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

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.