On the Cauchy problem of defocusing mKdV equation: long-time asymptotics under step-like initial data

Taiyang Xu, Engui Fan

Published 2022-04-04Version 1

We investigate the long time asymptotics for the Cauchy problem of the defocusing modified Kortweg-de Vries (mKdV) equation with step-like initial data, i.e., $q_{0}(x)=q(x,t=0)=C_{L}$, as $x<0$ and $q_{0}(x)=C_{R}$ as $x>0$, where $C_{L}>C_{R}>0$ are arbitrary positive real numbers. We firstly develop the direct scattering theory to establish the Riemann-Hilbert (RH) problem associated with step-like initial data. Then by introducing the related $g$ function in different space-time regions and using the steepest descent analysis, we deform the original matrix valued RH problem to explicitly solving models. Finally we obtain the different long-time asymptotic behavior of the solution of the Cauchy problem for defocusing mKdV equation in four different space-time regions $\mathcal{R}_{\xi,I}, \mathcal{R}_{\xi,II},\mathcal{R}_{\xi,III}$ and $\mathcal{R}_{\xi,IV} $ in the half-plane, where $\mathcal{R}_{\xi,I}$ and $ \mathcal{R}_{\xi,IV}$ are far left field regions; and $\mathcal{R}_{\xi,II}$ and $ \mathcal{R}_{\xi,III}$ are rarefaction wave regions.