Sundman-like transformations and the NRT nonlinear Schrödinger equation
P. R. Gordoa, A. Pickering, D. Puertas-Centeno, E. V. Toranzo
TL;DR
This work introduces a four-parameter generalization of Sundman transformations that act on both a function and its derivative, together with an inverse, to transform and solve autonomous ODEs and to analyze the NRT nonlinear Schrödinger equation. The authors show that certain autonomous second-order equations can be mapped to autonomous first-order equations in transformed variables, enabling quadrature-based solutions, and they demonstrate this framework on traveling-wave reductions of the NRT equation. A central contribution is expressing Lorentzian traveling-wave solutions for the NRT equation with arbitrary nonlinearity parameter $q$ as GSTs of linear or simpler nonlinear solutions, and showing that composing a GST with its inverse shifts the nonlinearity parameter $q$. Additionally, a β=0 subfamily of GSTs forms a group under composition and connects to generalized trigonometric functions, suggesting deep algebraic structure and potential for broader analytic applications. The results provide a unified machinery to encode nonlinear effects into transformations, facilitate exact solutions, and set the stage for extensions to higher-order ODEs and wider NRT models.
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 Schrödinger equation. More precisely, we employ them to analyze a family of ordinary differential equations which includes the Lorentzian solutions of the NRT-nonlinear Schrödinger 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.