### Abstract

This paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear Schrödinger (NLS) equation as well as other related systems starting from a model coming from the gravity-capillary water wave system in the long-wave limit. Our main goal is to understand resonances and their effects on having the nonlinear Schrödinger approximation or modification of it or having other models to describe the limit equation. In this first paper, our goal is not to derive NLS but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time. This yields a new class of large data for which we have a large time of existence.

Original language | English (US) |
---|---|

Pages (from-to) | 1202-1240 |

Number of pages | 39 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 66 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*66*(8), 1202-1240. https://doi.org/10.1002/cpa.21464

**Multifrequency NLS scaling for a model equation of gravity-capillary waves.** / Masmoudi, Nader; Nakanishi, Kenji.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 66, no. 8, pp. 1202-1240. https://doi.org/10.1002/cpa.21464

}

TY - JOUR

T1 - Multifrequency NLS scaling for a model equation of gravity-capillary waves

AU - Masmoudi, Nader

AU - Nakanishi, Kenji

PY - 2013/8

Y1 - 2013/8

N2 - This paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear Schrödinger (NLS) equation as well as other related systems starting from a model coming from the gravity-capillary water wave system in the long-wave limit. Our main goal is to understand resonances and their effects on having the nonlinear Schrödinger approximation or modification of it or having other models to describe the limit equation. In this first paper, our goal is not to derive NLS but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time. This yields a new class of large data for which we have a large time of existence.

AB - This paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear Schrödinger (NLS) equation as well as other related systems starting from a model coming from the gravity-capillary water wave system in the long-wave limit. Our main goal is to understand resonances and their effects on having the nonlinear Schrödinger approximation or modification of it or having other models to describe the limit equation. In this first paper, our goal is not to derive NLS but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time. This yields a new class of large data for which we have a large time of existence.

UR - http://www.scopus.com/inward/record.url?scp=84878415225&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878415225&partnerID=8YFLogxK

U2 - 10.1002/cpa.21464

DO - 10.1002/cpa.21464

M3 - Article

AN - SCOPUS:84878415225

VL - 66

SP - 1202

EP - 1240

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 8

ER -