Global well-posedness of periodic KP-I initial value problem in the energy space

TL;DR

This paper proves the global well-posedness of the periodic KP-I initial value problem in the energy space, establishing existence, uniqueness, and continuous dependence of solutions for initial data in this space.

Contribution

It demonstrates the global well-posedness of the periodic KP-I equation in the energy space, a significant extension of previous local results.

Findings

Global well-posedness in the energy space $E^1$

Solutions exist uniquely and depend continuously on initial data

Applicable to initial data with zero mean in the spatial domain

Abstract

The periodic KP-I initial value problem $\partial_t u+\partial_x^3 u-\partial_x^{-1}\partial_y^2 u+\partial_x (u^2/2)=0$ on $T_{x,y}^2\times R_t, $u(0)=\phi$ is globally well-posed in the energy space $E^1 = E^1 (T^2)=\phi: T^2\to R:\hat\phi(0,n)=0$ for all $n\in Z \ 0$ and $||\phi||_{E^1 (T^2)}=||\hat{\phi}(m,n)(1+|m|+|n/m|)||_{l^2(Z^2)}<\infty$.