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

Abstract

Abstract. The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.2000 Mathematics Subject Classification. 53A30, 35Q58. Introduction.Due to the (re-)discovery of a variety of important connections between the differential geometry of surfaces and integrable systems, (classical and modern) differential geometry has been widely recognised as an integral part of soliton theory (see, e.g., [1]-[3]). 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 . In particular, we establish important connections with both discrete and continuous isothermic surfaces and generalised Clifford point-circle configurations. We also show that the Schwarzian Davey-Stewartson II hierarchy explicitly defines integrable deformations of conformal immersions in R 4 .

Discrete holomorphic geometry I. Darboux transformations and spectral curves

“…Alternatively, we can characterize the Darboux transformation by a non-linear, Möbius invariant zero curvature relation which can be expressed as a multi-ratio condition. This condition already appeared in a number of instances, including the characterization of integrable triangular lattices investigated by Bobenko, Hoffmann and Suris [1] and in the space and time discrete versions of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equation introduced by Konopelchenko and Schief [25,26]. From our point of view the former correspond to 3-periodic sequences of Darboux transforms and the latter to a sequence of iterated Darboux transforms.…”

“…Like the Darboux transformation for immersions of smooth surfaces is a time-discrete version of the Davey-Stewartson flow, see [6], one expects the Darboux transformation of discrete surfaces to be a space-and time-discrete version of the Davey-Stewartson flow. This is indeed the case: for immersed discrete surfaces with regular combinatorics, the Darboux transformation can be interpreted as the space-and time-discrete Davey-Stewartson flow introduced by Konopelchenko and Schief [26]. The case of immersions into S 2 = CP 1 is a special reduction of the theory corresponding to the double discrete KP equation [25].…”

“…As an example we briefly discuss a special case of our theory. Let W be the complex holomorphic line bundle over the vertices of the regular triangulation of the plane all of whose holomorphic sections are of the form ψ = ϕf , where ϕ is a trivializing holomorphic section and f is a complex function that maps all black triangles to positive equilateral triangles in C. [26] if and only if it satisfies the multi ratio equation…”

Möbius--Lie Geometry and Its Extension

“…The key analytic properties of continued fractions-their convergence-can be linked to asymptotic behaviour of such an infinite ensemble [13]. (4) A remarkable relation exists between discrete integrable systems and Möbius geometry of finite configurations of cycles [16,[55][56][57]69]. It comes from "reciprocal force diagrams" used in 19th-century statics, starting with J.C. Maxwell.…”

Hirota equation and the quantum plane

“…In these geometry books, which summarize classical XIX-th century style developments in theory of submanifolds and their transformations, one can recognize many fundamental facts from the theory of integrable partial differential equations. In looking for analogous geometric interpretation of integrable partial difference systems we have found that very often their integrability features are encoded in incidence geometry theorems of Pappus, Desargues, Pascal, Miquel and others [21,31,32,34,36], compare also works [15,17,16,50,55,56,59,60,61] written in a similar spirit; for introduction to projective geometry and its subgeometries see [24,83].Hirota's discrete Kadomtsev-Petviashvili (KP) equation [52] may be considered as the Holy Grail of integrable systems theory, both on the classical and the quantum level [65]. In the present paper, based on our earlier publications [36,37,42], we review geometric aspects of the non-commutative Hirota system within the framework of projective geometry over division rings.…”