Invariant 2-tori in the time-dependent Ginzburg-Landau equation

Publisher Website
Google Scholar
Abstract
Numerical evidence shows the existence of 2- and 3-tori in the time-dependent complex Ginzburg-Landau equation (CGL) in one spatial dimension with periodic boundary conditions. The author employs standard bifurcation methods, such as the Lyapunov-Schmidt reduction, to prove the existence of 2-tori by rigorous analytical tools. Bifurcations involve multi-dimensional (real or complex) eigenspaces. The procedure enables one to obtain a unique smooth bifurcating manifold the dimension of which equals the dimension of the eigenspace. In particular, primary bifurcations from the zero solution yield periodic orbits, and secondary ones from rotating waves yield 2-tori. Their stability is discussed by means of centre manifold theory. While the rotating waves are represented by a wavefunction with temporally and spatially constant modulus, the bifurcating 2-tori have a wavefunction the modulus of which is a travelling wave. In superconductivity the square modulus of the wavefunction is proportional to the density of superconducting electrons.