Long-time asymptotics of the defocusing mKdV equation with step initial data

TL;DR

This paper rigorously analyzes the long-time behavior of solutions to the defocusing mKdV equation with step initial data, revealing a tripartite structure and confirming results with numerical simulations.

Contribution

It generalizes the construction of the g-function and introduces a genus reduction method for the Riemann-Hilbert problem in this context.

Findings

Solution exhibits three distinct regions: decaying plane wave, dispersive shock wave, and constant state.

Asymptotic results match numerical simulations closely.

Provides a detailed description of the solution's long-time behavior for step initial data.

Abstract

This work investigates the long-time asymptotics of solution to defocusing modified Korteweg-de Vries equation with a class of step initial data. A rigorous asymptotic analysis is conducted on the associated Riemann-Hilbert problem by applying Deift-Zhou nonlinear steepest descent method. In this process, the construction of odd-symmetry g-function is generalized and the method of genus reduction on the Riemann-theta function is proposed via conformal transformation and symmetries. It is revealed that for sufficiently large time, the solution manifests a tripartite spatiotemporal structure, i.e., in the left plane-wave region, the solution decays to a modulated plane wave with oscillatory correction; in the central dispersive shock wave region, the solution is governed by a modulated elliptic periodic wave; in the right plane wave region, the solution converges exponentially to a constant. The results from the long-time asymptotic analysis have been shown to match remarkably well with that obtained by direct numerical simulations.