Instability mechanism for bright solitary-wave solutions to the cubic–quintic Ginzburg–Landau equation

We address the stability of solitary waves to the complex cubic–quintic Ginzburg–Landau equation near the nonlinear Schrödinger limit. It is shown that the adiabatic method does not capture all possible instability mechanisms. The solitary wave can destabilize owing to discrete eigenvalues that move out of the continuous spectrum upon adding nonintegrable perturbations to the nonlinear Schrödinger equation. If an eigenvalue does move out of the continuous spectrum, then we say that an edge bifurcation has occurred. We present a novel analytical technique that allows us to determine whether eigenvalues arise in such a fashion, and if they do, to locate them. Using this approach, we show that Hopf bifurcations can arise in the cubic–quintic Ginzburg–Landau equation.