Institute of Mathematics


Modul:   MAT675  PDE and Mathematical Physics

Bloch theory and spectral gaps for the linearized water wave system

Prof. Dr. Catherine Sulem talk

Speaker invited by: Prof. Dr. Thomas Kappeler

Date: 12.12.19  Time: 18.10 - 19.10  Room: Y27H35/36

We examine the movement of a free surface of a two-dimensional fluid over a variable bottom. We assume that the bottom has a periodic profile and we study the water wave system linearized near the stationary state of water at rest. The latter reduces to a spectral problem for the Dirichlet-Neumann operator in a fluid domain with a periodic bottom and a flat surface elevation. We approach this question using Bloch spectral decomposition which is a classical tool for problems in periodic geometries or equivalently differential operators with periodic coefficients. We show that the spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. We find that, generically, the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. The theory provides also a basis for perturbative calculations that give rise to explicit formulas for the spectral gaps for small bottom variations.