Regular spatial structures emerge in a wide range of different dynamics characterized by local and/or nonlocal coupling terms. In several research fields this has spurred the study many models, which can explain pattern formation. The modulations patterns, occurring on long and temporal scales, cannot be captured linear approximation analysis. Here, we show that, starting from general model wit...

We consider the discretization of a special class of nonlinear parabolic equations, including the complex Ginzburg–Landau equation, by implicit–explicit multistep methods and establish stability under a best possible linear stability condition.

The variational approach to weakly nonlocal thermodynamic theories is critically revisited in the light of modern nonequilibrium thermody-namics. The example of Ginzburg-Landau equation is investigated in detail.

We report dynamic regimes supported by a sharp quasi-one-dimensional (1D) ("razor"), pyramid-shaped ("dagger"), and conical ("needle") potentials in the 2D complex Ginzburg-Landau (CGL) equation with cubic-quintic nonlinearity. This is a model of an active optical medium with respective expanding antiwaveguiding structures. If the potentials are strong enough, they give rise to continuous gener...

This paper studies questions related to the dynamic transition between local and global minimizers in the Ginzburg-Landau theory of superconductivity. We derive a heuristic equation governing the dynamics of vortices that are close to the boundary, and of dipoles with small inter vortex separation. We consider a small random perturbation of this equation, and study the asymptotic regime under w...

We study, analytically, the discrete complex cubic Ginzburg–Landau (dCCGL) equation. We derive the energy balance equation for the dCCGL and consider various limiting cases. We have found a set of exact solutions which includes as particular cases periodic solutions in terms of elliptic Jacobi functions, bright and dark soliton solutions, and constant magnitude solutions with phase shifts. We h...

In the framework of a 2D Fermi-gas with short-range repulsive interaction we find the critical temperature of the superfluid phase transition based on Kohn-Luttinger effect, and analyze its dependence on magnetic field. We calculate strong-coupling corrections to the Ginzburg-Landau free energy functional and analyze the staibility of 2D superfluid phases. Possible applications of the results t...

It is an open challenge to analyze the crystalline color superconducting phases that may arise in cold dense, but not asymptotically dense, three-flavor quark matter. At present the only approximation within which it seems possible to compare the free energies of the myriad possible crystal structures is the Ginzburg-Landau approximation. Here, we test this approximation on a particularly simpl...

We show numerically that the one-dimensional quintic complex Ginzburg–Landau equation admits four different types of stable hole solutions. We present a simple analytic method which permits to calculate the region of existence and approximate shape of stable hole solutions in this equation. The analytic results are in good agreement with numerical simulations.

We study a complex Ginzburg-Landau equation in the plane, which has the form of a Gross-Pitaevskii equation with some dissipation added. We focus on the regime corresponding to well-prepared unitary vortices and derive their asymptotic motion law. 2000 Mathematics Subject Classification: 35B20,35B40,35Q40,82D55.