Dissipative effects in nonlinear Klein-Gordon dynamics

We consider dissipation in a recently proposed nonlinear Klein-Gordon dynamics that admits exact time-dependent 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 generalization of the celebrated telegraph equation, unifying within one single theoretical framework the nonlinear Klein-Gordon equation, a nonlinear Schrödinger equation, and the power-law diffusion (porous-media) equation. The associated dynamics exhibits physically appealing traveling solutions of the q-plane wave form with a complex frequency ω and a q-Gaussian square modulus profile