The Effect of Nonlocal Interactions on the Dynamics of the Ginzburg-Landau Equation

Nonlocal amplitude equations of the complex Ginzburg-Landau type arise in a few physical contexts, such as in ferromagnetic systems. In this paper, we study the effect of the nonlocal term on the global dynamics by considering a model nonlocal complex amplitude equation. First, we discuss the global existence, uniqueness and regularity of solutions to this equation. Then we prove the existence of the global attractor, and of a finite dimensional inertial manifold. We provide upper and lower bounds to their dimensions, and compare them with those of the cubic complex Ginzburg-Landau equation. It is observed that the nonlocal term plays a stabilizing or destabilizing role depending on the sing of the real part of its coefficient. Moreover, the nonlocal term affects not only the diameter of the attractor but also its dimension. To appear: Zeitschrift fur Angewandte Mathematik und Physik, (ZAMP) 1 California Institute of Technology, Applied Mechanics 104-44, Pasadena, CA 91125, USA. E-Mai..