Modified Energy Functionals and the NLS Approximation

TL;DR

This paper provides a simplified and strengthened proof that wave packet solutions of a water wave model are well approximated by the nonlinear Schrödinger equation, extending the validity to the full existence interval.

Contribution

It introduces a new proof method using a modified energy functional, avoiding inversion issues and improving previous results on NLS approximation for water wave models.

Findings

Approximation holds for the full interval of existence.

Proof simplifies and strengthens previous results.

Avoids normal form transform inversion problems.

Abstract

We consider a model equation from [14] that captures important properties of the water wave equation. We give a new proof of the fact that wave packet solutions of this equation are approximated by the nonlinear Schrodinger equation. This proof both simplifies and strengthens the results of [14] so that the approximation holds for the full interval of existence of the approximate NLS solution rather than just a subinterval. Furthermore, the proof avoids the problems associated with inverting the normal form transform in [14] by working with a modified energy functional motivated by [1] and [8].