Stability and Evolution of Solitary Waves in Perturbed Generalized Nonlinear Schrödinger Equations

In this paper, we study the stability and evolution of solitary waves in perturbed generalized nonlinear Schrödinger (NLS) equations. Our method is based on the completeness of the bounded eigenstates of the associated linear operator in L2 space and a standard multiple-scale perturbation technique. Unlike the adiabatic perturbation method, our method details all instability mechanisms caused by perturbations of such equations and explicitly specifies when such instabilities will occur. In particular, our method uncovers the instability caused by bifurcation of nonzero discrete eigenvalues of the linearization operator. As an example, we consider the perturbed cubic-quintic NLS equation in detail and determine the stability regions of its solitary waves. In the instability region, we also specify where the solitary waves decay, collapse, develop moving fronts, or evolve into a stable spatially localized and temporally periodic state. The generalization of this method to other perturbed nonlinear wave systems is also discussed.