Dispersive estimates for full dispersion KP equations

TL;DR

This paper establishes dispersive estimates for the linear part of the Full Dispersion KP equations, leading to local well-posedness results in certain Sobolev spaces, using advanced asymptotic analysis techniques.

Contribution

It introduces new dispersive estimates for the Full Dispersion KP equations and proves local well-posedness in $H^s(R^2)$ for $s > 7/4$, improving understanding of these models.

Findings

Dispersive estimates are proven for the linear full dispersion KP equations.

The initial value problem is shown to be locally well-posed in $H^s(R^2)$ for $s > 7/4$.

The proof combines stationary phase method with sharp asymptotics on asymmetric Bessel functions.

Abstract

We prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev-Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev-Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev-Petviashvili is locally well-posed in $H^s(\mathbb R^2)$, for $s>\frac74$, in the capillary-gravity setting.