Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation

A nonlinear theory describing the long-term dynamics of unstable solitons in the generalized nonlinear Schr\"odinger (NLS) equation is proposed. An analytical model for the instability-induced evolution of the soliton parameters is derived in the framework of the perturbation theory, which is valid near the threshold of the soliton instability. As a particular example, we analyze solitons in the NLS-type equation with two power-law nonlinearities. For weakly subcritical perturbations, the analytical model reduces to a second-order equation with quadratic nonlinearity that can describe, depending on the initial conditions and the model parameters, three possible scenarios of the long-term soliton evolution: (i) periodic oscillations of the soliton amplitude near a stable state, (ii) soliton decay into dispersive waves, and (iii) soliton collapse. We also present the results of numerical simulations that confirm excellently the predictions of our analytical theory. {} 1996 The American Physical Society.