On the integrability aspects of nonparaxial nonlinear Schr\"odinger equation and the dynamics of solitary waves

TL;DR

This paper investigates the integrability and solitary wave dynamics of a nonparaxial nonlinear Schrödinger equation, showing it is non-integrable but admits stable bright solitary wave solutions with interesting interaction behaviors.

Contribution

It demonstrates the non-integrability of the NNLS equation via Painlevé analysis and constructs explicit bright solitary wave solutions using Hirota's method, analyzing their stability and interactions.

Findings

NNLS equation fails the Painlevé integrability test.

Stable bright solitary waves are constructed and numerically verified.

Energy switching occurs during solitary wave collisions due to nonparaxial effects.

Abstract

The integrability nature of a nonparaxial nonlinear Schr\"odinger (NNLS) equation, describing the propagation of ultra-broad nonparaxial beams in a planar optical waveguide, is studied by employing the Painlev\'e singularity structure analysis. Our study shows that the NNLS equation fails to satisfy the Painlev\'e test. Nevertheless, we construct one bright solitary wave solution for the NNLS equation by using the Hirota's direct method. Also, we numerically demonstrate the stable propagation of the obtained bright solitary waves even in the presence of an external perturbation in a form of white noise. We then numerically investigate the coherent interaction dynamics of two and three bright solitary waves. Our study reveals interesting energy switching among the colliding solitary waves due to the nonparaxiality.