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

TL;DR

This paper investigates how regularity properties of solutions to the KPII equation propagate over time, showing that initial regularity on a half-line extends to later times across the entire domain.

Contribution

It establishes new results on the propagation of regularity for solutions of the KPII equation, particularly for initial data with localized regularity.

Findings

Regularity propagates from initial data on a half-line to the entire domain over time.

Solutions maintain higher regularity in the spatial variable for all positive times.

The results apply to initial data in specific Sobolev spaces with localized regularity.

Abstract

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 $t>0$.