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.