Asymptotic reductions and solitons of nonlocal nonlinear Schr\"{o}dinger equations

TL;DR

This paper derives asymptotic reductions of a 3+1D nonlocal nonlinear Schrödinger model to Boussinesq and KP equations, predicting stable and unstable solitary wave solutions with numerical validation.

Contribution

It introduces a novel asymptotic reduction framework for nonlocal NLS equations in higher dimensions, linking them to well-known integrable models and analyzing wave stability.

Findings

Dark anti-dark solitary waves depend on nonlocality strength.

Stable ring-shaped solitons are supported by strong nonlocality.

Numerical simulations confirm analytical predictions.

Abstract

Asymptotic reductions of a defocusing nonlocal nonlinear Schr\"{o}dinger model in $(3+1)$-dimensions, in both Cartesian and cylindrical geometry, are presented. First, at an intermediate stage, a Boussinesq equation is derived, and then its far-field, in the form of a variety of Kadomtsev-Petviashvilli (KP) equations for right- and left-going waves, is found. KP models include versions of the KP-I and KP-II equations, in Cartesian and cylindrical geometry. Solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are also predicted to occur. Their nature and stability is determined by a parameter defined by the physical parameters of the original nonlocal system. It is thus found that (dark) anti-dark solitary waves are only supported by a weak (strong) nonlocality, and are unstable (stable) in higher-dimensions. Our analytical predictions are corroborated by direct numerical simulations.