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

Abstract

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

CLC Number:

O175.2

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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Figures/Tables 14

References 45

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Inverse Scattering Transform for the Focusing Kundu-Eckhaus Equation: Long-time Dynamics of the Steplike Oscillating Background

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