On a coupled system of KP-type

TL;DR

This paper introduces a new modified KP system called the Non-KP model, which addresses initial data restrictions and analyzes its variational, Hamiltonian, and linear properties.

Contribution

The paper proposes the Non-KP model, a novel KP-type system that removes non-physical initial data restrictions and explores its mathematical structure.

Findings

Variational derivation of the Non-KP model

Analysis of the Hamiltonian structure

Establishment of linear estimates in Bourgain spaces

Abstract

A defining characteristic of the Kadomstev-Petviashvili (KP) model equation is that the well-posedness results are subject to the restriction that at all transverse positions, the mass $\int u \,dx = \text{constant independent of $y$}.$ In 2007, for a rather general class of equations of KP type, it was shown that the zero-mass (in $x$) constraint is satisfied at any non-zero time even if it is not satisfied at initial time zero. To remedy this ``odd'' behavior, a model modification is introduced which does not impose non-physical restrictions upon the initial data. In this article, we introduce a new modified KP system, named the Non-KP model equation. After providing a variational derivation of the Non-KP model, we analyze its Hamiltonian evolutionary structure. Furthermore, we prove linear estimates in the Bourgain spaces $X^{s,b}$ corresponding to the integral equation arising from the Duhamel formulation of system.