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

TL;DR

This paper analyzes the long-time behavior of solutions to the derivative nonlinear Schrödinger equation with nonzero boundary conditions using the $ar{ ext{∂}}$ steepest descent method, confirming soliton resolution and providing detailed asymptotics.

Contribution

It applies the $ar{ ext{∂}}$ steepest descent method to derive long-time asymptotics for the derivative NLS with nonzero boundary conditions, including soliton resolution in a specific region.

Findings

Confirmed soliton resolution conjecture in the specified region.

Derived detailed asymptotic expansion with soliton interactions.

Achieved residual error estimate of order $ ext{O}(t^{-3/4})$.

Abstract

In this paper, we apply $\overline\partial$ steepest descent method to study the Cauchy problem for the derivative nonlinear Schr\"odinger equation with nonzero 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 $|q_\pm|=1$. Based on the spectral analysis of the Lax pair, we express the solution of the derivative nonlinear Schr\"odinger 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 $N(\Lambda)$-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.