The paper deals with the nonlinear Schrödinger equation perturbed by a nonlinear dissipative term, which, however, conserves the plasmon number (wave action). Such terms describe the nonlinear Landau damping in plasmas or the Raman-scattering-induced dissipation in a nonlinear optical medium. The perturbing term drives a soliton as a constant force, leaving its amplitude unchanged. It is demonstrated that collisions between the driven solitons give rise to radiative losses of the plasmon number. Next, the kinetics of a rarefied soliton gas is analyzed, and it is shown that collisions result in complete decay of solitons into radiation. An asymptotic law of the decay is found.