Dispersive Estimates for Full Dispersion KP Equations

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 $$H^s(\mathbb R^2)$$ H s ( R 2 ) , for $$s>\frac{7}{4}$$ s > 7 4 , in the capillary-gravity setting.

On the full dispersion Kadomtsev–Petviashvili equations for dispersive elastic waves

“…This is similar to how the Whitham equation is a fully dispersive version of the KdV equation [22]. The full dispersion KP model has been studied from a number of viewpoints; see for instance [16,19,20] and the references therein. The main conclusion to be drawn from those studies is that the full dispersion KP equation provides a more accurate asymptotic description of the weakly transverse waves than the KP equation and it imposes a weaker constraint than the one for the KP equation.…”

“…In order to remedy such shortcomings of the KP equation, a full dispersion KP equation for water waves was proposed in [18]. We refer the reader to [16,19,20] for various properties of the full dispersion KP equation. The most important aspect of the full dispersion KP equation is that the linear dispersion relation is the same as the exact dispersion relation of water waves model.…”

Dispersive estimates for linearized water wave type equations in $\mathbb R^d$

“…Estimates for the first and second order derivatives of this function is derived recently in [9]. In general, we have T (j) (r) = S 2 • P j−1 (S, T ) (j 1)…”

Comparison between Boussinesq‐ and Whitham–Boussinesq‐type systems

“…They also proved global well‐posedness of the Cauchy problem for sufficiently small initial data in the $$$ {L}^2\left(\mathbb{R}\right)\times {H}^{\frac{1}{2}}\left(\mathbb{R}\right) $$$ norm. See also Pilod et al and Tesfahun 5,6 for two‐dimensional systems.…”