Critical lengths for the linear Kadomtsev-Petviashvili II equation

TL;DR

This paper investigates the critical length phenomenon for the linear Kadomtsev-Petviashvili II equation, establishing controllability and stabilization results that depend on domain length avoiding specific critical values.

Contribution

It introduces the first identification of critical lengths for the two-dimensional KP-II equation, extending the understanding of length-dependent properties from 1D to 2D.

Findings

Established observability inequalities for KP-II

Derived boundary controllability and stabilization results

Identified critical lengths based on the Paley-Wiener theorem

Abstract

The critical length phenomenon of the Korteweg-de Vries equation is well known; however, in higher dimensions, it is unknown. This work explores this property in the context of the Kadomtsev-Petviashvili equation, a two-dimensional generalization of the Korteweg-de Vries equation. Specifically, we demonstrate observability inequalities for this equation, which allow us to deduce the exact boundary controllability and boundary exponential stabilization of the linear system, provided that the spatial domain length avoids certain specific values, a direct consequence of the Paley-Wiener theorem. To the best of our knowledge, our work introduces new results by identifying a set of critical lengths for the two-dimensional Kadomtsev-Petviashvili equation.