# The Fifth Order KP--II Equation on the Upper Half--plane

## TL;DR

This paper investigates the fifth order KP--II equation on the upper half-plane, establishing low regularity local well-posedness and demonstrating the nonlinear solution's increased smoothness compared to initial data.

## Contribution

It introduces new techniques for analyzing the fifth order KP--II equation on a half-plane, including low regularity well-posedness and smoothing properties of solutions.

## Key findings

- Established low regularity local well-posedness using Bourgain's restricted norm method.
- Proved the nonlinear part of solutions is smoother than initial data.
- Applied Fourier-Laplace method to solve initial and boundary value problems.

## Abstract

In this paper we study the fifth order Kadomtsev--Petviashvili II (KP--II) equation on the upper half-plane $U=\{(x,y)\in \R^2: y>0\}$. In particular we obtain low regularity local well-posedness using the restricted norm method of Bourgain and the Fourier-Laplace method of solving initial and boundary value problems. Moreover we prove that the nonlinear part of the solution is in a smoother space than the initial data.