Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data

• Published in: Journal of Differential Equations Vol. 386; pp. 113 - 163
• Main Authors: Geng, Xianguo, Liu, Wenhao, Li, Ruomeng
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 25.03.2024
• Subjects: Coupled complex short-pulse equation; Long-time asymptotics; Nonlinear steepest decent method; Riemann-Hilbert problem; Nonlinear steepest decent method; Long-time asymptotics; Riemann-Hilbert problem; Coupled complex short-pulse equation
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2023.12.019

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.