An analysis of spatiotemporal localized solutions in the variable coefficients (3+1)-dimensional nonlinear Schr\"{o}dinger equation with six different forms of dispersion parameters

TL;DR

This paper constructs and analyzes spatiotemporal localized solutions of a (3+1)-dimensional nonlinear Schrödinger equation with variable coefficients, exploring their behavior across six different dispersion profiles to understand rogue wave phenomena.

Contribution

It introduces a similarity transformation method to derive localized solutions for a variable-coefficient (3+1)D nonlinear Schrödinger equation and studies their dynamics across diverse dispersion profiles.

Findings

Localized solutions include rogue waves and breathers with distinct characteristics.

Dispersion profiles significantly influence the trajectory and intensity of localized waves.

Localized solutions coexist with collapsing solutions in the equation.

Abstract

We construct spatiotemporal localized envelope solutions of a (3+1)-dimensional nonlinear Schr\"{o}dinger equation with varying coefficients such as dispersion, nonlinearity and gain parameters through similarity transformation technique. The obtained localized rational solutions can serve as prototypes of rogue waves in different branches of science. We investigate the characteristics of constructed localized solutions in detail when it propagates through six different dispersion profiles, namely constant, linear, Gaussian, hyperbolic, logarithm and exponential. We also obtain expressions for the hump and valleys of rogue wave intensity profiles for these six dispersion profiles and study the trajectory of it in each case. Further, we analyze how the intensity of another localized solution, namely breather, changes when it propagates through the aforementioned six dispersion profiles. Our studies reveal that these localized solutions co-exist with the collapsing solutions which are already found in the (3+1)-dimensional nonlinear Schr\"{o}dinger equation. The obtained results will help to understand the corresponding localized wave phenomena in related fields.