Nonlinear Evolution Equations

Peter Takác
Fachbereich Mathematik
Universität Rostock, Rostock, Germany

Shouhong Wang
Department of Mathematics
Indiana University, Bloomington, IN

We are interested in the analysis and prediction of dynamic phenomena in complex systems. Complex systems are characterized by many degrees of freedom and nonlinear interactions among the component parts. Their behavior is the result of a complicated interplay between the internal degrees of freedom and the various external forces at work on the system.

As mathematicians, we use universal models, which are supposed to describe generic behavior of complex systems. Examples are the Ginzburg-Landau equations and other so-called amplitude equations. These equations describe the nonlinear behavior of perturbations of a weakly unstable basic state of an autonomous evolutionary system.

The steady state is usually the simplest state in which a system can exist. Under certain circumstances, a steady state is stable - that is, if one perturbs the system a little, it will eventually return to this steady state. But sometimes the family of steady states is stable only up to a critical value of a parameter, at which point a new family of steady solutions branches off (bifurcation). One can study this bifucation more closely in a small neighborhood of the critical point, where one can construct a (small) correction to the steady state. The amplitude of this correction varies slowly in time; its evolution is governed by an amplitude equation. An amplitude equation often takes the form of a Ginzburg-Landau equation, dA/dt = aA + b|A|²A, where A is the amplitude (generally a complex vector-valued function of space and time), a and b are complex constants, and t is the slow time variable.

Our research is motivated by applications in materials science, primarily in superconductivity. Vortices in type-II superconductors constitute a complex system, whose evolution is governed by equations of the Ginzurg-Landau type. We study these equations for the properties of their solutions. In particular, we are interested in questions of existence, uniqueness, regularity, long-time dynamic behavior, etc.

The figures above, taken from Ref. [4] below, summarize the bifurcation of vortex solutions of the complex Ginzburg-Landau equation, an equation for a complex scalar-valued function in one space dimension with a complex bifurcation parameter rho. The figures give the real and imaginary part of the solution U as a function of the argument of rho (top) and the modulus of rho (bottom).

The following recent publications are available on the web:

1. J. Fleckinger-Pellé, H. G. Kaper, and P. Takác, “Dynamics of the Ginzburg-Landau Equations of Superconductivity,” Nonlinear Anal. - Theory, Methods & Applic. 32 (1998), 647-665
2. H. G. Kaper and P. Takác, “Ginzburg-Landau Equations with a Time-Dependent Magnetic Field,” Nonlinearity 11 (1998), 291-305
3. H. G. Kaper, B. Wang, and S. Wang, “Determining Nodes for the Ginzburg-Landau Equations of Superconductivity,” Discrete and Continuous Dynamical Systems 4 (1998), 205-224
4. H. G. Kaper and P. Takác, “Bifurcating Vortex Solutions of the Complex Ginzburg-Landau Equations,” submitted for publication. Preprint ANL/MCS-P725-0998

Contact:

Hans G. Kaper, MCS Division
Argonne National Laboratory
Argonne, Illinois 60439
E-mail: kaper@mcs.anl.gov
(630) 252-7160

[ MCS Division - Applied Mathematics | MCS Division - Research | MCS Division | Hans G. Kaper - Home Page ]

Last update: October 22, 1998 (HGK)