Nonlinear Wave Equations Related to Nonextensive Thermostatistics

Angel Plastino,Roseli Wedemann

doi:10.3390/e19020060

Abstract

We advance two nonlinear wave equations related to the nonextensive thermostatistical formalism based upon the power-law nonadditive S q entropies. Our present contribution is in line with recent developments, where nonlinear extensions inspired on the q-thermostatistical formalism have been proposed for the Schroedinger, Klein–Gordon, and Dirac wave equations. These previously introduced equations share the interesting feature of admitting q-plane wave solutions. In contrast with these recent developments, one of the nonlinear wave equations that we propose exhibits real q-Gaussian solutions, and the other one admits exponential plane wave solutions modulated by a q-Gaussian. These q-Gaussians are q-exponentials whose arguments are quadratic functions of the space and time variables. The q-Gaussians are at the heart of nonextensive thermostatistics. The wave equations that we analyze in this work illustrate new possible dynamical scenarios leading to time-dependent q-Gaussians. One of the nonlinear wave equations considered here is a wave equation endowed with a nonlinear potential term, and can be regarded as a nonlinear Klein–Gordon equation. The other equation we study is a nonlinear Schroedinger-like equation.

Highlights

We introduce and investigate some features of two nonlinear wave equations related to the nonextensive thermostatistical formalism [1,2,3]
The wave function φ( x, t) corresponding to these solutions depends on the spatial coordinate x and on time t through the quantity x − vt. This means that the evolution of the wave function is given by a uniform translation at a constant speed v without change in shape. These exact solutions have the form of q-plane waves, which can be regarded as complex valued analogues of the q-exponential distributions appearing at the core of the nonextensive thermostatistics [2]
We proposed and explored some properties of two nonlinear wave equations admitting exact analytical solutions related to the q-Gaussian form. q-Gaussians play a central role within the nonextensive themrostatistical formalism, and it is of interest to explore all the possible dynamical mechanisms that may lead to q-Gaussians

Summary

Introduction

We introduce and investigate some features of two nonlinear wave equations related to the nonextensive thermostatistical formalism [1,2,3]. The power-law NLFP equation admits exact time-dependent solutions, of the q-Gaussian form. The most important solutions of the NRT equation, which highlight its close connection with the nonextensive thermostatistical formalism, are the q-plane waves These solutions are written in terms of q-exponentials with non-real arguments, φ( x, t). Our aim in the present effort is to introduce two wave equations related to the nonextensive thermostatistics that, in contrast with the NRT equation, admit exact analytical time-dependent solutions involving q-Gaussian functions. This is a property that the wave equations advanced here share with the NLFP equation. The proposed wave equations differ from the NLFP in one essential aspect: they describe a conservative dynamics

Nonlinear Wave Equation with q-Gaussian Solutions

Conclusions