Traveling Solitons in the Damped-driven Nonlinear Schrödinger Equation∗

The well-known effect of the linear damping on the moving nonlinear Schrödinger soliton (even when there is a supply of energy via the spatially homogeneous driving) is to quench its momentum to zero. Surprisingly, the zero momentum does not necessarily mean zero velocity. We show that two or more parametrically driven damped solitons can form a complex traveling with zero momentum at a nonzero constant speed. All traveling complexes we have found so far have turned out to be unstable. Thus, the parametric driving is capable of sustaining the uniform motion of damped solitons, but some additional agent is required to stabilize it.