Equilibrium states of a variational formulation for the Ginzburg-Landau equation

Abstract

Periodic boundary value problem for one of the versions of the complex Ginzburg- Landau equation, which is commonly called the variational Ginzburg-Landau equation are studied. Questions of existence and stability in the sense of Lyapunov, and also the local bifurcations problem of spatially nonhomogeneous equilibrium states are investigated. Three types of such solutions for the given problem are indicated. The exact formulas of the solutions for the first two types are suggested. Equilibrium states of the second type are expressed through elliptic functions. The third type of equilibrium states appears as a result of bifurcations of automodel equilibrium states, i.e., solutions of the first type in the case when the stability changes. It is shown that equilibrium states of the second and third types are unstable.