Optical-fiber transmission of pulses can be modeled with the complex Ginzburg–Landau equation. We find novel stable soliton pairs and trains, which are relevant in this case, and analyze them. We suggest that the distance between the pulses and the phase difference between them is defined by energy and momentum balance equations, rather than by equations of standard perturbation theory. We pres...

We present evidence for a deep connection between the zero-temperature coarsening of both the two-dimensional time-dependent Ginzburg-Landau equation and the kinetic Ising model with critical continuum percolation. In addition to reaching the ground state, the time-dependent Ginzburg-Landau equation and kinetic Ising model can fall into a variety of topologically distinct metastable stripe stat...

Results from a comprehensive analytical and numerical study of nonequilibrium dynamics in the two-dimensional complex Ginzburg-Landau equation have been presented. In particular, spiral defects have been used to characterize the domain growth law and the evolution morphology. An asymptotic analysis of the single-spiral correlation function shows a sequence of singularities-analogous to those se...

One-dimensional Ginzburg–Landau equations with derivatives of noninteger order are considered. Using psi-series with fractional powers, the solution of the fractional Ginzburg–Landau (FGL) equation is derived. The leading-order behaviours of solutions about an arbitrary singularity, as well as their resonance structures, have been obtained. It was proved that fractional equations of order α wit...

In this paper, two methods are employed to investigate for which values of the parameters, if any, the two-dimensional real Landau-Ginzburg equation possesses the Painlevé property. For an ordinary differential equation to have the Painlevé property, all of its solutions must be meromorphic but for partial differential equations there are two inequivalent definitions, one a direct investigation...

Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [22] as phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the f...

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. ...

The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to secondorder phase transitions, from superconductivity, superfluidity and Bose-Einstein condensation to liquid crystals and strings in field theory. Our goal is to give an overview of various phenomena described the c...

The Ginzburg–Landau equation has been used as a mathematical model for various pattern formation systems in mechanics, physics and chemistry. In this paper, we study the complex Ginzburg–Landau equation in two spatial dimensions with periodical boundary conditions. The method numerically approximates the solution by collocation method based on radial basis functions (RBFs). To improve the numer...

We study a parametrically driven Ginzburg-Landau equation model with nonlinear management. The system is made of laterally coupled long active waveguides placed along a circumference. Stationary solutions of three kinds are found: periodic Ising states and two types of Bloch states, staggered and unstaggered. The stability of these states is investigated analytically and numerically. The nonlin...