## Mathematical Problems in the Theory of Water Waves: A Workshop on the Problems in the Theory of Nonlinear Hydrodynamic Waves, May 15-19, 1995, Luminy, FranceThe proceedings featured in this book grew out of a conference attended by 40 applied mathematicians and physicists which was held at the International Center for Research in Mathematics in Luminy, France, in May 1995. This volume reviews recent developments in the mathematical theory of water waves. The following aspects are considered: modeling of various wave systems, mathematical and numerical analysis of the full water wave problem (the Euler equations with a free surface) and of asymptotic models (Korteweg-de Vries, Boussinesq, Benjamin-Ono, Davey-Stewartson, Kadomtsev-Petviashvili, etc.), and existence and stability of solitary waves. Features the latest developments in the theory of water waves, rigorous and formal results, and papers from world-renowned experts in the field. |

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### Contents

Radiation and modulational instability described by the fifthorder Korteweg De Vries equation | 1 |

Numerical simulation of singular solutions of the generalized KortewegdeVries equation | 17 |

Spatially quasiperiodic capillarygravity waves | 31 |

On modulations of weakly nonlinear water waves | 47 |

Birkhoff normal forms for water waves | 57 |

Remarks on the stability of generalized KP solitary waves | 75 |

On the asymptotic integrability of a generalized Burgers equation | 85 |

Spatial instabilities and chaos in highorder Hamiltonian standing water waves | 99 |

Solitary wave interactions for the fifthorder KdV equation | 133 |

Forced BenjaminOno equations and related topics | 145 |

The frequency downshift phenomenon | 157 |

Solitary waves on the free surface of a rotating cylindrical flow | 173 |

On a generalized KadomtsevPetviashvili equation | 193 |

A phenomenological description of soliton splitting during run up | 211 |

Asymptotic stability of nonlinear bound states in conservative dispersive systems | 223 |

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1000 time,periods amplitude analysis approximation asymptotic stability blow-up Bona computed constant corresponding decay defined denote depth derived dimensional dispersion relation dynamics E-mail address eigenvalues energy envelope solitary waves evolution equations Evolution of mode fBO equation Figure finite Floquet multipliers flow Fluid Mech free surface frequency down-shift function given gravity waves Hamiltonian system initial data integrable interaction Kadomtsev-Petviashvili equation Korteweg–de Vries equation Lemma linear logscale long-time evolution Math Mathematics Subject Classification method mode 12 modulus of mode nonlinear Schrödinger equation normal forms transformations numerical obtained parameter periodic orbit periodic solutions phase Phys proof quasi-periodic recurrence region resonance overlap satisfies sideband singularity solitary wave solitary wave solutions soliton spatial instability standing Stokes wave standing waves subspace surface tension Theorem theory thermalization time,periods Evolution topography truncation variables velocity water wave problem wave of type wavenumber Zakharov zero