Using a Lagrangian approach, which is generalized by the introduction of a dissipation function, we develop a Collective-Coordinate-Theory which yields ODEs for the collective coordinates position, velocity, amplitude and phase of the soliton. The ODEs are solved analytically and numerically for the following cases: constant, harmonic and biharmonic K(t). In the first case the spatial average of f(x) vanishes, nevertheless the soliton performs on the average a unidirectional motion. Here the amplitude of oscillations around the average motion is much smaller than the period of f(x). In the biharmonic case the soliton performs a ratchet motion.

We conjecture a new stability criterion: if the curve P(V), where P(t) and V(t) are the soliton momentum and velocity, has a branch with negative slope, the soliton is predicted to become unstable which is confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the shorter the branch with negative slope is, the longer the soliton lives.