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.

where W : R2 → R is positive function with three local minima. A similar result was proved by P.Sternberg in [17] in the case that W has two minima. Moreover, Bronsard, Gui and Schatzman ([5]) proved existence of solution to (1)-(2) when W is equivariant by the symmetry group of the equilateral triangle. Our interest in this problem is originated in some models of three-boundary motion. Materia...

A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg–Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extende...

In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg–Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feed...

This paper is a continuation of [ER], where a model of interface dynamics was analyzed. This model is based on the Ginzburg-Landau equation in an unbounded one-dimensional domain. A similar model had originally been studied on a finite interval subject to Neumann boundary conditions by J. Carr and R.L. Pego, [CP1,CP2]. For a physical motivation and a discussion of related models, see Bray, [B],...

We study the initial boundary value problem for a time-dependent Ginzburg-Landau model of superconductivity. First, we prove the uniform boundedness of strong solutions with respect to diffusion parameter > 0 in the case of Coulomb gauge for 2D case. Our second result is the uniqueness of axially symmetric weak solutions in 3D with L2 initial data under Lorentz gauge. Mathematics Subject Classi...

We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg-Landau equation. We then proceed to show that this approximation also extends to the invariant measures of these equations.