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

In this article, dissipative perturbations of the nonlinear Schrödinger equation (NLS) are considered. For dissipative equations, when determining the stability of a solitary wave, one must locate both the point spectrum and the continuous spectrum. If the wave is to be stable, all the spectrum must reside in the left-half plane, except for the translational eigenvalue(s) at the origin. However, for the NLS the continuous spectrum is located on the imaginary axis, as the NLS can be thought of as an infinite-dimensional Hamiltonian system. Since dissipative perturbations will destroy this feature, it is then possible for eigenvalues to bifurcate out of the continuous spectrum and into the right-half plane, leading to an unstable wave. Here we show that the Evans function can be extended across the continuous spectrum, and hence it can be used to track these bifurcating eigenvalues. The extension is done for a general class of equations, and the result should therefore be useful for a larger class of problems than that presented here. Using the extended Evans function, we are then able to locate the spectrum for bright solitary-wave solutions to various perturbed nonlinear Schrödinger equations, and discuss their stability. In addition, we discuss the existence and stability of multi-bump solitary waves for a particular perturbation, the parametrically forced NLS equation