Minisymposium: Calculs rigoureux en systèmes dynamiques
Existence of modulated fronts for the modified Swift-Hohenberg equation: a computer-assisted proof
Deschênes, Andréa
Université Laval

Over the past few decades, the spontaneous formation of patterns such as spatially periodic rolls, hexagonal cell structures, and spiral waves in spatially extended systems has attracted much attention. In the context of the modified Swift-Hohenberg PDE, some of these interesting interfaces can be modelled as modulated fronts, i.e. as waves that are time-periodic in an appropriate co-moving coordinate frame. Via the appropriate change of coordinates introduced in [Doelman, Sandstede, Scheel and Schneider. European J. of Appl. Math. 14 (1), 2003], it is known that computing the modulated fronts reduces to compute heteroclinic orbits between equilibria of a given system of second order ODEs. In this talk, we introduce a computational method to prove existence of some of these connecting orbits, hence leading to rigorous statements about co-existence of different type of non trivial patterns for the original PDE. The rigorous method combines linear splines, the parameterization method of invariant manifold, fixed point theory and interval arithmetics. This is joint work with J.B. van den Berg, J.-P. Lessard and J.D. Mireles James.

Mardi, 18 juin, 16h30
Salle Des Plaines A