Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schr\"{o}dinger equation with time-dependent nonlinearity

TL;DR

This paper investigates a nonlinear Schrödinger equation with time-dependent nonlinearity, revealing its integrability, deriving explicit soliton solutions, analyzing their interactions, and studying modulational instability under perturbations.

Contribution

It establishes the complete integrability of the equation, constructs explicit multi-soliton solutions, and analyzes their stability and interactions.

Findings

The equation admits an infinite number of conservation laws.

Explicit bright multi-soliton solutions are derived.

Modulational instability is analyzed under small perturbations.

Abstract

In this paper, we consider a general form of nonlinear Schr\"{o}dinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schr\"{o}dinger equation is identified by admitting an infinite number of conservation laws. Using the Darboux transformation method, we obtain some explicit bright multi-soliton solutions in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.