Dissipative Effects in Nonlinear Klein-Gordon Dynamics

TL;DR

This paper investigates dissipative effects in a nonlinear Klein-Gordon framework that admits soliton solutions with properties consistent with relativistic relations, unifying several nonlinear wave and diffusion equations.

Contribution

It introduces a nonlinear evolution equation combining Klein-Gordon, Schrödinger, and diffusion dynamics, incorporating dissipation and $q$-exponential solutions within nonextensive thermostatistics.

Findings

Solutions behave like free particles with relativistic relations.

The equation unifies Klein-Gordon, Schrödinger, and diffusion equations.

The dynamics support soliton-like traveling solutions with $q$-Gaussian profiles.

Abstract

We consider dissipation in a recently proposed nonlinear Klein-Gordon dynamics that admits soliton-like solutions of the power-law form $e_q^{i(kx-wt)}$, involving the $q$-exponential function naturally arising within the nonextensive thermostatistics [$e_q^z \equiv [1+(1-q)z]^{1/(1-q)}$, with $e_1^z=e^z$]. These basic solutions behave like free particles, complying, for all values of $q$, with the de Broglie-Einstein relations $p=\hbar k$, $E=\hbar \omega$ and satisfying a dispersion law corresponding to the relativistic energy-momentum relation $E^2 = c^2p^2 + m^2c^4 $. The dissipative effects explored here are described by an evolution equation that can be regarded as a nonlinear version of the celebrated telegraphists equation, unifying within one single theoretical framework the nonlinear Klein-Gordon equation, a nonlinear Schroedinger equation, and the power-law diffusion (porous media) equation. The associated dynamics exhibits physically appealing soliton-like traveling solutions of the $q$-plane wave form with a complex frequency $\omega$ and a $q$-Gaussian square modulus profile.