Travelling water-waves, as a paradigm for bifurcations in reversible infinite dimensional ``dynamical'' systems
The author is interested in bifurcations of solutions lying in a neighborhood \(O\). To this end at first the author restricts himself to reversible finite dimensional systems for which he studies periodic, quasiperiodic, homoclinic to \(O\), and homoclinic to periodic solutions. Using center manifold reduction and normal form theory the author manages to prove the persistence of large classes of reversible solutions under higher order terms. The obtained results are applied to water-wave problems, where 2D travelling waves in a potential flow are considered. Indeed, in case of finite depth layers, the author shows that the problem of finding small bounded solutions is reducible to a finite dimensional center manifold, on which the system reduces to reversible ODE. Some open problems are discussed as well.
scientific article; zbMATH DE number 1184398