Quintero, Niurka R.; Bernardo Sanchez-Rey; Fred Cooper and Franz G. Mertens
Soliton dynamics in the damped and parametrically driven nonlinear Dirac equation, in 1 + 1 dimension with scalar-scalar self-interaction is analysed. The considered parametric force has the spatial period λ. A variational approach using collective coordinates for studying the time dependent response of the solitary waves to this parametric force is developed. The dynamical equations for the collective coordinates are also obtained by an alternative method, namely the method of moments. The soliton dynamics depends crucially on the competition between two length scales: the spatial period λ and the width of the soliton ls. For λ ≫ ls the soliton oscillates in an effective potential, while for λ ≪ ls it moves uniformly as a free particle. The transition between these two regimes occurs when λ is comparable to the soliton width. This match enhances the soliton instabilities so that even small values of the perturbation are enough to modify drastically the soliton shape and destroy it for long times.