On the commuting charges and the Bethe ansatz for the quantised Kadomtsev-Petviashvili equation

We study the quantised version of the Kadomtsev-Petviashvili equation in the formulation proposed in [4]. We analyse two aspects. One is the conservation of the first higher charge $H_3$ (besides the total mass $H_0$, the momentum $H_1$ and the Hamiltonian $H_2$). We find the expression of $H_3$ and prove commutativity, provided we regularise a specific ultraviolet divergence. The result then does not depend on the choice of a regulator, under fairly general assumptions. We then study the Bethe ansatz in finite volume and obtain a general solution to the two-body problem. We have used AI supervised by us to aid the proofs.