Uniqueness of lump solutions of KP-I equation

TL;DR

This paper classifies lump solutions of the KP-I equation, showing they are rational, polynomial tau functions, and proves their uniqueness as ground states, extending previous classifications of rational solutions.

Contribution

It provides a complete classification of lump solutions of the KP-I equation using inverse scattering transform, establishing their rationality, polynomial tau functions, and uniqueness.

Findings

Lump solutions are rational functions with polynomial tau functions of degree k(k+1).

Lump solutions are the unique ground states of the KP-I equation.

The classification generalizes previous results on rational solutions for KdV.

Abstract

The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution. This is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify the lump type solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller, we show that the lump type solutions are rational and their tau function has to be a polynomial of degree $k(k+1)$. In particular, this implies that the lump solution is the unique ground state of the KP-I equation (as conjectured by Klein and Saut in \cite{Klein0}). Our result generalizes a theorem by Airault-McKean-Moser on the classification of rational solutions for the KdV equation.