Existence and stability of standing hole solutions to complex Ginzburg–Landau equations

We consider the existence and stability of the hole, or dark soliton, solution to a Ginzburg–Landau perturbation of the defocusing nonlinear Schrödinger equation (NLS), and to the nearly real complex Ginzburg–Landau equation (CGL). By using dynamical systems techniques, it is shown that the dark soliton can persist as either a regular perturbation or a singular perturbation of that which exists for the NLS. When considering the stability of the soliton, a major difficulty which must be overcome is that eigenvalues may bifurcate out of the continuous spectrum, i.e. an edge bifurcation may occur. Since the continuous spectrum for the NLS covers the imaginary axis, and since for the CGL it touches the origin, such a bifurcation may lead to an unstable wave. An additional important consideration is that an edge bifurcation can happen even if there are no eigenvalues embedded in the continuous spectrum. Building on and refining ideas first presented by Kapitula and Sandstede (1998 Physica D 124 58–103) and Kapitula (1999 SIAM J. Math. Anal. 30 273–97), we use the Evans function to show that when the wave persists as a regular perturbation, at most three eigenvalues will bifurcate out of the continuous spectrum. Furthermore, we precisely track these bifurcating eigenvalues, and thus are able to give conditions for which the perturbed wave will be stable. For the NLS the results are an improvement and refinement of previous work, while the results for the CGL are new. The techniques presented are very general and are therefore applicable to a much larger class of problems than those considered here. AMS classification scheme numbers: 30B10, 30B40, 34A05, 34A26, 34A47, 34C35, 34C37, 34D15, 34E05, 35K57, 35P15, 35Q51, 35Q55, 78A60