Generalized and new solutions of the NRT nonlinear Schr\"odinger equation

TL;DR

This paper derives new analytical solutions for the NRT nonlinear Schrödinger equation using Lie symmetry reductions, including elliptic, Bessel, and inverse error functions, and presents a general solution for any nonlinearity index.

Contribution

The paper introduces a comprehensive set of solutions for the NRT nonlinear Schrödinger equation, including a general solution for any nonlinearity index using generalized trigonometric functions.

Findings

Solutions involving elliptic, Bessel, and inverse error functions.

A closed-form general solution for any nonlinearity index.

Application of Lie symmetry reductions to derive solutions.

Abstract

In this paper we present new solutions of the non-linear Schr\"oodinger equation proposed by Nobre, Rego-Monteiro and Tsallis for the free particle, obtained from different Lie symmetry reductions. Analytical expressions for the wave function, the auxiliary field and the probability density are derived using a variety of approaches. Solutions involving elliptic functions, Bessel and modified Bessel functions, as well as the inverse error function are found, amongst others. On the other hand, a closed-form expression for the general solution of the traveling wave ansatz (see Bountis and Nobre) is obtained for any real value of the nonlinearity index. This is achieved through the use of the so-called generalized trigonometric functions as defined by Lindqvist and Dr\'abek, the utility of which in analyzing the equation under study is highlighted throughout the paper.