Collapse for the higher-order nonlinear Schr\"odinger equation
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.
Cite
@article{arxiv.1505.04378,
title = {Collapse for the higher-order nonlinear Schr\"odinger equation},
author = {V. Achilleos and S. Diamantidis and D. J. Frantzeskakis and T. P. Horikis and N. I. Karachalios and P. G. Kevrekidis},
journal= {arXiv preprint arXiv:1505.04378},
year = {2015}
}
Comments
19 pages, 10 figures. To appear in Physica D