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In this paper, we apply $\overline\partial$ steepest descent method to studythe Cauchy problem for the derivative nonlinear Schr\"odinger equation withnonzero boundary conditions \begin{align}&iq_{t}+q_{xx}+i\sigma(|q|^2q)_{x}=0,\\ & (x,0) = q_0(x),\quad\lim_{x\to\pm\infty} q_0(x) = q_\pm,\end{align} where $|q_\pm|=1$. Basedon the spectral analysis of the Lax pair, we express the solution of thederivative nonlinear Schr\"odinger equation in terms of solutions of aRiemann-Hilbert problem.In a fixed space-time solitonic region $-3<x/t<-1$, wecompute the long time asymptotic expansion of the solution $q(x,t)$,whichimplies soliton resolution conjecture and can be characterized with an$N(\Lambda)$-soliton whose parameters are modulated bya sum of localizedsoliton-soliton interactions as one moves through the region; the residualerror order $\mathcal{O}( t^{-3/4})$ from a $\overline\partial$ equation.

Long time asymptotic behavior for the derivative Schr\"odinger equation with nonzero boundary conditions