Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices

The time evolution of several interacting Ginzburg-Landau vortices according to an equation of Schrödinger type is approximated by motion on a finite-dimensional manifold. That manifold is defined as an unstable manifold of an auxiliary dynamical system, namely the gradient flow of the Ginzburg-Landau energy functional. For two vortices the relevant unstable manifold is constructed numerically and the induced dynamics is computed. The resulting model provides a complete picture of the vortex motion for arbitrary vortex separation, including well-separated and nearly coincident vortices. AMS classification scheme numbers: 35Q55, 37K05, 70K99 1 Ginzburg-Landau vortices and their dynamics Vortices play a fundamental role in a large variety of physical systems, ranging from ordinary fluids over condensed matter to the early universe. This variety is reflected in the mathematical models used to describe the formation, structure and dynamics of vortices. In fluid dynamics, the basic ingredient of the mathematical model is the velocity field of the fluid, and the vortex is a particular configuration of that velocity field. In condensed matter theory, the mathematical models are field theories. The basic field of such models is a complex valued scalar field, and vortices are particular configurations of that field. One important difference between vortices in ordinary fluids and those in condensed matter systems is that the total vorticity can take address since Dec. 2001: Max-Planck-Institute for Biophysical Chemistry, Theoretical Molecular Biology Group, D-37077 Göttingen, Germany; e-mail [email protected] e-mail [email protected]