Null controllability of the complex Ginzburg–Landau equation

The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators.     

Controllability of the Ginzburg–Landau equation

This Note investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Null-controllability results are derived from a Carleman estimate and an analysis based upon the theory of sectorial operators. To cite this article: L. Rosier, B.-Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Dark soliton dynamics under the complex Ginzburg–Landau equation

The dynamical properties of the complex Ginzburg–Landau equation are considered in the defocusing (normal dispersion) regime. It is found that under appropriate conditions stable evolution of dark solitons can occur. These conditions are derived using a newly developed perturbation theory that also reveals an important aspect of the dynamics: the formation of a shelf that accompanies the soliton and is an intricate part of its evolution. Further conditions to suppress this effect are also derived. These analytical predictions are found to be in excellent agreement with direct numerical simulations.     

Well-posedness of the fractional Ginzburg–Landau equation

In this paper, we investigate the well-posedness of the real fractional Ginzburg–Landau equation in several different function spaces, which have been used to deal with the Burgers’ equation, the semilinear heat equation, the Navier–Stokes equations, etc. The long time asymptotic behavior of the nonnegative global solutions is also studied in details.     

Soliton-like solutions for the modified variable-coefficient Ginzburg–Landau equation

This paper is devoted to studying the solutions of the modified Ginzburg–Landau equation with variable coefficient. Based on computerized symbolic computation, several families of new exact dark- and bright-soliton-like solutions are presented. Moreover, we derive some similarity solutions, which can be illustrated in terms of the elliptic and the second kind of Painlevé transcendent equation.     

Finite-time blowup for a complex Ginzburg–Landau equation with linear driving

In this paper, we consider the complex Ginzburg–Landau equation ${u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}$ on ${\mathbb{R}^N}$ , where ${\alpha > 0,\,\gamma \in \mathbb{R}}$ and ${-\pi /2 < \theta < \pi /2}$ . By convexity arguments, we prove that, under certain conditions on ${\alpha,\theta,\gamma}$ , a class of solutions with negative initial energy blows up in finite time.     

Exact chirped solitary-wave solutions for Ginzburg–Landau equation

In this short letter, by applying specially envelope transform and direct ansatz approach to (1 + 1)D Ginzburg–Landau equation the authors obtain a new type of exact solitary wave solution including chirped bright solitary-wave and chirped dark solitary-wave solutions.     

Breather type of chirped soliton solutions for the 2D Ginzburg–Landau equation

New exact solutions including bright soliton solutions, breather and periodic types of chirped soliton solutions, kink-wave and homoclinic wave solutions for the 2D Ginzburg–Landau equation are obtained using the special envelope transform and the auxiliary function method. It is shown that the specially envelope transform and the auxiliary function method provide a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.     

Polynomial mixing for the complex Ginzburg–Landau equation perturbed by a random force at random times

In this paper we study the problem of ergodicity for the complex Ginzburg–Landau (CGL) equation perturbed by an unbounded random kick-force. Randomness is introduced both through the kicks and through the times between the kicks. We show that the Markov process associated with the equation in question possesses a unique stationary distribution and satisfies a property of polynomial mixing.      

Error estimates of the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation

In this paper, we consider the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation. Semidiscrete and fully discrete spectral collocation schemes are given. In the fully discrete case, a three-level spectral collocation scheme is considered. An energy estimation method is used to obtain error estimates for the approximate solutions. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

Recurrent solutions of the derivative Ginzburg–Landau equation with boundary forces

In this paper, we will establish the existence of the bounded solutions, periodic solutions, quasi-periodic solutions and almost periodic solutions for the derivative Ginzburg–Landau equation with time-dependent boundary external forces. A smoothing effect for the semigroup associated with linear Ginzburg–Landau operator is the crucial tool in establishing the main results.