Constructive proofs of existence in differential equations on R^n

CDSNS Colloquium
Friday, April 26, 2024 - 3:30pm for 1 hour (actually 50 minutes)
Skiles 254
Matthieu Cadiot – McGill University – matthieu.cadiot@mail.mcgill.ca
Alex Blumenthal

Please Note: Zoom link to attend remotely:

In this talk I will present a computer-assisted method to study solutions vanishing at infinity in differential equations on R^n. Such solutions arise naturally in various models, in the form of traveling waves or localized patterns for instance, and involve multiple challenges to address both on the numerical and on the analytical side. Using spectral techniques, I will explain how Fourier series can serve as an approximation of the solution as well as an efficient mean for the construction of a fixed-point operator for the proof. To illustrate the method, I will present applications to the constructive proof of localized patterns in the 2D Swift-Hohenberg equation and in the Gray-Scott model. The method extends to non-local equations and proofs of solitary travelling waves in the (capillary-gravity) Whitham equation will be exposed.