Quantisation of Kadomtsev-Petviashvili equation

TL;DR

This paper introduces a quantum version of the Kadomtsev-Petviashvili (KP) equation on a cylinder, modeling an infinite system of bosons with mass-dependent interactions, and verifies its integrability up to a certain mass sector.

Contribution

It proposes a novel quantisation of the KP equation involving mass-carrying bosons and constructs Bethe eigenfunctions, demonstrating quantum integrability up to mass 8.

Findings

Hamiltonian is Galilei-invariant with split and merge interaction terms.

Bethe eigenfunctions are explicitly constructed for the model.

Quantum integrability is verified up to the mass M=8 sector.

Abstract

A quantisation of the KP equation on a cylinder is proposed that is equivalent to an infinite system of non-relativistic one-dimensional bosons carrying masses $m=1,2,\ldots$ The Hamiltonian is Galilei-invariant and includes the split $\Psi^\dagger_{m_1}\Psi^\dagger_{m_2}\Psi_{m_1+m_2}$ and merge $\Psi^\dagger_{m_1+m_2}\Psi_{m_1}\Psi_{m_2}$ terms for all combinations of particles with masses $m_1$, $m_2$ and $m_1+m_2$, with a special choice of coupling constants. The Bethe eigenfunctions for the model are constructed. The consistency of the coordinate Bethe Ansatz, and therefore, the quantum integrability of the model is verified up to the mass $M=8$ sector.