On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach

TL;DR

This paper proves that solutions to a fifth-order Kadomtsev-Petviashvili-II equation are Gevrey regular in time with a sharper order than previous results, using a new analytical approach applicable to dispersive equations.

Contribution

It introduces a novel analytical method to establish Gevrey regularity in time for dispersive equations, extending and sharpening prior results for the fifth-order KP-II equation.

Findings

Solutions are Gevrey regular of order 5σ in time.

The solution does not belong to G^z for any 1 ≤ z < 5σ.

A new method applies to a class of dispersive equations with polynomial nonlinearities.

Abstract

In this paper, we improve and extend the results obtained by Boukarou et al. \cite{boukarou1} on the Gevrey regularity of solutions to a fifth-order Kadomtsev-Petviashvili-II equation. We establish Gevrey regularity in the time variable for solutions in $2+1$ dimensions, providing a sharper result obtained through a new analytical approach. Assuming that the initial data are Gevrey regular of order $\sigma \geq 1$ in the spatial variables, we prove that the corresponding solution is Gevrey regular of order $5 \sigma$ in time. Moreover, we show that the function $u(x, y, t)$, viewed as a function of $t$, does not belong to $G^z$ for any $1 \leq z<5 \sigma$. Our proof introduces a new analytical method that establishes a general principle for dispersive equations of the form $ \partial_t u = \pm\partial_x^\alpha u + P(u),$ where $\partial_x^\alpha $ is the highest spatial derivative and $P(u)$ a polynomial in spatial derivatives of total order at most $\alpha-1$, the solution cannot belong to the Gevrey class $G^z$ in time for any $z$ satisfying $1 \leq z<\alpha \sigma$.