Dissipative effects in nonlinear Klein-Gordon dynamics

Abstract

We consider dissipation in a recently proposed nonlinear Klein-Gordon dynamics that admits soliton-like solutions of the power-law form e i(kx−wt) q , involving the q-exponential function naturally arising within the nonextensive thermostatistics [e z q ≡ [1 + (1 − q)z] 1/(1−q) , with e z 1 = e z ]. These basic solutions behave like free particles, complying, for all values of q, with the de Broglie-Einstein relations p = k, E = ω and satisfying a dispersion law corresponding to the relativistic energymomentum 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 ω and a q-Gaussian square modulus profile.

Lorentzian solitary wave in a generalised nonlinear Schrödinger equation

“…We have found a travelling-wave solution of a generalised NLSE (6) with an additional term Γ(ψ(x, t)) = λψ(x, t) q and of its coupled equation (7). The density of energy consistent with this generalised NLSE (6) is obtained from classical field theory (16). For specific values of the constants of this solution, the density of energy (18) behaves as a Lorentzian solitary wave (22) of total energy E s = 2 1+q 2 k 2 2m , with q > −1, q = 1, and momentum P s = k. Notice that the total energy of this solution as well as the velocity of the solitary wave depends on the value of q.…”

New research based on the new high-order generalized uncertainty principle for Klein–Gordon equation

“…By substituting equation (11) into equation (10) and considering the new high-order generalized uncertainty principle, the above equation can be rewritten as where the parameters κ and δ have the following form:…”

Analytic approaches of the anomalous diffusion: A review

“…On the other hand the q-exponential function has had several implications for diffusive phenomena as well as for other contexts in physics. In particular, Tsallis and others have used this formalism to find a class of solutions for nonlinear versions of the Schrödinger, Klein-Gordon, and Dirac equations [86,87]. The q-exponential ansatz was presented in literature in 1988 by physicist C. Tsallis in Ref.…”