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...

Relative equilibria are special solutions of partial differential equations (PDEs), which are stationary in an appropriate comoving frame of reference. Such solutions occur frequently in biological and chemical models, e.g. when describing pattern formation of reaction-diffusion equations. Examples are traveling waves in 1d, planar and spiral waves in 2d and scroll waves in 3d. If the equation ...

Attractivity of the Ginz-burg-Landau mode distribution for a pattern forming system with marginally stable long modes. Justification of the 2D NLS equation – Quadratic resonances do not matter in case of analytic initial conditions. Justification of the Nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth. approximation of time osc...

Using the Lagrangian formalism, with a simple trial function for dissipative optical two-dimensional 2D soliton beams, we show that there are two disjoint sets of stationary soliton solutions of the complex cubicquintic Ginzburg-Landau equation, with concave and convex phase profiles, respectively. These correspond to continuously self-focusing and continuously self-defocusing types of 2D solit...

We determine the semiclassical energy levels for the φ field theory in the broken symmetry phase on a 2D cylindrical geometry with antiperiodic boundary conditions by quantizing the appropriate finite–volume kink solutions. The analytic form of the kink scaling functions for arbitrary size of the system allows us to describe the flow between the twisted sector of c = 1 CFT in the UV region and ...

We develop an optimized perturbation theory for the Ginzburg Landau description of thermal fluctuations effects in the vortex liquids. Unlike the high temperature expansion which is asymptotic, the optimized expansion is convergent. Radius of convergence on the lowest Landau level is aT = −3 in 2D and aT = −5 in 3D. It allows a systematic calculation of magnetization and specific heat contribut...

We investigate a gauged matrix model in the large N limit which is closely related to the superconductor fluctuation and the flux lattice melting in two dimensions. With the use of saddle point method the free energy is expanded up to eighth order for the coupling constant g. In the case that the coefficient of quadratic term of the Ginzburg-Landau matrix model is negative, a critical point g =...

We study directed percolation on a 2D triangular lattice where the forward direction is perpendicular to one of the lattice axes. In addition to the geometrical description, an equivalent dynamic description of the model is presented. Simulations are performed based on an algorithm from the dynamic description to obtain the critical probability of bond occupation as pc ≈ 0.522. A mean field the...

ABSTRACT. For nonlinear parabolic evolution equations, it is proved that, under the assumptions oflocal Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. I...

Magnetic properties of layered high temperature superconductors with a weak interlayer coupling in the region of the critical fluctuations are investigated in the framework of the Ginzburg–Landau approach. The sample magnetization is calculated within a perturbative approach to the second order in the interlayer coupling constant. The first order correction can be incorporated into the non-inte...