Numerical inverse scattering transform for the coupled modified Korteweg-de Vries equation

TL;DR

This paper introduces a numerical inverse scattering transform framework for the coupled mKdV equation, enabling direct computation of solutions and long-time asymptotics without time-stepping.

Contribution

It extends the NIST method to a coupled matrix system, addressing the complexities of the associated Riemann-Hilbert problem.

Findings

Successfully computes scattering data using Chebyshev collocation.

Deforms the Riemann-Hilbert problem via nonlinear steepest descent.

Accurately captures asymptotic features in long-time simulations.

Abstract

This paper develops the numerical inverse scattering transform (NIST) framework for the coupled modified Korteweg-de Vries (mKdV) equation based on its associated Riemann-Hilbert problem. The coupled system gives rise to a $3\times3$ matrix-valued Riemann-Hilbert problem, whose jump matrix and scattering data have a more involved structure than in the scalar case. This matrix setting makes the extension of NIST to the coupled system nontrivial, both in the direct scattering computation and in the numerical solution of the inverse problem. Within this framework, the scattering data are first computed by solving the matrix direct scattering problem using a Chebyshev collocation method with suitable mappings. The Deift-Zhou nonlinear steepest descent method is then used to analyze and deform the oscillatory Riemann-Hilbert problem. In particular, the phase function admits two stationary points symmetric about the origin, and the analysis leads to a division of the $(x,t)$-plane into three regions with corresponding contour deformations. Compared with traditional numerical methods, the NIST computes the solution directly at prescribed spatial and temporal points without relying on time-stepping. Numerical experiments illustrate the performance of the proposed NIST in long-time simulations and indicate that it captures the main asymptotic features of the coupled mKdV solutions.