Existence and stability of 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 Schroedinger 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 in Kapitula and Sandstede (Physica D, 1998) and Kapitula (SIAM J. Math. Anal., 1999), we 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.Comment: 41 pages, 4 figures, submitte

Analyticity of Essentially Bounded Solutions to Semilinear Parabolic Systems and Validity of the Ginzburg-Landau Equation

Some analytic smoothing properties of a general strongly coupled, strongly parabolic semilinear system of order $2m$ in $realnos^D times (0,T)$ with analytic entries are investigated. These properties are expressed in terms of holomorphic continuation in space and time of essentially bounded global solutions to the system. Given $0 < T' < T le infty$, it is proved that any weak, essentially bounded solution ${bold u} = (u_1,dots,u_N)$ in $realnos^Dtimes (0,T)$ possesses a bounded holomorphic continuation $bold u (x+iy,sigma + itau )$ into a region in $complexnos^Dtimescomplexnos$ defined by $(x,sigma )in realnos^Dtimes (T',T)$, $|y| < A'$ and $|tau | < B'$, where $A'$ and $B'$ are some positive constants depending upon $T'$. The proof is based on analytic smoothing properties of a parabolic Green function combined with a contraction mapping argument in a Hardy space $H^infty$. Applications include weakly coupled semilinear systems of complex reaction-diffusion equations such as the complex Ginzburg-Landau equations. Special attention is given to the problem concerning the validity of the derivation of amplitude equations which describe various instability phenomena in hydrodynamics