Standing waves for a two-way model system for water waves

Abstract

In this paper, we prove the existence of a large family of non-trivial bifurcating standing waves for a model system which descri bes two-way propagation of water waves in a channel of finite depth or in the near shore zone. In particular, it is shown that, contrary to the classical standing gravity wave problem on a fluid layer of finite depth, the Lyapunov-Schmidt method applies to find the bifurcation equation. The bifurcation set is formed with the discrete union of Whitney's umbrellas in the three-dimensional space formed with 2 parameters representing the time-period and the wave length, and the average of one of the amplitudes.