Title:

Ginzburg-Landau Approximation for Pattern-Forming Systems

Abstract:

Modulation equations are simple scalar equations that accurately capture the evolution of the entire system in certain regimes. In my presentation, I will focus on dissipative systems in which a linearly stable equilibrium bifurcates into an unstable one as a parameter of the system crosses a critical value. For PDE systems, this type of bifurcation, known as a Turing or Turing-Hopf bifurcation, constitutes evidence of the formation of regular spatio-temporal patterns in the solution. Many reaction-advection-diffusion models arising for instance in chemistry and ecology, are pattern-forming systems. For a wide class of pattern-forming systems the corresponding modulation equation is the Ginzburg-Landau equation which paints a rich picture of the type of patterns that can arise.

Starting from the 1990s onward, proving the validity of modulation equations for various complex systems has been an important object of investigation. A very general framework is known for semilinear systems, but the method fails in general for quasilinear ones. In a joint work with Guido Schneider, we proved that a general approximation result for quasilinear systems holds.

To present some of the main ideas of the theory, I will present an (overly) simple validity result for bifurcating ODEs, I will highlight the issues that appear when we adapt the method to PDE systems and I will discuss the detailed picture that the Ginzburg-Landau equation provides.

Speaker: Théo Belin, Postdoc at Algebra, Analysis and Dynamical Systems