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Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

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 $f(x) =r\cos(K x)$, 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 $λ=2 π/K$, and the soliton width $l_s$, which is a function of the nonlinearity $κ$ as well as the initial frequency $ω_0$ of the solitary wave, there is a transition (at fixed $ω_0$) from trapped to unbound behavior of the soliton, which depends on the parameters $r$ and $K$ of the external force and the nonlinearity parameter $κ$. We previously studied this phenomena when $κ=1$ (2019 J. Phys. A: Math. Theor. {\bf 52} 285201) where we showed that for $λ\gg l_s$ the soliton oscillates in an effective potential, while for $λ\ll l_s$ 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 $ω_0$. Since the solitary wave gets narrower for fixed $ω_0$ as a function of $κ$, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase $κ$.