Nonlinear Wave Equations Related to Nonextensive Thermostatistics

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.

Keywords

• nonlinear wave equations
• nonextensive entropies
• nonextensive thermostatistics