### 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

### Cite this

**A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem.** / Totz, Nathan; Wu, Sijue.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 310, no. 3, pp. 817-883. https://doi.org/10.1007/s00220-012-1422-2

}

TY - JOUR

T1 - A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem

AU - Totz, Nathan

AU - Wu, Sijue

PY - 2012/3/1

Y1 - 2012/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84863279498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863279498&partnerID=8YFLogxK

U2 - 10.1007/s00220-012-1422-2

DO - 10.1007/s00220-012-1422-2

M3 - Article

AN - SCOPUS:84863279498

VL - 310

SP - 817

EP - 883

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -