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arXiv:2609.15316

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

Chiara Paletta, Алессандро Торриелли

Published Sep 14, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2609.15316

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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 H3H_3H3​ (besides the total mass H0H_0H0​, the momentum H1H_1H1​ and the Hamiltonian H2H_2H2​). We find the expression of H3H_3H3​ 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.
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,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms Ψm1†Ψm2†Ψm1+m2\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}Ψm1​†​Ψm2​†​Ψm1​+m2​​ and Ψm1+m2†Ψm1Ψm2\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}Ψm1​+m2​†​Ψm1​​Ψm2​​ 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.
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A general formalism for the study of excitations above equilibrium in Bethe ansatz solvable models is presented. Nonzero temperature expressions for dressed energy, momentum, spin and charge are obtained, and it is found that the dressed spin and charge are in general momentum dependent. For an electronic model one may only have spin-charge separation at zero temperature where the ground state is half-filled and has zero magnetisation. Finally, the excitations of the Hubbard-Shastry models are examined in detail.
Recent interest in discrete, classical integrable systems has focused on their connection to quantum integrable systems via the Bethe equations. In this note, solutions to the rational nested Bethe ansatz (RNBA) equations are constructed using the ``completed Calogero-Moser phase space'' of matrices which satisfy a finite dimensional analogue of the canonical commutation relationship. A key feature is the fact that the RNBA equations are derived only from this commutation relationship and some elementary linear algebra. The solutions constructed in this way inherit continuous and discrete symmetries from the CM phase space.