Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data
Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data
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DOI：
10.1016/j.jde.2023.12.019
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发表时间：
2024-03
影响因子：
2.4
通讯作者：
Xianguo Geng;Wenhao Liu;Ruomeng Li
中科院分区：
文献类型：
--
作者：
Xianguo Geng;Wenhao Liu;Ruomeng Li
我们描述了与 4×4 矩阵谱问题相关的耦合复短脉冲方程的初值问题解的长时渐近行为。由于能量依赖势和WKI类型的存在，4×4矩阵谱问题的谱分析非常困难。我们采用的方法是逆散射变换和Deift-Zhou非线性最速下降法的结合。从与耦合复短脉冲方程相关的Lax对出发，通过引入一些适当的谱函数变换导出基本黎曼-希尔伯特问题，并通过谱变量在k→0处的渐近行为重构基本黎曼-希尔伯特问题解的参数化势。最终通过一系列Deift-Zhou轮廓变形获得耦合复短脉冲方程解的首阶渐近行为。
We characterize the long-time asymptotic behavior of the solution of the initial value problem for the coupled complex short-pulse equation associated with the 4× 4 matrix spectral problem. The spectral analysis of the 4× 4 matrix spectral problem is very difficult because of the existence of energy-dependent potentials and the WKI type. The method we adopted is a combination of the inverse scattering transform and Deift-Zhou nonlinear steepest descent method. Starting from the Lax pair associated with the coupled complex short-pulse equation, we derive a basic Riemann-Hilbert problem by introducing some appropriate spectral function transformations, and reconstruct the potential parameterized from the solution of the basic Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at k→ 0. We finally obtain the leading order asymptotic behavior of the solution of the coupled complex short-pulse equation through a series of Deift-Zhou contour deformations.