Long time asymptotic behavior for the nonlocal nonlinear Schr\"odinger equation with weighted Sobolev initial data

TL;DR

This paper extends the $ar{ ext{d}}$ steepest descent method to analyze the long-time asymptotic behavior of solutions to the nonlocal nonlinear Schrödinger equation with weighted Sobolev initial data, revealing soliton interactions and dispersion effects.

Contribution

It develops a novel spectral analysis and Riemann-Hilbert problem approach for the NNLS equation, providing detailed asymptotic expansions including effects of nonlocality.

Findings

Asymptotic expansion includes soliton and interaction terms.

Second and third order terms are influenced by a function related to stationary phase.

Results differ from classical NLS due to nonlocal effects.

Abstract

In this paper, we extend $\overline\partial$ steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schr\"odinger (NNLS) equation with weighted Sobolev initial data %and finite density initial data \begin{align*} &iq_{t}+q_{xx}+2\sigma q^2(x,t)\overline{q}(-x,t)=0, & q(x,0)=q_0(x), \end{align*} where $ q_0(x)\in L^{1,1}(\mathbb{R})\cap L^{2,1/2}(\mathbb{R})$. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem is expressed in terms of solutions of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, we obtain the asymptotic expansion of the Cauchy problem for the NNLS equation in solitonic region. The leading order term is soliton solutions, the second term is the error term is the interaction between solitons and dispersion, the error term comes from the corresponding $\bar{\partial}$ equation. Compared to the asymptotic results on the classical NLS equation, the major difference is the second and third terms in asymptotic expansion for the NNLS equation were affected by a function $ {\rm Im}\nu(\xi)$ for the stationary phase point $\xi$.