Long-time asymptotics for the modified complex short pulse equation

Mingming Chen; Xianguo Geng; Kedong Wang

doi:10.3934/dcds.2022060

Article Contents

2022, Volume 42, Issue 9: 4439-4470. Doi: 10.3934/dcds.2022060

This issue Previous Article Critical gauged Schrödinger equations in

$ \mathbb{R}^2 $

with vanishing potentials Next Article Comparison principles for nonlocal Hamilton-Jacobi equations

Long-time asymptotics for the modified complex short pulse equation

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

^*Corresponding author: Xianguo Geng
^*Corresponding author: Xianguo Geng

Received: November 30, 2021

Early access: April 2022

Published: August 2022

The second author is supported by the National Natural Science Foundation of China (Grant Nos. 11931017, 11871440)

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Abstract

Based on the spectral analysis and the inverse scattering method, by introducing some spectral function transformations and variable transformations, the initial value problem for the modified complex short pulse (mCSP) equation is transformed into a $ 2\times2 $ matrix Riemann-Hilbert problem. It is proved that the solution of the initial value problem for the mCSP equation has a parametric expression related to the solution of the matrix Riemann-Hilbert problem. Various Deift-Zhou contour deformations and the motivation behind them are given. Through several appropriate transformations and strict error estimates, the original matrix Riemann-Hilbert problem can be reduced to the model Riemann-Hilbert problem, whose solution can be solved explicitly in terms of the parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the initial value problem for the mCSP equation is obtained by using the nonlinear steepest decent method.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35Q15.

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Figure 1. The oriented contour on $ \mathbb{R} $

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Figure 2. The signature table for Re $ (i\theta) $ in the complex $ k $-plane

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Figure 3. The oriented contour on $ \mathbb{R} $

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Figure 4. The oriented contour $ \Sigma $

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Figure 5. The oriented contour $ \Sigma^\prime $

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Figure 6. The oriented contour $ \Sigma_A $ or $ \Sigma_B $ (reoriented)

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