  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.
