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 ${(x,t): \vert \xi \vert<6}$ 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.