Generalized isothermic lattices

Doliwa, Adam

doi:10.1088/1751-8113/40/42/s03

Cited by 12 publications

(21 citation statements)

References 43 publications

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“…Given integrable reduction of the 3D Moutard system (for example, the generalized isothermic lattice [15]), by restricting it to the quasiregular rhombic tiling one can obtain the corresponding integrable system on the triangular or on the honeycomb lattice. In particular, to obtain the constant angles circle patterns [5] one starts from further reduction of the generalized isothermic lattice.…”

Section: Discussionmentioning

confidence: 99%

Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

Doliwa

Nieszporski

Santini

2007

Journal of Mathematical Physics

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Abstract. An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete 3D system is constructed.

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Section: Discussionmentioning

confidence: 99%

Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

Doliwa

Nieszporski

Santini

2007

Journal of Mathematical Physics

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“…At this point we may study the quadrilateral lattice maps and the corresponding discrete Darboux equations, their Darboux-type transformations and reductions; apart from above cited works see also [10,58,3,41,93] for geometric but also analytic (in the case of the field of complex or real numbers) tools to study such maps and corresponding solutions of the discrete Darboux system. We remark that the pioneering works of A. Bobenko and U. Pinkall and their collaborators on discrete isothermic surfaces [12,51,88] can be directly incorporated in the theory of multidimensional lattices of planar quadrilaterals [15,33]. Also discrete pseudospherical surfaces [11] together with more general discrete asymptotic surfaces [85] can be considered as reductions of quadrilateral lattices indirectly via the Plücker embedding [30]; see also other related works [57,13,87,75,43,91].…”

Section: Desargues Maps and Multidimensional Quadrilateral Latticesmentioning

confidence: 90%

Hirota equation and the quantum plane

Doliwa¹

2013

Algebraic and Geometric Aspects of Integrable Systems and Random Matrices

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We discuss geometric integrability of Hirota's discrete KP equation in the framework of projective geometry over division rings using the recently introduced notion of Desargues maps. We also present the Darboux-type transformations, and we review symmetries of the Desargues maps from the point of view of root lattices of type A and the action of the corresponding affine Weyl group. Such a point of view facilities to study the relation of Desargues maps and the discrete conjugate nets. Recent investigation of geometric integrability of Desargues maps allowed to introduce two maps satisfying functional pentagon equation. Moreover, the ultra-locality requirement imposed on the maps leads to Weyl commutation relations. We show that the pentagonal property of the maps allows to define a coproduct in the quantum plane bi-algebra, which can be extended to the corresponding Hopf algebra.The relevance of a geometric theorem is determined by what the theorem tells us about space, and not by the eventual difficulty of the proof. The Desargues' theorem of projective geometry comes as close as a proof can to the Zen ideal. It can be summarized in two words: "I see!" Nevertheless, Desargues' theorem, far from trivial despite the simplicity of its proof, has many more applications both in geometry and beyond ... Gian Carlo Rota, The Phenomenology of Mathematical Proof IntroductionEverybody interested in the (pre)history of soliton theory should consult monographs of Bianchi [8], Darboux [25,26], Eisenhart [47] or Tzitzéica [96]. 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. The crucial notion here is that of Desargues maps, where the underlying geometric property is collinearity of three points, which gives the linear problem for the Hirota system. This should be considered as further simplification of (already rather non-complicated) approach to integrable discrete geometry via the theory of multidimensional quadrilateral lattices [39] based on coplanarity of four points. We remark that the quadrilateral lattice is the integrable discrete analogue [84,28] of the co...

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“…Roughly speaking, it is associated with the existence of a huge amount of internal or hidden symmetry and this fact explains quite well the predictability and regularity which characterize the integrable systems. Lattice equations (partial difference or P∆E's), which exhibit a higher complexity, attracted many studies in the last years [1][2][3][4][5]. The main progress was possible due to appearance of some efficient tools in proving integrability such as singularity confinement [6,7], "cube consistency" [8][9][10] and complexity growth [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

On some new forms of lattice integrable equations

Babalic

Cârstea

2014

Open Physics

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Abstract:Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon -type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings. PACS

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Generalized isothermic lattices

Cited by 12 publications

References 43 publications

Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

Hirota equation and the quantum plane

On some new forms of lattice integrable equations

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