On non-uniqueness of solitary waves on two-dimensional rotational flow

Abstract

We consider solitary water waves on a rotational, unidirectional flow in a two-dimensional channel of finite depth. Ovsyannikov has conjectured in 1983 that the solitary wave is uniquely determined by the Bernoulli constant, mass flux and by the flow force. This conjecture was disproved by Plotnikov in 1992 for the ir-rotational flow. In this paper we show that this conjecture is wrong also for rotational flows. Moreover we prove that in any neighborhood of the first bifurcation point on the branch of solitary waves, approaching the extreme wave, there are infinitely many pairs of solitary waves corresponding to the same Bernoulli constant. We give a description of the structure of this set of pairs. The proof is based on a bifurcation analysis of the global branch of solitary waves which is of independent interest.