Spectra of Short Pulse Solutions of the Cubic-Quintic Complex Ginzburg-Landau Equation near Zero Dispersion

We describe a computational method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic-quintic complex Ginzburg-Landau equation about numerically determined stationary solutions. We compare the results of the method to a formula for an edge bifurcation obtained using the small dissipation perturbation theory of Kapitula and Sandstede. This comparison highlights the importance for analytical studies of perturbed nonlinear wave equations of using a pulse ansatz in which the phase is not constant, but rather depends on the perturbation parameter. In the presence of large dissipative effects, we discover variations in the structure of the spectrum as the dispersion crosses zero that are not predicted by the small dissipation theory. In particular, in the normal dispersion regime we observe a jump in the number of discrete eigenvalues when a pair of real eigenvalues merges with the intersection point of the two branches of the continuous spectrum. Finally, we contrast the method to computational Evans function methods.