Dissipative effects in nonlinear Klein-Gordon dynamics

Abstract

We consider dissipation in a recently proposed nonlinear Klein-Gordon dynamics that admits exact time-dependent solutions of the power-law form ei(kx>wt) q , involving the q-exponential function naturally arising within the nonextensive thermostatistics (ezq ≡ [1∗ (1>q)z]1/(1>q), with ez 1 < ez). These basic solutions behave like free particles, complying, for all values of q, with the de Broglie-Einstein relations p < >k, E < >> and satisfying a dispersion law corresponding to the relativistic energy-momentum relation E2 < c2p2 ∗ m2c4. 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 Schrodinger equation, and the power-law diffusion (porousmedia) equation. The associated dynamics exhibits physically appealing traveling solutions of the q-plane wave form with a complex frequency > and a q-Gaussian square modulus profile.