Gain of Regularity for the KP-I Equation

In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation’s dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data $\varphi$ possesses certain regularity and sufficient decay as $x\to \mathrm{\infty }$, then the solution $u(t)$ will be smoother than $\varphi$ for $0<t\le T$ where $T$ is the existence time of the solution.