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.     

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.     

Existence of standing waves for the complex Ginzburg–Landau equation

We study the existence of standing wave solutions of the complex Ginzburg–Landau equation

equation(GL)

_φt−e^iθ(ρI−Δ)φ−e^iγ|φ|^αφ=0${\phi }_t-e^{i\theta }(\rho I-\mathrm{\Delta })\phi -e^{i\gamma }{|\phi |}^{\alpha }\phi =0$

in R^N${\mathbb{R}}^N$, where α>0$\alpha >0$, (N−2)α<4$(N-2)\alpha <4$, ρ>0$\rho >0$ and θ,γ∈R$\theta ,\gamma \in \mathbb{R}$. We show that for any θ∈(−π/2,π/2)$\theta \in (-\pi /2,\pi /2)$ there exists ε>0$\epsilon >0$ such that (GL) has a non-trivial standing wave solution if |γ−θ|<ε$|\gamma -\theta |<\epsilon$. Analogous result is obtained in a ball Ω∈R^N$\Omega \in {\mathbb{R}}^N$ for ρ>−_λ1$\rho >-{\lambda }_1$, where _λ1${\lambda }_1$ is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.     

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.     

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.     

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.     

A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation

In this paper, we study a linearized Crank–Nicolson Galerkin finite element method for solving the nonlinear fractional Ginzburg–Landau equation. The boundedness, existence and uniqueness of the numerical solution are studied in details. Then we prove that the optimal error estimates hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Finally, numerical tests are investigated to support our theoretical analysis.     

Random attractors for a Ginzburg–Landau equation with additive noise

The existence of a compact random attractor for the random dynamical system generated by the complex Ginzburg–Landau equation with additive white noise has been proved. And a precise estimate of the upper bound of the Hausdorff dimension of the random attractor is obtained.     

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.     

Nehari method for the generalized Ginzburg–Landau system

The Dirichlet problem for the generalized Ginzburg–Landau system is considered. The existence of positive vector solutions is proved in the following three cases: (1) the cross term has weak growth; (2) the interaction constant is large enough; and (3) the cross term has strong growth and the interaction constant is positive and close to zero.     

Local bifurcations and a global attractor for two versions of the weakly dissipative Ginzburg–Landau equation

Theoretical and Mathematical Physics - We consider the periodic boundary value problem for two variants of a weakly dissipative complex Ginzburg–Landau equation. In the first case, we study a...     

Inviscid limit for the derivative Ginzburg–Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation $u_t=(\nu +\mathrm{i})\mathrm{\Delta }u+{\overrightarrow{\lambda }}_1?\mathrm{?}({|u|}^{2}u)+({\overrightarrow{\lambda }}_2?\mathrm{?}u){|u|}^2+\alpha {|u|}^{2\delta }u$, where $\delta \in \mathbb{N}$, ${\overrightarrow{\lambda }}_1,{\overrightarrow{\lambda }}_2$ are complex constant vectors, $\nu \in [0,1]$, $\alpha \in \mathbb{C}$. For $n\ge 3$, we show that it is uniformly global well posed for all $\nu \in [0,1]$ if initial data $u_0$ in modulation space $M_{2,1}^s$ and Sobolev spaces $H^{s+n/2}$ ($s>3$) and $6u_0{6}_{L^2}$ is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in $C(0,T;L^2)$ if $\nu \to 0$ and $u_0$ in $M_{2,1}^s$ or $H^{s+n/2}$ with $s>4$. For $n=2$, we obtain the local well-posedness results and inviscid limit with the Cauchy data in $M_{1,1}^s$ ($s>3$) and $6u_0{6}_{L^1}?1$.