Yang-Baxter maps and symmetries of integrable equations on quad-graphs

Papageorgiou, V.; Tongas, Anastasios; Веселов, А. П.

doi:10.1063/1.2227641

Cited by 87 publications

(176 citation statements)

References 31 publications

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“…The map (43) is identical to the Harrison-type map (47) in [18], up to rescaling of the variables. This map is in fact related to the A…”

Section: Bbss and Yb Maps The Classical Bbs Due To Takahashi And Satmentioning

confidence: 97%

“…However, difficulties arise when one tries to interpret the lattice obtained from (29), and it is not clear if it corresponds to a YB map at all. Interestingly enough, the lattices that do correspond to YB maps bear no obvious relation to those discussed in [18] in relation to quadgraphs. These problems, as well as the study of the BBSs related to the above YB maps deserve further attention.…”

Section: Bbss and Yb Maps The Classical Bbs Due To Takahashi And Satmentioning

confidence: 99%

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Yang–baxter Maps and the Discrete Kp Hierarchy

Kakei¹,

Nimmo²,

Willox³

2009

Glasgow Math. J.

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Abstract. We present a systematic construction of the discrete KP hierarchy in terms of Sato-Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang-Baxter maps is explained in two explicit examples.2000 Mathematics Subject Classification. 39A10, 39A12. Introduction.A problem widely recognized as one of the most fundamental open problems in the theory of integrable systems is that of the relation between classical and quantum integrable systems with infinitely many degrees of freedom. Recently a possible route emerged through which a firm connection between the classical and quantum settings might be established: the ultradiscretization of (classical) discrete integrable systems [8,22]. Crystal bases, arising in the zero-temperature limit of quantum enveloping algebras, have been shown to play an important rôle in the description of the dynamical properties of so-called box and ball systems (BBSs) [20], which in turn can be obtained from 1+1-dimensional discrete integrable systems through a special limiting procedure: the ultradiscrete limit [22]. On the other hand, geometric crystals [1] are classical analogues of such crystals. These are such that (quantum) crystal bases can be obtained from them through a limiting procedure very much like the ultradiscrete limit. Geometric crystals have been studied extensively in connection with the combinatorial properties of BBSs [10, 11], but, most importantly, they also offer prime examples of Yang-Baxter (YB) maps [7], i.e. of set-theoretical solutions to the YB equation [6,26]. In particular, it can be shown that the R-matrix associated to such a YB map (or with the tropical version thereof, obtained from the geometric crystal) in the ultradiscrete limit turns into the combinatorial R-matrix that governs the time evolution of the BBS [8]. This R-matrix, in turn, is directly related to the crystal associated with that BBS. The question that remains however,

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“…The map (43) is identical to the Harrison-type map (47) in [18], up to rescaling of the variables. This map is in fact related to the A…”

Section: Bbss and Yb Maps The Classical Bbs Due To Takahashi And Satmentioning

confidence: 97%

Section: Bbss and Yb Maps The Classical Bbs Due To Takahashi And Satmentioning

confidence: 99%

Yang–baxter Maps and the Discrete Kp Hierarchy

Kakei¹,

Nimmo²,

Willox³

2009

Glasgow Math. J.

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“…Since this Poisson structure is obtained from the Sklyanin bracket, we can show that every transfer mapT n ofR α,β will be Poisson with respect to (16) and the integrals that are derived from the trace of the corresponding monodromy matrix will be in involution. Now, we observe that by setting x 1 = y 1 = 0 atR α,β , we derive u 1 = v 1 = 0.…”

Section: Transfer Maps As Reductions Of Higher-dimensional Integrablementioning

confidence: 99%

“…[7,11,14,15,16]). In these cases the YB property reflects the 3-dimensional (3D) consistency property of the corresponding equation.…”

Section: Introduction To Yang-baxter Mapsmentioning

confidence: 99%

Relativistic collisions as Yang–Baxter maps

Kouloukas

2017

Physics Letters A

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We prove that one-dimensional elastic relativistic collisions satisfy the set-theoretical Yang-Baxter equation. The corresponding collision maps are symplectic and admit a Lax representation. Furthermore, they can be considered as reductions of a higher dimensional integrable Yang-Baxter map on an invariant manifold. In this framework, we study the integrability of transfer maps that represent particular periodic sequences of collisions.

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“…The dBKP-system has many equivalent forms and appears in very different contexts. In addition to the known geometric interpretations and a reformulation as Yang-Baiter system ( [8]), the dBKP-system may be considered as a nonlinear superposition principle for the classical 2-dimensional Moutard transformations ( [5]). The expression (5) enjoys an extra symmetry property: the equation Q (−−−) = 0 is invariant under the action of the SL 2 (C) group of fractional-linear transformations…”

mentioning

confidence: 99%

Classification of Three-Dimensional Integrable Scalar Discrete Equations

Tsarëv

Wolf

2008

Lett Math Phys

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We classify all integrable 3-dimensional scalar discrete quasilinear equations Q 3 = 0 on an elementary cubic cell of the lattice Z 3 . An equation Q 3 = 0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the lattice Z 4 . Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system. MSC: 37K10, 52C99

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Yang-Baxter maps and symmetries of integrable equations on quad-graphs

Cited by 87 publications

References 31 publications

Yang–baxter Maps and the Discrete Kp Hierarchy

Yang–baxter Maps and the Discrete Kp Hierarchy

Relativistic collisions as Yang–Baxter maps

Classification of Three-Dimensional Integrable Scalar Discrete Equations

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