On the first subharmonic bifurcations in a branch of Stokes water waves

Abstract

Steady surface waves in a two-dimensional channel are considered. We study bifurcations, which occur on a branch of Stokes water waves starting from a uniform stream solution. Two types of bifurcations are considered: bifurcations in the class of Stokes waves (Stokes bifurcation) and bifurcations in a class of periodic waves with the period M times the period of the Stokes wave (M-subharmonic bifurcation). If we consider the first Stokes bifurcation point then there are no M-subharmonic bifurcations before this point and there exists M-subharmonic bifurcation points after the first Stokes bifurcation for sufficiently large M, which approach the Stokes bifurcation point when M→∞. Moreover the set of M-subharmonic bifurcating solutions is a closed connected continuum. We give also a more detailed description of this connected set in terms of the set of its limit points, which must contain extreme waves, or overhanging waves, or solitary waves or waves with stagnation on the bottom, or Stokes bifurcation points different from the initial one.