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Multifrequency NLS Scaling for a Model Equation of Gravity‐Capillary Waves
Author(s) -
Masmoudi Nader,
Nakanishi Kenji
Publication year - 2013
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21464
Subject(s) - limit (mathematics) , nonlinear system , mathematics , scaling , schrödinger's cat , gravitational wave , series (stratigraphy) , mathematical analysis , class (philosophy) , nonlinear schrödinger equation , nls , modulation (music) , schrödinger equation , classical mechanics , mathematical physics , physics , geometry , quantum mechanics , computer science , paleontology , acoustics , biology , biochemistry , chemistry , artificial intelligence , cytoplasm , nuclear localization sequence
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. © 2012 Wiley Periodicals, Inc.