A Two-Component Sasa–Satsuma Equation: Large-Time Asymptotics on the Line

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2024-02-24 DOI:10.1007/s00332-024-10015-9

引用次数: 0

Abstract

We consider the initial value problem for a two-component Sasa–Satsuma equation associated with a \(4\times 4\) Lax pair with decaying initial data on the line. By utilizing the spectral analysis, the solution of the two-component Sasa–Satsuma system is transformed into the solution of a \(4\times 4\) matrix Riemann–Hilbert problem. Then, the long-time asymptotics of the solution is obtained by means of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. We show that there are three main regions in the half-plane \(-\infty<x<\infty \) , \(t>0\) , where the asymptotics has qualitatively different forms: a left fast decaying sector, a central Painlevé sector where the asymptotics is described in terms of the solution to a system of coupled Painlevé II equations, which is related to a \(4\times 4\) matrix Riemann–Hilbert problem, and a right slowly decaying oscillatory sector.

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双分量萨萨摩方程：线性大时间渐近线

摘要我们考虑了与线上衰减初始数据的 \(4\times 4\) Lax 对相关的双分量 Sasa-Satsuma 方程的初值问题。通过利用谱分析，两分量 Sasa-Satsuma 系统的解被转化为一个（4 次）矩阵 Riemann-Hilbert 问题的解。然后，通过 Deift 和 Zhou 针对振荡黎曼-希尔伯特问题的非线性最陡下降方法，得到了解的长时渐近线。我们表明，在半平面 \(-\infty<x<\infty \) , \(t>0\) 中存在三个主要区域，其渐近线具有质的不同形式：一个左侧快速衰减扇区，一个中心 Painlevé 扇区，其渐近线用一个耦合 Painlevé II方程组的解来描述，该方程组与一个 \(4\times 4\) 矩阵 Riemann-Hilbert 问题相关，以及一个右侧缓慢衰减振荡扇区。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464