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

Abstract

Abstract.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 in Kapitula and Sandstede [35] and Kapitula [32], 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.

Dark solitons in external potentials

“…Definition 3.13 Let φ ǫ (x) be the kink mode of Definition 2.8. The kink mode is said to be spectrally unstable in the time evolution of the GP equation (1.2) if there exists an eigenvector (u, w) ∈ L 2 (R, C 2 ) of the spectral problem 19) for the eigenvalue λ with Re(λ) > 0, where…”

“…Among others, several analytical results were important in the development of dark solitons in recent years: perturbation theory based on renormalized power [24] and momentum [23], orbital stability of dark solitons [1,2], completeness of eigenfunctions in the cubic NLS [8,15], inverse scattering for the vector cubic NLS equation [41], construction of the Evans function for dark solitons in the perturbed cubic NLS [19], asymptotic analysis of the radiation and dynamics of dark solitons [26,27,35], and spectral analysis of transverse instabilities of one-dimensional dark solitons [25,36].…”

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

“…This has been investigated in a variety of previous studies, including [Leg01,PSAK95,CM92,KR00,SS05,LF97]. Partial analytical results can be found in [KR00,SS05]. Numerical and asymptotic evidence in [CM92,PSAK95] suggests that the sources are stable in an open region of parameter space near the NLS limit of (1.1), which corresponds to the limit |α|, |β| → ∞ and γ 1 , γ 2 → 0.…”

“…To find a spectrally stable source, one needs to find parameter values for which both the unstable eigenvalue (from the rGL limit) and the perturbed zero eigenvalue (from the cCGL limit) become stable. This has been investigated in a variety of previous studies, including [Leg01,PSAK95,CM92,KR00,SS05,LF97]. Partial analytical results can be found in [KR00,SS05].…”

Absolute instabilities of standing pulses

“…The stability of the Nozaki-Bekki holes has been investigated in [7]; however, the result stated in [7] is incorrect, and we shall give a corrected statement, and its proof, in Appendix A.…”