Long time asymptotics for the defocusing mKdV equation under nonzero boundary conditions in solitonic region without phase points

Zechuan Zhang, Taiyang Xu, Engui Fan

Published 2021-08-08Version 1

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries equation for finite density type initial data. With the $\bar{\partial}$ generalization of the nonlinear steepest descent method of Deift and Zhou, we extrapolate the leading order approximation to the solution of mKdV for large time in the solitonic region of space-time, $|x/t|<6$, and we give bounds for the error which decay as $t\rightarrow\infty$ for a general class of initial data whose difference from the non-vanishing background possesses a fixed number of finite moments. Our results provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for mKdV equation.