Stability and evolution of solitary waves in perturbed generalized nonlinear Schrodinger equations

In this paper, we study the stability and evolution of solitary waves in pe rturbed generalized nonlinear Schrodinger (NLS) equations. Our method is ba sed on the completeness of the bounded eigenstates of the associated linear operator in L-2 space and a standard multiple-scale perturbation technique . Unlike the adiabatic perturbation method, our method details all instabil ity mechanisms caused by perturbations of such equations and explicitly spe cies when such instabilities will occur. In particular, our method uncovers the instability caused by bifurcation of nonzero discrete eigenvalues of t he linearization operator. As an example, we consider the perturbed cubic-q uintic NLS equation in detail and determine the stability regions of its so litary waves. In the instability region, we also specify where the solitary waves decay, collapse, develop moving fronts, or evolve into a stable spat ially localized and temporally periodic state. The generalization of this m ethod to other perturbed nonlinear wave systems is also discussed.