Long time asymptotic behavior for the derivative Schrödinger equation with nonzero boundary conditions

Abstract

In this paper, we apply $\overline{\partial }$ steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with nonzero boundary conditions \begin{align} &iq_{t}+q_{xx}+iσ(|q|^2q)_{x}=0,\\ & (x,0) = q_0(x), \quad\lim_{x\to\pm\infty} q_0(x) = q_\pm,\end{align} where $|q_{\pm }|=1$. Based on the spectral analysis of the Lax pair, we express the solution of the derivative nonlinear Schrödinger equation in terms of solutions of a Riemann-Hilbert problem.In a fixed space-time solitonic region $-3<x/t<-1$, we compute the long time asymptotic expansion of the solution $q(x,t)$,which implies soliton resolution conjecture and can be characterized with an $N(\hat{I}›)$-soliton whose parameters are modulated bya sum of localized soliton-soliton interactions as one moves through the region; the residual error order $\mathcal{O}(t^{-3/4})$ from a $\overline{\partial }$ equation.

47 pages. arXiv admin note: text overlap with arXiv:2005.12208