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

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):|ξ|<6} with ξ:=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):|ξ|>6,|ξ|=O(1)} for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics is in compatible with the “background solution” and the error terms are derived via rigorous analysis.