Sundman-like transformations and the NRT nonlinear Schr\"odinger equation

TL;DR

This paper introduces a generalized Sundman transformation involving derivatives, applied to analyze solutions of the NRT nonlinear Schrödinger equation, including Lorentzian solitary waves, and reveals a group structure in parameter shifts.

Contribution

It develops a new class of transformations that encapsulate nonlinear behaviors and enables parameter shifts, applied specifically to the NRT nonlinear Schrödinger equation solutions.

Findings

Explicit Lorentzian solutions for all q values

Transformation-based parameter shift in solutions

Identification of a group structure in transformations

Abstract

We present a new generalization of the well-known power-type Sundman transformation, involving not only powers of the function but also of its derivative, along with its inverse. Our aim is to explore the use of such transformations in the derivation of solutions of ordinary differential equations and in the study of their properties. We then show their usefulness in the framework of the nonlinear Nobre--Reigo-Monteiro--Tsallis (NRT) nonlinear Schr\"odinger equation. More precisely, we employ them to analyze a family of ordinary differential equations which includes the Lorentzian solutions of the NRT-nonlinear Schr\"odinger equation for a constant potential. Moreover, an explicit expression for the Lorentzian solitary wave solutions is given, for any real value of the non-linearity parameter q, in terms of a transformation depending on q applied to the classical Lorentzian solution with q = 1, i.e., we succeed in encapsulating the whole nonlinear behavior in the new transformations. In addition, the composition of this transformation with its inverse (with different parameters) allows us to perform a shift in the nonlinearity parameter q. Moreover, a certain subfamily of our generalized transformations, which perform a shift on the non-linearity parameter q of the Lorentzian solutions, is found to have a group structure. The same subfamily of transformations allows us, again, to perform a shift in the non-linearity parameter q, but in this case in the traveling wave solution for a free particle.