Long-time asymptotic behavior of the nonlocal nonlinear Schr\"{o}dinger equation with finite density type initial data

TL;DR

This paper analyzes the long-time behavior of solutions to the nonlocal nonlinear Schrödinger equation with finite density initial data using advanced Riemann-Hilbert and steepest descent methods, revealing detailed asymptotic structures.

Contribution

It develops a $ar{ ext{D}}$-steepest descent approach for the NNLS equation, providing precise asymptotics in the soliton region with error estimates.

Findings

Asymptotic solution characterized by soliton and continuous spectrum contributions.

Error bounds established for the asymptotic approximation as time tends to infinity.

Confirmation of soliton structures via discrete spectrum analysis.

Abstract

In this work, we employ the $\bar{\partial}$-steepest descent method to investigate the Cauchy problem of the nonlocal nonlinear Schr\"{o}dinger (NNLS) equation with finite density type initial conditions in weighted Sobolev space $\mathcal{H}(\mathbb{R})$. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the NNLS equation with the finite density type initial boundary value condition. By developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution $q(x,t)$ in soliton region of space-time, $\left(\frac{x}{2t}\right)=\xi$ for any fixed $\xi=\in (1,K)$($K$ is a sufficiently large real constant), and give bounds for the error decaying as $|t|\rightarrow\infty$. Based on the resulting asymptotic behavior, the asymptotic approximation of the NNLS equation is characterized with the soliton term confirmed by $N(\Lambda)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.