Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

Abstract

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter $\kappa$ 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 $\lambda =2\pi /K$, and the soliton width $l_s$, which is a function of the nonlinearity $\kappa$ as well as the initial frequency ${\omega }_0$ of the solitary wave, there is a transition (at fixed ${\omega }_0$) from trapped to unbound behavior of the soliton, which depends on the parameters $r$ and $K$ of the external force and the nonlinearity parameter $\kappa$. We previously studied this phenomena when $\kappa =1$ (2019 J. Phys. A: Math. Theor. {\bf 52} 285201) where we showed that for $\lambda \gg l_s$ the soliton oscillates in an effective potential, while for $\lambda \ll l_s$ it moves uniformly as a free particle. In this paper we focus on the $\kappa$ 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 $\kappa =1/2,1,2$ at fixed value of the frequency ${\omega }_0$. Since the solitary wave gets narrower for fixed ${\omega }_0$ as a function of $\kappa$, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase $\kappa$.

arXiv comment13 pages, 4 figures

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• Solitary wave solutions of the 2+1 and 3+1 dimensional nonlinear Dirac equation constrained to planar and space curves