The identical synchronization of two discrete Ginzbug-Landau equations using local coupling is proved in the theory. It is based on the theory of infinite dimensional dynamical system. It is proved that the two discrete Ginzburg-Landau equations have absorbing sets and attractors. The analytic expression of Lyapunov exponent is obtained and the results are consistent with the V.Parlitz’s numeri...

We study solutions of the 2D Ginzburg-Landau equation −∆u + 1 ε2 u(|u| − 1) = 0 subject to “semi-stiff” boundary conditions: Dirichlet conditions for the modulus, |u| = 1, and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in t...

We introduce a model of a two-dimensional (2D) optical waveguide with Kerr nonlinearity, linear and quintic losses, cubic gain, and temporal-domain filtering. In the general case, temporal dispersion is also included, although it is not necessary. The model provides for description of a nonlinear planar waveguide incorporated into a closed optical cavity. It takes the form of a 2D cubic-quintic...

We show that dissipative systems can have a multiplicity of stationary solutions in the form of both stable and unstable solitons. As a model equation, we use the complex cubic–quintic Ginzburg–Landau equation. For a given set of the equation parameters, this equation has many coexisting soliton solutions. Our stability results show that although most of them are unstable, they can have stable ...