Computation, Dimensionality, and Zero-Dissipation limit of the Ginzburg-Landau Wave Equation

In this paper, convergence is proved and optimal error estimatesare obtained for the Galerkin approximation scheme for computingspatially period solutions of the Ginzburg-Landau wave equation.Using this convergent scheme, we carry out a series of numericalexperiments towards an understanding of the finite dimensionalityof the wave motion evolving from a small neighbourhd of thezero solution. In the zerodissipation Limit, we show that thesolutions approach those of the cubic SchrÖdinger equation.