On Negatons of the Toda Lattice

Abstract

Möbius invariant versions of the discrete Darboux, KP, BKP and CKP equations are derived by imposing elementary geometric constraints on an (irregular) lattice in a three-dimensional Euclidean space. Each case is represented by a fundamental theorem of plane geometry. In particular, classical theorems due to Menelaus and Carnot are employed. An interpretation of the discrete CKP equation as a permutability theorem is also provided.

The B-quadrilateral lattice, its transformations and the algebro-geometric construction

“…During last years the many results describing the well known connection between integrable partial differential equations and the differential geometry of submanifolds have been transfered to the discrete (difference) level, see for example [4,19,43]. The interest in such a research is stimulated from various fields, like computer visualization, combinatorics, lattice models in statistical mechanics and quantum field theory, and recent developments in quantum gravity.…”

“…The linear problem and the Darboux-type (Moutard) transformations for the discrete BKP equation (1.1) were constructed in [38]. In literature there are known several geometric interpretations of the discrete BKP equation in terms of the reciprocal figures and inversive geometry [29,43], or in terms of the trapezoidal nets [4]. It should be also mentioned that the discrete BKP equation has been recently investigated, under the name of the cube recurrence, in combinatorics [40,25,8].…”

Billiard Algebra, Integrable Line Congruences, and Double Reflection Nets

“…We show that double reflection nets induce a subclass of the Grassmannian Darboux nets from [3] associated with pencils of quadrics. As we know after Schief (see [18]) the Darboux nets are associated to discrete integrable hierarchies. In Sec.…”

Conformal Geometry of the (Discrete) Schwarzian Davey-Stewartson Ii Hierarchy

“…However, the fundamental nature of geometry in the context of integrable systems is a subject of ongoing research and recent investigations have uncovered unexpected geometric links. For instance, it has been established that Hirota's master equation [4] in its Schwarzian form and the associated scalar Schwarzian Kadomtsev-Petviashvili (SKP) hierarchy are encapsulated in Menelaus' classical theorem of plane geometry [5]- [7].…”

“…However, the fundamental nature of geometry in the context of integrable systems is a subject of ongoing research and recent investigations have uncovered unexpected geometric links. For instance, it has been established that Hirota's master equation [4] in its Schwarzian form and the associated scalar Schwarzian Kadomtsev-Petviashvili (SKP) hierarchy are encapsulated in Menelaus' classical theorem of plane geometry [5]- [7].In the present paper, we embark on a study of the geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue, the quaternionic discrete SKP (qdSKP) equation. We demonstrate that the qdSKP equation and various associated continuum limits are canonical objects of conformal (Möbius) geometry in R 4 .…”