Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

Abstract

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter κ is analyzed, when the external force is periodic in space and given by f(x) = r cos(Kx), both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force λ = 2π/K, and the soliton width ls, which is a function of the nonlinearity κ as well as the initial frequency ω0 of the solitary wave, there is a transition (at fixed ω0) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter κ. We previously studied this phenomena when κ = 1 (2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for λ ≫ ls the soliton oscillates in an effective potential, while for λ ≪ ls it moves uniformly as a free particle. In this paper we focus on the κ dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when κ = 1/2,1,2 at fixed value of the frequency ω0. Since the solitary wave gets narrower for fixed ω0 as a function of κ, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase κ.