Blow-up results and soliton solutions for a generalized variable coefficient nonlinear Schr\"{o}dinger equation

TL;DR

This paper derives exact analytical solutions, including solitons and blow-up solutions, for a generalized variable coefficient nonlinear Schrödinger equation relevant in optics and quantum fluids, using similarity transformations and multiparameter systems.

Contribution

It introduces a novel multiparameter approach to find explicit solutions with singularities and oscillating behaviors for a generalized nonlinear Schrödinger equation with variable coefficients.

Findings

Existence of explicit solutions with singularities.

Family of oscillating periodic soliton solutions.

Demonstration of bright, dark, and Peregrine-type solitons.

Abstract

In this paper, by means of similarity transformations we study exact analytical solutions for a generalized nonlinear Schr$\ddot{\mbox{o}}$dinger equation with variable coefficients. This equation appears in literature describing the evolution of coherent light in a nonlinear Kerr medium, Bose-Einstein condensates phenomena and high intensity pulse propagation in optical fibers. By restricting the coefficients to satisfy Ermakov-Riccati systems with multiparameter solutions, we present conditions for existence of explicit solutions with singularities and a family of oscillating periodic soliton-type solutions. Also, we show the existence of bright-, dark- and Peregrine-type soliton solutions, and by means of a computer algebra system we exemplify the nontrivial dynamics of the solitary wave center of these solutions produced by our multiparameter approach.