We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.

We have found new dissipative solitons of the complex cubic-quintic Ginzburg-Landau equation with extreme amplitudes and short duration. At certain range of the equation parameters, these extreme spikes appear in pairs of slightly unequal amplitude. The bifurcation diagram of pulse amplitude versus dispersion parameter is constructed. c © 2015 Optical Society of America OCIS codes: 060.5530, 14...

We prove the validity of an averaging principle for a class of systems of slow-fast reactiondiffusion equations with the reaction terms in both equations having polynomial growth, perturbed by a noise of multiplicative type. The models we have in mind are the stochastic Fitzhugh-Nagumo equation arising in neurophysiology and the Ginzburg-Landau equation arising in statistical mechanics.

We study the asymptotic limit of solutions of the Ginzburg-Landau equations in two dimensions with or without magnetic field. We first study the Ginzburg-Landau system with magnetic field describing a superconductor in an applied magnetic field, in the “London limit” of a Ginzburg-Landau parameter κ tending to ∞. We examine the asymptotic behavior of the “vorticity measures” associated to the v...

We prove the validity of an averaging principle for a class of systems of slow-fast reaction-diffusion equations with the reaction terms in both equations having polynomial growth, perturbed by a noise of multiplicative type. The models we have in mind are the stochastic Fitzhugh– Nagumo equation arising in neurophysiology and the Ginzburg–Landau equation arising in statistical mechanics.

This paper deals with the numerical analysis of the focalization of a beam of particles. In particular, this model can be useful to check whether or not the cut-off Boltzmann equation leads to some kind of smoothing effect as for the Fokker-Planck-Landau equation.

A model Boltzmann equation (see formulas (1.1.6) { (1.1.9) below) without Grad's angular cutoo assumption is considered. One proves 1. the instantaneous smoothing in both position and velocity variables by the evolution semigroup associated to the Cauchy problem for this model; 2. the derivation of the analogue of the Landau-Fokker-Planck equation in the limit when grazing collisions prevail.

The Ginzburg}Landau model for the radiation "eld of a free-electron laser (FEL) was originally derived for a high-gain ampli"er (S.Y. Cai and A. Bhattacharjee, Phys. Rev. A 43 (1991) 6934). With a view to making precise comparisons with experimental data from the long-pulse FEL oscillator at the University of California at Santa Barbara (UCSB) (L.R. Elias, G. Ramian, J. Hu, A. Amir, Phys. Rev. ...

It is shown that the complex Ginzburg-Landau (CGL) equation on the real line admits nontrivial 2-periodic vortex solutions that have 2n simple zeros (\vortices") per period. The vortex solutions bifurcate from the trivial solution and inherit their zeros from the solution of the linearized equation. This result rules out the possibility that the vortices are determining nodes for vortex solutio...

We have found a dissipative soliton resonance which applies to nonlinear dynamical systems governed by the complex cubic-quintic Ginzburg-Landau equation. Specifically, for particular values of the equation parameters, the soliton energy increases indefinitely. These equation parameters can easily be found using approximate methods, and the results agree very well with numerical ones. The pheno...