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We consider the Cauchy problem for the defocusing modified Korteweg-de Vries(mKdV) equation with finite density type initial data. With the$\bar{\partial}$ generalization of the nonlinear steepest descent method ofDeift and Zhou, we extrapolate the leading order approximation to the solutionof mKdV for large time in the solitonic space-time region $|x/t+4|<2$, and wegive bounds for the error which decay as $t\rightarrow\infty$ for a generalclass of initial data whose difference from the non-vanishing backgroundpossesses a fixed number of finite moments. Our results provide a verificationof the soliton resolution conjecture and asymptotic stability of $N$-solitonsolutions for mKdV equation with finite density type initial data.

Soliton resolution and asymptotic stability of $N$-soliton solutions for the defocusing mKdV equation with finite density type initial data