Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.02309

On a certain system of discrete equations with the Laurent property

Andrei K. Svinin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider a system of two bilinear discrete equations with four parameters. It defines a certain discrete dynamical system. Each of its solutions is represented as a pair of sequences. It is proved that for arbitrary initial data all terms of both sequences are Laurent polynomials in the initial data with integer coefficients. This property is called the Laurent property. The proof relies on an established connection between the system and the Somos-5 equation. It is shown that both components of the solution satisfy the same equation with the same parameters. To prove the main theorem, the strong Laurent property for the Somos-5 equation is used, which makes it possible to control negative powers of the initial data. An analogue of this property is obtained directly for the system under study. As corollaries, sufficient conditions for the integrality of solutions are found. An example of a pair of integer sequences illustrating the theorem on the strong Laurent property is given.

Explore related subjects

nlin.SImath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andrei K. Svinin. 2026-11-01. On a certain system of discrete equations with the Laurent property. https://arxiv.org/abs/2611.02309

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The $\mathrm{al}_{ab}$ functions of hyperelliptic curves: a generalization of Jacobi's elliptic functions and explicit solutions of a degenerate Krichever-Novikov equation

The Jacobi elliptic functions $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ and their ratios are expressed by Weierstrass's functions $\mathrm{al}_a=\sqrt{x-b_a}$ of an elliptic curve. For a hyperelliptic curve of genus $g$, Weierstrass's $\mathrm{al}_a$ functions generalize $\mathrm{sn}$, $\mathrm{cn}$ and $\mathrm{dn}$, and the modified Korteweg-de Vries and sine-Gordon equations stand behind them. In this paper we study the $\mathrm{al}_{ab}$ functions, which are defined by the sigma function shifted by the sum of two half-periods and which, together with the ratios of the $\mathrm{al}_a$ functions, generalize the remaining Jacobi elliptic functions. We determine the exact lattice of periods of $\mathrm{al}_{12}$ and show that it is single-valued on the unramified double covering of the Jacobian corresponding to the double covering $\hat{w}^2=(x-b_1)(x-b_2)$ of the curve. We show that $\mathrm{al}_{12}$ is the Wronskian of the band-edge eigenfunctions $\mathrm{al}_1$ and $\mathrm{al}_2$, so that the half-period translations realize Darboux transformations. As the main result, $\mathrm{al}_{12}$, $\mathrm{al}_2/\mathrm{al}_1$ and $1/\mathrm{al}_{12}$ satisfy one and the same degenerate Krichever-Novikov equation, whose polynomial has the double roots $0$ and $\infty$, and the half-periods act on these solutions as the Möbius symmetries of the equation. Thus the Krichever-Novikov equation stands behind the $\mathrm{al}_{ab}$ functions. In genus one these functions reduce to $\mathrm{sn}$, $\mathrm{ns}$ and $\mathrm{dc}$.

nlin.SI↗
arXiv2026

The integer spectrum of a superintegrable quantum field theory

In our previous article CMP 407 (2026) 131, we explored the rich algebraic structures arising from a specific quantum many-body Hamiltonian quartic in the bosonic creation-annihilation operators. This Hamiltonian originated from the works of Gérard and Grellier on classical turbulent energy transfer in a Lax-integrable setting, and its quantum version respects hierarchies of polynomial conservation laws all of which possess purely integer spectra, as does the Hamiltonian itself. The latter feature, known from numerical studies, has previously remained unproved. Here, we present an elementary solution of the quantum Gérard-Grellier Hamiltonian. The eigenstates are labelled by partitions and the corresponding eigenvalues admit simple explicit expressions in terms of the principal hook lengths. The key ingredient in this construction is a specific basis we introduce, inspired by the classical multisoliton coherent states, in which the Hamiltonian is upper-triangular. The expressions for the eigenvalues then immediately follow, while the full eigenbasis can be recovered via a Gram-Schmidt procedure.

nlin.SI↗
arXiv2026

Ground state and excitations of the \(C_2^{(1)}\) spin chain with non-diagonal boundaries

Understanding boundary effects in higher-rank integrable quantum spin chains requires connecting exact spectral relations with the root configurations underlying their energies and excitations. With non-diagonal boundaries, determining these configurations remains challenging even when an exact spectral solution is available, hindering the direct extraction of thermodynamic properties. We address this problem for the rational $C_2^{(1)}$ chain by combining DMRG in the tensor-network framework with the quantum inverse scattering method. The numerically determined zeros of the fundamental and fused transfer-matrix eigenvalues reveal three bulk-string distributions, labeled by $x$, $y$, and $w$, together with boundary strings and additional roots. Substituting these configurations into the operator identities relating the two transfer matrices yields the ground-state energy and the surface energy in the thermodynamic limit under non-diagonal boundary conditions. The analytical surface energy agrees with the numerical results up to $O(\tfrac1N)$ corrections. From the regularity conditions of the operator identities, we derive a boundary-string selection rule and numerically identify allowed configurations beyond the ground state. For excitations preserving the boundary-string configuration, we identify three elementary hole excitations, $X$, $Y$, and $W$, in the $x$-, $y$-, and $w$-string distributions, respectively, and derive their energies. Based on the symmetry of the zero roots, we also present a bulk-excitation selection rule governing the allowed combinations of these elementary excitations in physical eigenstates. The framework developed here can therefore be used to systematically analyze boundary effects, bulk excitations, and related physical properties in other high-rank integrable quantum models with non-diagonal boundary conditions.

nlin.SI↗