Quantization of the Kadomtsev–Petviashvili equation

We propose a quantization of the Kadomtsev–Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses $m=1,2,\dots$. The Hamiltonian is Galilei-invariant and includes the split and merge terms $\Psi^{\dagger}_{m_1}\Psi^{\dagger}_{m_2} \Psi_{m_1+m_2}$ and $\Psi^{\dagger}_{m_1+m_2}\Psi_{m_1}\Psi_{m_2}$ for all combinations of particles with masses $m_1$, $m_2$, and $m_1+m_2$ for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass $M=8$ sector.