Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations

Abstract

The propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear Schrödinger equation (NLS). When considering realistic fibers one must examine perturbed NLS equations, with the particular perturbation depending on the physical situation that is being modeled. A common example is the complex Ginzburg-Landau equation (CGL), which is a dissipative perturbation. It is known that some of the stable bright solitons of the NLS survive a dissipative perturbation such as the CGL. Given that a wave persists, it is then important to determine its stability with respect to the perturbed NLS. A major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum. Since the essential spectrum of the NLS is located on the imaginary axis, such eigenvalues may lead to an unstable wave. In fact, eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum. Here we present a technique which can be used to track these bifurcating eigenvalues. As a consequence, we are able to locate the spectrum for bright solitary-wave solutions to various perturbed nonlinear Schrödinger equations, and determine precise conditions on parameters for which the waves are stable. In addition, we show that a particular perturbation, the parametrically forced NLS equation, supports stable multi-bump solitary waves. The technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here.