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

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Nonlinear Science 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.

查看原文本刊更多论文

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。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of Nonlinear Science 数学-力学

CiteScore

5.00

自引率

3.30%

发文量

审稿时长

4.5 months

期刊介绍： The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be. All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.