Nonlinear Dirac equation solitary waves under a spinor force with different components

Abstract: We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction in the presence of external forces as well as damping of the form $\gamma^0 f(x,t) - i \mu \gamma^0 \Psi$, where both $f, {f_j = r_i e^{i K_j x} }$ and $\Psi$ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. In our previous paper we assumed $K_j=K, ~ j=1,2$ which allowed a transformation to a simplifying coordinate system, and we also assumed the "small" component of the external force was zero. Here we include the effects of the small component and also the case $K_1 \neq K_2$ which dramatically modifies the behavior of the solitary wave in the presence of these external forces.