Dissipative Effects in Nonlinear Klein-Gordon Dynamics

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 [$e_q^z\equiv [1+(1-q)z{]}^{1\mathrm{/}(1-q)}$, 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 $p=\mathrm{\hslash }k$, $E=\mathrm{\hslash }\omega$ and satisfying a dispersion law corresponding to the relativistic energy-momentum relation $E^2=c^2{p}^2+m^2{c}^4$. 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.