Collapse for the higher-order nonlinear Schrödinger equation

• Conditions for collapse of a higher order nonlinear Schrödinger equation are derived.
• Collapse dynamical scenarios for a variety of initial excitations are demonstrated.
• A critical value of the linear gain, separating the possible decay of solutions from collapse is identified.
• The role of the higher-order effects to the transient dynamics is revealed.

We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schrödinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.