### Abstract

Steady periodic water waves on infinite depth, satisfying exactly the kinematic and dynamic boundary conditions on the free surface, with or without surface tension, are given by solutions of a rather tidy nonlinear pseudo-differential operator equation for a 2π-periodic function of a real variable. Being an Euler-Lagrange equation, this formulation has the advantage of gradient structure, but is complicated by the fact that it involves a non-local operator, namely the Hilbert transform, and is quasi-linear. This paper is a mathematical study of the equation in question. First it is shown that its W^{1,2} solutions are real analytic. Then bifurcation theory for gradient operators is used to prove the existence of (non-zero) small amplitude waves near every eigenvalue (irrespective of multiplicity) of the linearised problem. Finally it is shown that when surface tension is absent there are no sub-harmonic bifurcations or turning points at the outset of the branches of Stokes waves which bifurcate from the trivial solution.

Original language | English |
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Pages (from-to) | 207-240 |

Number of pages | 34 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 152 |

Issue number | 3 |

DOIs | |

Publication status | Published - 5 Jun 2000 |

### ASJC Scopus subject areas

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

### Cite this

*Archive for Rational Mechanics and Analysis*,

*152*(3), 207-240. https://doi.org/10.1007/s002050000086