Transformations of Quadrilateral Lattices

Abstract

Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogues: the multidimensional quadrilateral lattices, i.e. lattices x: Z^N -> R^M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analogue of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between "transformations" and "discretizations" is also investigated for quadrilateral lattices. We finally interpret these results within the D-bar formalism.

50 pages, 15 figures; minor corrections, added references

References in corpus (5)

• Multidimensional Quadrilateral Lattices are Integrable
• Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies
• Geometric Discretisation of the Toda System
• Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences
• Generalized integrable hierarchies and Combescure symmetry transformations

Cited by in corpus (35)

• The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
• Multivariate orthogonal polynomial and integrable systems
• The multicomponent 2D Toda hierarchy: Discrete flows and string equations
• Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy
• Desargues maps and the Hirota-Miwa equation
• The B-quadrilateral lattice, its transformations and the algebro-geometric construction
• The Hirota equation over finite fields. Algebro-geometric approach and multisoliton solutions
• The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota-Miwa system
• Singularity confinement for maps with the Laurent property
• The C-(symmetric) quadrilateral lattice, its transformations and the algebro-geometric construction
• Discrete KP equation with self-consistent sources
• Non-commutative lattice modified Gel'fand-Dikii systems
• Discrete asymptotic nets and W-congruences in Plucker line geometry
• Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrodinger operator
• Geometric discretization of the Koenigs nets
• Pseudospherical surfaces on time scales: a geometric definition and the spectral approach
• The pentagon relation and incidence geometry
• Asymptotic lattices and their integrable reductions I: the Bianchi and the Fubini-Ragazzi lattices
• Integrable Systems and Discrete Geometry
• Quantization of three-wave equations
• Discrete time Toda systems
• Discrete line complexes and integrable evolution of minors
• Generalized isothermic lattices
• Geometric discretization of the Bianchi system
• Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices
• Integrable dynamics of Toda-type on the square and triangular lattices
• Darboux transformations for linear operators on two dimensional regular lattices
• Discrete Darboux system with self-consistent sources and its symmetric reduction
• Integrable discrete nets in Grassmannians
• Discrete $\Omega$-nets and Guichard nets via discrete Koenigs nets
• Infinitesimal Darboux transformation and semi-discrete mKdV equation
• Darboux transformations for a 6-point scheme
• On $\tau$-function of the quadrilateral lattice
• Discrete minimal nets with symmetries
• Bäcklund transformations as integrable discretization. The geometric approach