Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

DOI10.1063/1.2406056zbMath1121.37058arXivnlin/0410046OpenAlexW2029487626WikidataQ104434197 ScholiaQ104434197MaRDI QIDQ3442188

Maciej Nieszporski, Adam Doliwa, Petr G. Grinevich, Paolo Maria Santíni

Publication date: 16 May 2007

Published in: Journal of Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/nlin/0410046

Mathematics Subject Classification ID

Groups and algebras in quantum theory and relations with integrable systems (81R12) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry (37K25) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions (37K20) Surfaces in Euclidean and related spaces (53A05) Multiplicative and other generalized difference equations (39A20)

Related Items

Moutard transform approach to generalized analytic functions with contour poles ⋮ Moutard transform for generalized analytic functions ⋮ The B-quadrilateral lattice, its transformations and the algebro-geometric construction ⋮ The Green function of the discrete finite-gap one-energy two-dimensional Schrödinger operator on the quad graph ⋮ Isothermic surfaces in sphere geometries as Moutard nets ⋮ A sufficient nonsingularity condition for a discrete finite-gap one-energy two-dimensional Schrödinger operator on the quad-graph ⋮ Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences ⋮ The Green's function of a five-point discretization of a two-dimensional finite-gap Schrödinger operator: the case of four singular points on the spectral curve ⋮ Creation and annihilation of point-potentials using Moutard-type transform in spectral variable ⋮ The dressing chain and one-point commuting difference operators of rank 1 ⋮ Analytic solutions and integrability for bilinear recurrences of order six ⋮ Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants ⋮ Multi-component extension of CAC systems

Cites Work

• Unnamed Item
• Unnamed Item
• Basic properties of discrete analytic functions
• Ocneanu cell calculus and integrable lattice models
• Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
• Two-dimensional ``inverse scattering problem for negative energies and generalized-analytic functions. I: Energies below the ground state
• Quantum difference nonlinear Schrödinger equation
• The Laplacian and Dirac operators on critical planar graphs
• The symmetric, \(d\)-invariant and Egorov reductions of the quadrilateral lattice
• Exactly solvable SOS models. Local height probabilities and theta function identities
• On Hirota's difference equations
• Tata lectures on theta. I: Introduction and motivation: Theta functions in one variable. Basic results on theta functions in several variables. With the assistance of C. Musili, M. Nori, E. Previato, and M. Stillman
• Theta functions on Riemann surfaces
• Classification of integrable equations on quad-graphs. The consistency approach
• The discrete Korteweg-de Vries equation
• Projective generalizations of Lelieuvre's formula
• Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrödinger operator
• Superposition principles associated with the Moutard transformation: an integrable discretization of a (2+1)–dimensional sine–Gordon system
• Integrable and solvable systems, and relations among them
• Bäcklund transformations and nonlinear differential difference equations
• Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds
• Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function
• Potential theory on a rhombic lattice
• THE SELF-ADJOINT 5-POINT AND 7-POINT DIFFERENCE OPERATORS, THE ASSOCIATED DIRICHLET PROBLEMS, DARBOUX TRANSFORMATIONS AND LELIEUVRE FORMULAE
• Discrete Riemann surfaces and the Ising model.