  The Lugiato-Lefever equation is a damped and driven version of the well-known
nonlinear Schr\"odinger equation. It is a mathematical model describing complex
phenomena in dissipative and nonlinear optical cavities. Within the last two
decades, the equation has gained a wide attention as it becomes the basic model
describing optical frequency combs. Recent works derive the Lugiato-Lefever
equation from a class of damped driven $\phi^4$ equations closed to resonance.
In this paper, we provide a justification of the envelope approximation. From
the analysis point of view, the result is novel and non-trivial as the drive
yields a perturbation term that is not square integrable. The main approach
proposed in this work is to decompose the solutions into a combination of the
background and the integrable component. This paper is the first part of a
two-manuscript series.
