Spectral analysis and long-time asymptotics of complex mKdV equation

TL;DR

This paper derives the long-time asymptotic behavior of the complex mKdV equation using the Deift-Zhou nonlinear steepest descent method by transforming the problem into a Riemann-Hilbert problem and solving it.

Contribution

It introduces a novel application of the Deift-Zhou method to analyze the complex mKdV equation's long-time behavior through Riemann-Hilbert problem techniques.

Findings

Long-time asymptotics of complex mKdV derived

Transformation into Riemann-Hilbert problem established

Solution of model problem yields asymptotic results

Abstract

In this paper, we obtain the long-time asymptotics of complex mKdV equation via Defit-Zhou method (Non-linear steepest descent method). The Cauchy problem of complex mKdV equation is transformed into the corresponding Riemann-Hilbert problem on the basis of the Lax pair and the scattering matrix. After that Riemann-Hilbert problems are converted through a decomposition of the matrix-valued spectral function and factorizations of the jump matrix for Riemann-Hilbert problem. Finally, by solving the last model problem, the long-time asymptotics of complex mKdV equation are derived.