DICHOTOMIES OF SOLUTIONS TO THE STOCHASTIC GINZBURG -- LANDAU EQUATION ON A TORUS

Abstract

We consider a stochastic analogue of the Ginzburg -- Landau equation in spaces of differential forms defined on a two-dimensional smooth compact oriented manifold without boundary. When studying the stability of solutions, the Ginzburg -- Landau equation is considered as a special case of a stochastic linear Sobolev-type equation. All considerations are carried out in spaces of random $K$-variables and $K$-"noises" on the manifold. As a manifold, we consider a two-dimensional torus, which is a striking example of a smooth compact oriented manifold without boundary. Under certain conditions imposed on the coefficients of the equation, we prove the existence of stable and unstable invariant spaces and exponential dichotomies of solutions. We develop an algorithm to illustrate the results obtained. Since there exists a smooth diffeomorphism between a map and a manifold, we reduce the question of stability of solutions on a two-dimensional torus to the same question on one of its maps. The developed algorithm is implemented in the Maple software environment. The results of the work are presented in the form of graphs of stable and unstable solutions, which are obtained for various values of the parameters of the Ginzburg -- Landau equation.