### Abstract

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data of the form ε B (ε α)e ^{ikα} for k > 0, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrödinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times of order O(ε ^{-2}) provided the initial data differs from the wave packet by at most O(ε ^{3/2}) in Sobolev spaces. These results are obtained by directly applying modulational analysis to the evolution equation with no quadratic nonlinearity constructed in Wu (Invent Math 117(1):45-135, 2009) and by the energy method.

Original language | English (US) |
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Pages (from-to) | 817-883 |

Number of pages | 67 |

Journal | Communications in Mathematical Physics |

Volume | 310 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2012 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics