 κ R 1 ∪Y 3 κ R 1 . Hence we define the complex-valued function δ˜(z) ≡ δ˜(ξ, z) by δ˜(z) = exp    1 2πi Z Y 3 κ R 1 ln¡ 1+|r (s)| 2 ¢ s − z d s − 1 2πi Z Y 6 κ R 1 ln¡ 1+|r (s)| 2 ¢ s − z d s    , z ∈ C\ ³ Y 3 κ R 1 ∪Y 6 κ R 1 ´ , (4.51) where the branch of the logarithm is chosen such that ln¡… view at source ↗
Figure 17
Figure 17. Figure 17: The contour Lj , j = 1,2,3,4 in the for the RH problem B.1. 2. For each z ∈ L, the boundary values NP ± (λ) satisfy the jump relation N P + (λ) = N P − (λ)e −i ¡ 4 3 λ 3+sλ ¢ σˆ 3P(λ), where JP (λ) =    Ã 1 0 ρ 1 ! , λ ∈ L1, Ã 1 0 −ρ 1 ! , λ ∈ L2, Ã 1 −ρ 0 1 ! λ ∈ L3, Ã 1 ρ 0 1! , λ ∈ L4. 3. N P (λ) is bounded near the origin. Then u(p) = 2 ¡ N P 1 ¢ 12 = 2 ¡ N P 1 ¢ 21 solves the Painlevé II e… view at source ↗
Figure 18
Figure 18. Figure 18: The jump contour Yj , j = 1,2,3,4 for the RH problem C.1. 1. N Ai(ζ) = I +O(z −1 ), |z| → ∞. 2. For each ζ ∈ Σ Ai , the boundary values N Ai ± (ζ) satisfy the jump relation N Ai + (ζ) = N Ai − (ζ)J Ai(ζ), ζ ∈ Y \{0}, where J Ai(ζ) :=    Ã 1 −e − 4 3 ζ 3/2 0 1 ! , ζ ∈ Y1, Ã 1 0 e 4 3 ζ 3/2 1 ! , ζ ∈ Y2 ∪Y4, Ã 0 1 −1 0! , ζ ∈ Y3. 3. For each integer N ≥ 0, the functions N Ai as,N (ζ) … view at source ↗
read the original abstract

We investigate the Cauchy problem for the focusing nonlinear Schr\"odinger (NLS) equation \begin{equation} iq_t(x,t)+q_{xx}(x,t)+2|q(x,t)|^2q(x,t)=0,\quad x\in\mathbb{R},\quad t\ge0,\nonumber \end{equation} subject to initial data $ q(x,0)$ satisfying the asymptotic boundary conditions \begin{equation}\label{eq:boundary} q(x,0) \sim q^{alg}(x,0) \quad \text{as} \quad x \to \pm\infty,\nonumber \end{equation} where $q^{alg}(x,t)$ denote finite-genus algebro-geometric quasi-periodic solutions of the focusing NLS equation. Employing the Riemann--Hilbert (RH) approach combined with the Deift--Zhou nonlinear steepest descent method, we analyze the long-time asymptotic behavior of solutions to this Cauchy problem. Our analysis distinguishes between two cases based on the genus $n$ of the underlying hyperelliptic Riemann surface: (i) Odd genus backgrounds: When the background solutions $q^{alg}(x,0)$ correspond to hyperelliptic curves of odd genus $n = 2s+1$ $(s \in \mathbb{N}_0)$, we identify distinct asymptotic regions in the $(x,t)$-plane characterized by the variable $\xi = x/t$, within which the leading-order asymptotics is expressed in terms of the second Painlev\'e transcendent. (ii)Even genus backgrounds: When the background solutions $q^{alg}(x,0)$ correspond to hyperelliptic curves of even genus $n = 2s$ $(s \in \mathbb{N})$, the asymptotic behavior in regions selected by $\xi$ is described in terms of parabolic cylinder functions. Specifically, we derive the leading-order asymptotics and establish explicit error bounds for the solution $q(x,t)$ as $t \to +\infty$, uniformly for $x \in \mathbb{R}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript analyzes the long-time asymptotics of the focusing NLS equation iq_t + q_xx + 2|q|^2 q = 0 with initial data q(x,0) that asymptotically matches a finite-genus algebro-geometric solution q^alg(x,0) as x → ±∞. Using the Riemann-Hilbert formulation combined with the Deift-Zhou nonlinear steepest descent method, it derives leading-order asymptotics as t → +∞ uniformly in x, distinguishing odd-genus (n=2s+1) cases yielding Painlevé II transcendent expressions in regions selected by ξ = x/t from even-genus (n=2s) cases involving parabolic cylinder functions, together with explicit error bounds.

Significance. If the derivations and error estimates hold, the result extends rigorous asymptotic analysis of the focusing NLS to a new class of non-decaying, quasi-periodic backgrounds, bridging soliton and finite-gap theories. The explicit uniform error bounds and separate treatment of odd/even genus via g-function construction and contour deformations constitute a technical advance with potential applications to modulated waves and integrable turbulence.

major comments (1)
  1. [Sections on RH problem and steepest descent (odd/even genus cases)] The central uniformity claim for all real x relies on partitioning the (x,t)-plane into bulk, transition, and far-field regions and controlling remainders in each; however, the manuscript does not explicitly verify that the scattering data for the perturbed finite-genus background permits the required contour deformations without pole crossings or additional restrictions on the reflection coefficient (see the discussion following the RH formulation and the steepest-descent analysis for odd genus).
minor comments (2)
  1. [Introduction and asymptotic regions] Notation for the hyperelliptic curve and the g-function construction could be clarified by adding a brief summary table comparing the odd-genus (Painlevé) and even-genus (parabolic cylinder) model problems.
  2. [Abstract and main theorems] A few typographical inconsistencies appear in the boundary condition equation labels and in the statement of the error bounds; these are easily corrected.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The single major comment is addressed point-by-point below; we will incorporate a clarifying addition to strengthen the presentation of the uniformity claim.

read point-by-point responses

  1. Referee: [Sections on RH problem and steepest descent (odd/even genus cases)] The central uniformity claim for all real x relies on partitioning the (x,t)-plane into bulk, transition, and far-field regions and controlling remainders in each; however, the manuscript does not explicitly verify that the scattering data for the perturbed finite-genus background permits the required contour deformations without pole crossings or additional restrictions on the reflection coefficient (see the discussion following the RH formulation and the steepest-descent analysis for odd genus).

    Authors: We appreciate this observation. The scattering data for the perturbed background is defined in the RH formulation (Section 2) via the jump matrix involving the reflection coefficient r(λ) associated with the difference q(x,0)−q^alg(x,0). Under the assumed decay of this difference at infinity, r(λ) is continuous on the real line, satisfies |r(λ)|<1 away from the branch cuts of the hyperelliptic curve, and introduces no additional poles. The g-function is built directly from the finite-genus spectral curve, and the steepest-descent contours (bulk, transition, and far-field) are chosen to lie in regions where Re(φ(λ;ξ))<0 while remaining disjoint from the discrete spectrum of the background; this construction is uniform in ξ because the stationary points and branch points are determined solely by the genus-n curve and the parameter ξ=x/t. Consequently, the deformations proceed without pole crossings for any real x. We agree, however, that an explicit verification of these properties was not stated as a separate lemma. In the revised manuscript we will insert a short remark (or appendix paragraph) confirming that the decay assumption on the perturbation guarantees the required analyticity and absence of crossings, thereby supporting the uniform error bounds. This is a minor clarification. revision: yes

Circularity Check

0 steps flagged

No circularity: standard RH steepest-descent applied to new initial-data class

full rationale

The derivation formulates the Riemann-Hilbert problem from the given asymptotic boundary conditions q(x,0) ~ q^alg(x,0) and applies the established Deift-Zhou nonlinear steepest descent method, partitioning the (x,t)-plane into bulk, transition and far-field regions to obtain model problems (Painlevé II for odd genus, parabolic cylinder for even genus) together with explicit error bounds. All steps rely on standard contour deformations and estimates from the literature; no equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The result is an independent extension of existing theory to the finite-genus perturbed case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on established theory of Riemann-Hilbert problems and the Deift-Zhou method for integrable systems; no free parameters, ad-hoc axioms, or new postulated entities are introduced in the abstract.

axioms (1)
  • domain assumption The Deift-Zhou nonlinear steepest descent method applies directly to the Riemann-Hilbert problem arising from the finite-genus background scattering data.
    Invoked without new justification in the abstract.

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Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Solitons and the Inverse Scattering Transform[M]

    ABLOWITZ M J, SEGUR H. Solitons and the Inverse Scattering Transform[M]. Philadelphia: SIAM, 1981

    work page 1981

  2. [2]

    Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation[M]//Annals of Mathematics Studies, 169

    KAMVISSIS S, MCLAUGHLIN K D T R, MILLER P D. Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation[M]//Annals of Mathematics Studies, 169. Princeton: Princeton University Press, 2003

    work page 2003

  3. [3]

    Applied Asymptotic Analysis[M]//Graduate Studies in Mathematics, 75

    MILLER P . Applied Asymptotic Analysis[M]//Graduate Studies in Mathematics, 75. Providence: American Mathematical Society, 2006

    work page 2006

  4. [4]

    Painlevé Transcendents: The Riemann-Hilbert Ap- proach[M]//Mathematical Surveys and Monographs, 128

    FOKAS A S, ITS A R, KAPAEV A A, et al. Painlevé Transcendents: The Riemann-Hilbert Ap- proach[M]//Mathematical Surveys and Monographs, 128. Providence: American Mathematical Society, 2006

    work page 2006

  5. [5]

    Long-Time Behavior of the Non-Focusing Nonlinear Schrödinger Equation: A Case Study[M]//Lectures in Mathematical Sciences, 5

    DEIFT P , ZHOU X. Long-Time Behavior of the Non-Focusing Nonlinear Schrödinger Equation: A Case Study[M]//Lectures in Mathematical Sciences, 5. Tokyo: Graduate School of Mathematical Sciences, Uni- versity of Tokyo, 1994

    work page 1994

  6. [6]

    The collisionless shock region for the long-time behavior of solutions of the KdV equation[J]

    DEIFT P , VENAKIDES S, ZHOU X. The collisionless shock region for the long-time behavior of solutions of the KdV equation[J]. Communications on Pure and Applied Mathematics, 1994, 47: 199-206

    work page 1994

  7. [7]

    A steepest descent method for oscillatory Riemann-Hilbert problems: Asymptotics for the MKdV equation[J]

    DEIFT P , ZHOU X. A steepest descent method for oscillatory Riemann-Hilbert problems: Asymptotics for the MKdV equation[J]. Annals of Mathematics, 1993, 137(2): 295-368

    work page 1993

  8. [8]

    New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems[J]

    DEIFT P , VENAKIDES S, ZHOU X. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems[J]. International Mathematics Research Notices, 1997, 6: 286-299. 47

    work page 1997

  9. [9]

    Far-field asymptotics for multiple-pole solitons in the large-order limit[J]

    BILMAN D, BUCKINGHAM R, WANG D S. Far-field asymptotics for multiple-pole solitons in the large-order limit[J]. Journal of Differential Equations, 2021, 297: 320-369

    work page 2021

  10. [10]

    Universal nature of the nonlinear stage of modulational instability[J]

    BIONDINI G, MANTZAVINOS D. Universal nature of the nonlinear stage of modulational instability[J]. Phys- ical Review Letters, 2016, 116(4): 043902

    work page 2016

  11. [11]

    Long-time asymptotics for the focusing nonlinear Schrödinger equa- tion with nonzero boundary conditions in the presence of a discrete spectrum[J]

    BIONDINI G, LI S, MANTZAVINOS D. Long-time asymptotics for the focusing nonlinear Schrödinger equa- tion with nonzero boundary conditions in the presence of a discrete spectrum[J]. Communications in Math- ematical Physics, 2021, 382(3): 1495-1577

    work page 2021

  12. [12]

    Extreme superposition: rogue waves of infinite order and the Painlevé- III hierarchy

    BILMAN, D, LING, L M, MILLER, P D. Extreme superposition: rogue waves of infinite order and the Painlevé- III hierarchy. Duke Math. J. 2020, 169: 671-760

    work page 2020

  13. [13]

    Long time asymptotic behavior of the focusing nonlinear Schrödinger equation[J]

    BORGHESE M, JENKINS R, MCLAUGHLIN K D T R. Long time asymptotic behavior of the focusing nonlinear Schrödinger equation[J]. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 2018, 35(4): 887-920

    work page 2018

  14. [14]

    The focusing NLS equation with step-like oscillating background: The genus 3 sector[J]

    BOUTET DE MONVEL A, LENalgS J, SHEPELSKY D. The focusing NLS equation with step-like oscillating background: The genus 3 sector[J]. Communications in Mathematical Physics, 2022, 390(3): 1081-1148

    work page 2022

  15. [15]

    Rarefaction waves of the Korteweg-de Vries equation via non- linear steepest descent[J]

    ANDRIEIEV K, EGOROVA I, LANGE T L, et al. Rarefaction waves of the Korteweg-de Vries equation via non- linear steepest descent[J]. Journal of Differential Equations, 2016, 261(10): 5371-5410

    work page 2016

  16. [16]

    Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions[J]

    BIONDINI G, KOVACIC G. Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions[J]. Journal of Mathematical Physics, 2014, 55(3): 031506

    work page 2014

  17. [17]

    Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability[J]

    BIONDINI G, MANTZAVINOS D. Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability[J]. Commu- nications on Pure and Applied Mathematics, 2017, 70(12): 2300-2365

    work page 2017

  18. [18]

    The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions[J]

    DEMONTIS F , PRINARI B, VAN DER MEE C, et al. The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions[J]. Journal of Mathematical Physics, 2014, 55: 101505

    work page 2014

  19. [19]

    Long-time asymptotics for Toda shock waves in the modulation region[J]

    EGOROVA I, MICHOR J, PRYIMAK A, et al. Long-time asymptotics for Toda shock waves in the modulation region[J]. Journal of Mathematical Physics, Analysis, Geometry, 2023, 19(2): 396-442

    work page 2023

  20. [20]

    Asymptotics of KdV shock waves via the Riemann-Hilbert ap- proach[J]

    EGOROVA I, PIORKOWSKI M, TESCHL G. Asymptotics of KdV shock waves via the Riemann-Hilbert ap- proach[J]. Indiana University Mathematics Journal, 2024, 73(2): 645-690

    work page 2024

  21. [21]

    On the asymptotic stability of N-soliton solutions of the defocusing nonlinear Schrödinger equation[J]

    CUCCAGNA S, JENKINS R. On the asymptotic stability of N-soliton solutions of the defocusing nonlinear Schrödinger equation[J]. Communications in Mathematical Physics, 2016, 347(2): 419-453

    work page 2016

  22. [22]

    The ¯∂steepest descent method for orthogonal polynomials on the real line with varying weights[J]

    MCLAUGHLIN K T R, MILLER P D. The ¯∂steepest descent method for orthogonal polynomials on the real line with varying weights[J]. International Mathematics Research Notices, 2008, 2008: rnn075

    work page 2008

  23. [23]

    Planar unimodular Baker-Akhiezer function for the nonlinear Schrödinger equation[J]

    KOTLYAROV V , SHEPELSKY D. Planar unimodular Baker-Akhiezer function for the nonlinear Schrödinger equation[J]. Annals of Mathematical Sciences and Applications, 2017, 2(2): 343-384

    work page 2017

  24. [24]

    Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background[J]

    FAN E, LI G, YANG Y, et al. Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background[J]. Mathematische Annalen, 2026, 394(2): 44

    work page 2026

  25. [25]

    The focusing NLS equation with step-like oscillat- ing background: Scenarios of long-time asymptotics[J]

    BOUTET DE MONVEL A, LENalgS J, SHEPELSKY D. The focusing NLS equation with step-like oscillat- ing background: Scenarios of long-time asymptotics[J]. Communications in Mathematical Physics, 2021, 383(2): 893-952

    work page 2021

  26. [26]

    Focusing NLS equation: Long-time dynamics of step-like initial data[J]

    BOUTET DE MONVEL A, KOTLYAROV V P , SHEPELSKY D. Focusing NLS equation: Long-time dynamics of step-like initial data[J]. International Mathematics Research Notices, 2011, 2011(7): 1613-1653

    work page 2011

  27. [27]

    Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space[J]

    DEIFT P , ZHOU X. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space[J]. Communications on Pure and Applied Mathematics, 2003, 56(8): 1029-1077

    work page 2003

  28. [28]

    Direct and inverse scattering transforms with arbitrary spectral singularities[J]

    ZHOU X. Direct and inverse scattering transforms with arbitrary spectral singularities[J]. Communications on Pure and Applied Mathematics, 1989, 42(7): 895-938

    work page 1989

  29. [29]

    Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data[J]

    BERTOLA M, MINAKOV A. Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data[J]. Analysis and Mathematical Physics, 2019, 9(4): 1761-1818. 48

    work page 2019

  30. [30]

    Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equation[J]

    KOTLYAROV V , MINAKOV A. Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equation[J]. Journal of Mathematical Physics, 2019, 60(12): 123501

    work page 2019

  31. [31]

    The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics[J]

    XU K, YANG Y, FAN E. The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics[J]. Journal of Differential Equations, 2024, 380: 24-91

    work page 2024

  32. [32]

    Long-time asymptotics of the Camassa-Holm equation on the line[C]//Integrable Systems and Random Matrices

    BOUTET DE MONVEL A, SHEPELSKY D. Long-time asymptotics of the Camassa-Holm equation on the line[C]//Integrable Systems and Random Matrices. Providence: American Mathematical Society, 2008: 99- 116

    work page 2008

  33. [33]

    Long-time asymptotics of the periodic Toda lattice under short-range perturba- tions[J]

    KAMVISSIS S, TESCHL G. Long-time asymptotics of the periodic Toda lattice under short-range perturba- tions[J]. Journal of Mathematical Physics, 2012, 53(7): 073206

    work page 2012

  34. [34]

    Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solu- tions[J]

    MIKIKITS-LEITNER A, TESCHL G. Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solu- tions[J]. Journal d’Analyse Mathématique, 2012, 116: 163-218

    work page 2012

  35. [35]

    A Riemann-Hilbert problem approach to infinite gap Hill’ s operators and the Korteweg-de Vries equation[J]

    MCLAUGHLIN K T R, NABELEK P V . A Riemann-Hilbert problem approach to infinite gap Hill’ s operators and the Korteweg-de Vries equation[J]. International Mathematics Research Notices, 2021, 2021(2): 1288- 1327

    work page 2021

  36. [36]

    A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation[J]

    HASTINGS S P , MCLEOD J B. A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation[J]. Archive for Rational Mechanics and Analysis, 1980, 73(1): 31-51. 49

    work page 1980