The subharmonic bifurcation of Stokes waves on vorticity flow

Abstract

Untill 1980 one of the main subjects of study in the theory of nonlinear water waves were the Stokes and solitary waves (regular waves). To that time small amplitude regular waves were constructed and the existence of large amplitude water waves of the same type was proved by using branches of water waves starting from a trivial (horizontal) wave and ending at extreme waves. Then in papers Chen & Saffman [7] and Saffman [32] numerical evidence was presented for existence of other type of waves as a result of bifurcations from a branch of ir-rotational Stokes waves on flow of infinite depth. It was demonstrated that the Stokes branch has infinitely many bifurcation points when it approaches the extreme wave and periodic waves with several crests of different height on the period bifurcate from the main branch. The only theoretical works dealing with this phenomenon are Buffoni, Dancer & Toland [4,5] where it was proved the existence of sub-harmonic bifurcations bifurcating from the Stokes branch for the ir-rotational flow of infinite depth approaching the extreme wave.The aim of this paper is to develop new tools and give rigorous proof of existence of subharmonic bifurcations in the case of rotational flows of finite depth. The whole paper is devoted to the proof of this result formulated in Theorem 5.4.