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arXiv · 2108.06284

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

Taiyang Xu·Zechuan Zhang·Engui Fan
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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 $\{(x,t): \vert ξ\vert<6\}$ 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 $\{(x,t): \vert ξ\vert>6, |ξ|=\mathcal{O}(1)\}$ 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.

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Taiyang Xu, Zechuan Zhang, Engui Fan. 2023-06-23. On the Cauchy problem of defocusing mKdV equation with finite density initial data: long time asymptotics in soliton-less regions. https://doi.org/10.1016/j.jde.2023.06.038

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