Long time asymptotics of large data in the Kadomtsev-Petviashvili models

TL;DR

This paper analyzes the long-time behavior of solutions to the Kadomtsev-Petviashvili equations with large initial data, providing asymptotic descriptions in regions free of solitons, using new virial identities.

Contribution

Introduces two novel virial identities for KP equations, enabling decay analysis without relying on integrability, applicable to large data and perturbations.

Findings

Asymptotic descriptions of solutions in large regions without lumps or solitons.

Decay results for solutions using new virial identities.

Applicable to both KP-I and KP-II equations with minimal regularity.

Abstract

We consider the Kadomtsev-Petviashvili (KP) equations posed on $\mathbb{R}^2$. For both equations, we provide sequential in time asymptotic descriptions of solutions, of arbitrarily large data, inside regions not containing lumps or line solitons, and under minimal regularity assumptions. The proof involves the introduction of two new virial identities adapted to the KP dynamics, showing decay in large regions of space, especially in the KP-I case, where no monotonicity property was previously known. Our results do not require the use of the integrability of KP and are adaptable to well-posed perturbations of KP.