Bifurcating bright and dark solitary waves for the perturbed cubic-quintic nonlinear Schrödinger equation

The existence of bright and dark multi-bump solitary waves for Ginzburg–Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three-dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N -bump dark solitary waves ( N ≧ 1) is shown via an application of the Exchange Lemma with Exponentially Small Error. N -bump bright solitary waves are shown to exist as a consequence of the work of Kapitula and Maier-Paape.