聚焦 Kundu-Eckhaus 方程的反散射变换法: 阶跃振荡背景下的长时间渐进性

摘要/Abstract

摘要：

该文利用非线性速降法研究了阶跃振荡背景下聚焦 Kundu-Eckhaus 方程解的长时间渐进性问题. 在稀疏情况下, 当解趋于 $x$ 轴时, 其渐进性以平面波的形式呈现;当解趋于 $t$ 轴时, 其渐进性以缓慢衰减的形式呈现; 而在两个过渡扇区, 解的渐进性可表示为调制椭圆波函数. 此外, 在激波情况下, 解的渐进性可由依赖于亏格为3的黎曼曲面的超椭圆函数表示.该文所得结论有助于解释存在五次非线性项以及自频移效应的调制不稳定性下的非线性阶段.

关键词: 聚焦 Kundu-Eckhaus 方程, 反散射变换法, Riemann-Hilbert 问题, 非线性速降法

Abstract:

In this paper, we study the long-time dynamics of the solution of the focusing Kundu-Eckhaus equation under steplike oscillating background via the nonlinear steepest descent method. In the rarefaction case, when the solution is near the $x$-axis, the form of the leading behavior is the plane waves, when the solution tends to the $t$-axis, the leading behavior decays slowly, and when the solution belongs to two transition sectors, the form of the leading behavior is the elliptic waves. Furthermore, in the shock case, the leading behavior is described by terms of hyperelliptic functions depended on a Riemann surface of genus 3. Our results may be useful to explain the nonlinear stage of modulation instability in presence of the the quintic nonlinear and the self-frequency shift effects.

Key words: The focusing Kundu-Eckhaus equation, Inverse scattering transform, Riemann-Hilbert problem, The nonlinear steepest descent method

中图分类号:

O175.2

王贵贤,王秀彬,韩波. 聚焦 Kundu-Eckhaus 方程的反散射变换法: 阶跃振荡背景下的长时间渐进性[J]. 数学物理学报, 2023, 43(4): 1085-1122.

Wang Guixian,Wang XiuBin,Han Bo. Inverse Scattering Transform for the Focusing Kundu-Eckhaus Equation: Long-time Dynamics of the Steplike Oscillating Background[J]. Acta mathematica scientia,Series A, 2023, 43(4): 1085-1122.

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