On the propagation of regularity of solutions of the Kadomtsev-Petviashvilli (KPII) equation

We shall deduce some special regularity properties of solutions to the IVP associated to the KPII equation. Mainly, for datum $u_0\in X_s(\mathbb R^2)$, $s>2$, (see (1.2) below) whose restriction belongs to $H^m((x_0,\infty)\times\mathbb R)$ for some $m\in\mathbb Z^+,\,m\geq 3,$ and $x_0\in \mathbb R$, we shall prove that the restriction of the corresponding solution $u(\cdot,t)$ belongs to $H^m((\beta,\infty)\times\mathbb R)$ for any $\beta\in \mathbb R$ and any $t>0$.