Abstract for Jon Wilkening

Jon Wilkening (University of California, Berkeley)

Friday, March 31, 2023
3:25-4:05pm

Spatially quasi-periodic bifurcations from periodic traveling water waves

Abstract: We present a numerical method for computing families of spatially quasi-periodic water waves that bifurcate from large-amplitude periodic traveling waves. The water wave equations are formulated in a conformal mapping framework on a higher-dimensional torus to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch. This yields traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. We also present a new method of locating bifurcation points and curves in multi-parameter bifurcation problems that allows us to visualize the bifurcation structure for the gravity-capillary wave problem.