Collapse for the higher-order nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the conditions under which solutions to a higher-order nonlinear Schr"odinger equation collapse in finite time, highlighting the influence of gain/loss parameters and confirming findings through numerical simulations.

Contribution

It provides a detailed analysis of collapse conditions in a generalized NLS equation with higher-order effects, extending understanding beyond the integrable case.

Findings

Collapse is mainly controlled by linear/nonlinear gain and loss parameters.

A critical linear gain value separates decay from collapse.

Numerical simulations agree with analytical predictions and show long-term stability of localized solutions.

Abstract

We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schr\"odinger (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.