An analytical approach to nucleation solutions and pulses in the one-dimensional real and complex Ginzburg-Landau equations

Abstract

We study the one-dimensional supercritical real and complex subcritical Ginzburg- Landau equations. In the real case, with periodic boundary conditions in a finite box, we construct analytically nucleation solutions allowing transitions between stable plane waves. Moreover we construct vortex solutions and study the Eckhaus bifurcation in a finite box. The Lyapounov functional for the above stationary solutions is also discussed. In the complex case we study stationary localized solutions in the region where there exist coexistence of homogeneous attractors. Using a matching approach we construct analytically pulses and show that their appearance is related to a saddle-node bifurcation. Our approximation scheme presented here is valid through the whole intermediate range of parameters between the variational and conservative limit. We perform numerical simulations which are in good agreement with our theoretical predictions. This review is based on the original works [42–44]. KeywordsPlane WaveContinuous LineBifurcation CurveLandau EquationPhase GradientThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.