Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse

Abstract

The sine-Gordon (SG) equation and perturbed nonlinear Schrödinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called light bullets) in nonlinear dispersive optical media. We begin with the (2 + 1) SG equation obtained as an asymptotic reduction in the two level dissipationless Maxwell-Bloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing-defocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet solutions of SG and perturbed NLS as well as critical NLS are reported, which validate that the solution of the perturbed NLS as well as its finite-term truncations are in qualitative and quantitative agreement with the solution of SG for the light bullets propagation even after the critical collapse of cubic focusing NLS. In contrast, standard critical NLS is in qualitative agreement with SG only before its collapse. As a benefit of such observations, pulse propagations are studied via solving the perturbed NLS truncated by reasonably many nonlinear terms, which is a much cheaper task than solving SG equation directly. © 2010 Elsevier B.V. All rights reserved.