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Generalized and new solutions of the NRT nonlinear Schrödinger equation

P. R. Gordoa, A. Pickering, D. Puertas-Centeno, E. V. Toranzo

TL;DR

The paper addresses the free-particle NRT-NLSE by employing Lie point symmetries to generate four similarity reductions, yielding new exact solutions expressed through Generalized Trigonometric Functions and other special functions such as elliptic and Bessel functions. A key result is a closed-form traveling-wave solution for any real nonlinearity index $q$, providing explicit forms for the wave $\Psi$, auxiliary field $\Phi$, and probability density $\rho$ via generalized trigonometric structures. The methods include mapping to polynomial ODEs, generalized tangent functions, Sundman-type transformations, and Abel-type reductions, revealing a rich landscape of oscillatory and nontrivial PDFs. These contributions broaden the solvable regime of the NRT-NLSE and offer analytic tools for analyzing nonextensive quantum dynamics, with future work planned for nonzero potentials and physical interpretation.

Abstract

In this paper we present new solutions of the non-linear Schröodinger 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 Drábek, the utility of which in analyzing the equation under study is highlighted throughout the paper.

Generalized and new solutions of the NRT nonlinear Schrödinger equation

TL;DR

The paper addresses the free-particle NRT-NLSE by employing Lie point symmetries to generate four similarity reductions, yielding new exact solutions expressed through Generalized Trigonometric Functions and other special functions such as elliptic and Bessel functions. A key result is a closed-form traveling-wave solution for any real nonlinearity index , providing explicit forms for the wave , auxiliary field , and probability density via generalized trigonometric structures. The methods include mapping to polynomial ODEs, generalized tangent functions, Sundman-type transformations, and Abel-type reductions, revealing a rich landscape of oscillatory and nontrivial PDFs. These contributions broaden the solvable regime of the NRT-NLSE and offer analytic tools for analyzing nonextensive quantum dynamics, with future work planned for nonzero potentials and physical interpretation.

Abstract

In this paper we present new solutions of the non-linear Schröodinger 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 Drábek, the utility of which in analyzing the equation under study is highlighted throughout the paper.

Paper Structure

This paper contains 12 sections, 135 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Generalized and new similarity reductions of the NRT-NLSE
  3. Travelling wave reduction
  4. Mapping to polynomial equations
  5. Solutions in terms of generalized trigonometric functions
  6. Scaling reduction
  7. The $\log(t)$ reduction
  8. The case $\mathbf{c_4=0}$
  9. The case $\mathbf{c_4\neq 0}$
  10. The $\log(x)$ reduction
  11. Conclusions
  12. A short introduction to generalized trigonometric functions

Figures (5)

  • Figure 1: Graphical representation of the traveling wave probability density (Eq. \ref{['eq:travel_density']}) as a function of the variable $x$ and for fixed $t=0$ and for $q=1.9$ ($\circ$) $q=2$ ([0.5ex]8mm0.8pt), $q=2.1$ ([0.5ex]8mm0.5pt3pt), $q=2.2$ ([0.5ex]8mm0.5pt1pt) , $q=2.3$ ([0.5ex]8mm0.5pt1.2mm 1pt 0.25mm 1pt) for $\alpha=\beta=i$, $K_1=sign(q-2)$ the sign function of $q-2$ and $K_2=0$.
  • Figure 2: In this graphic we represent the probability density in Eq. \ref{['eq:density_sec_4a']} as a function of $x$ for fixed $t=1$ and for $q=2$ .
  • Figure 3: In this graphic we represent generalized sine (continuous) and cosine (dashed) functions when $p=q=1.2$
  • Figure 4: In this graphic we represent generalized sine (continuous) and cosine (dashed) functions when $p=1.4$ and $q=2.2$
  • Figure 5: In this graphic we represent generalized sine (continuous) and cosine (dashed) functions when $p=q=3.5$