On the Cauchy problem of defocusing mKdV equation with finite density initial data: long time asymptotics in soliton-less regions

TL;DR

This paper analyzes the long-time behavior of solutions to the defocusing mKdV equation with finite density initial data, focusing on soliton-less regions and using Riemann-Hilbert problem techniques.

Contribution

It extends previous work by deriving the long-time asymptotics in soliton-less regions using Riemann-Hilbert analysis for the defocusing mKdV equation.

Findings

Asymptotics match the background solution in soliton-less regions

Error terms are rigorously derived

Results complement previous soliton resolution analysis

Abstract

We investigate the long-time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper [arXiv:2108.03650], which gives the soliton resolution for the defocusing mKdV equation in the central asymptotic sector $\{(x,t): \vert \xi \vert<6\}$ with $\xi:=x/t$. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the soliton-less regions $\{(x,t): \vert \xi \vert>6, |\xi|=\mathcal{O}(1)\}$ for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics are in compatible with the ``background solution'' and the error terms are derived via rigorous analysis.