ZN graded discrete Lax pairs and Yang–Baxter maps

Fordy, Allan P.; Xenitidis, Pavlos

doi:10.1098/rspa.2016.0946

Cited by 4 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In our calculations we also have to take into account that det(B) is a constant, an obvious consequence of (11) and our assumption that the Lax matrices have constant determinants.…”

Section: Integrability Of Difference Equationsmentioning

confidence: 99%

“…and the determining equations (11), which now become λL (1) + L (2) (p 00 , q10 ; α) λB (1) + B (0) = λS B (1) + S B (0) λL (1) + L (2) (p 00 , q 10 ; α) , (17a)…”

Section: Integrability Of Difference Equationsmentioning

confidence: 99%

“…[22,27,21,29], and relating them to the theory of Yang-Baxter and tetrahedron maps, e.g. [11,25]. Of particular interest are the so-called 3D-consistent equations which can be extended to the three-dimensional lattice in a consistent way.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A discrete Darboux-Lax scheme for integrable difference equations

Fisenko,

Konstantinou-Rizos,

Xenitidis

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We propose a discrete Darboux-Lax scheme for deriving auto-Bäcklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler-Yamilov type system which is related to the nonlinear Schrödinger (NLS) equation [19]. In particular, we construct an auto-Bäcklund transformation for this discrete system, its superposition principle, and we employ them in the construction of the one-and two-soliton solutions of the Adler-Yamilov system.

show abstract

“…In our calculations we also have to take into account that det(B) is a constant, an obvious consequence of (11) and our assumption that the Lax matrices have constant determinants.…”

Section: Integrability Of Difference Equationsmentioning

confidence: 99%

“…and the determining equations (11), which now become λL (1) + L (2) (p 00 , q10 ; α) λB (1) + B (0) = λS B (1) + S B (0) λL (1) + L (2) (p 00 , q 10 ; α) , (17a)…”

Section: Integrability Of Difference Equationsmentioning

confidence: 99%

See 1 more Smart Citation

A discrete Darboux-Lax scheme for integrable difference equations

Fisenko,

Konstantinou-Rizos,

Xenitidis

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…A quite general construction for tetrahedron maps first appeared in the work of Korepanov [35] in connection with integrable dynamical systems in discrete time. Presently, the relations of tetrahedron maps and Yang-Baxter maps with integrable systems (including PDEs and lattice equations) is a very active area of research (see, e.g., [1,2,7,9,12,15,23,26,27,29,34,35,38,42,46] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Tetrahedron maps, Yang-Baxter maps, and partial linearisations

Igonin,

Kolesov,

Konstantinou-Rizos

et al. 2021

Preprint

View full text Add to dashboard Cite

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang-Baxter maps, which are set-theoretical solutions to the quantum Yang-Baxter equation.In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Furthermore, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similar results on the differentials of Yang-Baxter and entwining Yang-Baxter maps are also presented.Using the obtained general results, we construct new examples of (parametric) Yang-Baxter and tetrahedron maps. The considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and Müller-Hoissen [9] in a study of soliton solutions of vector Kadomtsev-Petviashvili (KP) equations. Also, we present invariants for this nonlinear tetrahedron map.

show abstract

“…The benefit of such a construction is that all the integrable difference equations arising from such a framework can be written as quadrilateral coupled systems and one thus successfully avoids dealing with higherorder terms. Moreover, the integrability of these integrable difference equations also results in novel Yang-Baxter (YB) maps [22]. However, the exact solutions to the Fordy-Xenitidis (FX) models still needs to be clarified.…”

Section: Introductionmentioning

confidence: 99%

Direct linearization approach to discrete integrable systems associated with ℤ _𝒩 graded Lax pairs

2020

Proc. R. Soc. A.

View full text Add to dashboard Cite

Fordy and Xenitidis [ J. Phys. A: Math. Theor. 50 (2017) 165205. ( doi:10.1088/1751-8121/aa639a )] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of Z N graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel’fand–Dikii hierarchy.

show abstract

ZN graded discrete Lax pairs and Yang–Baxter maps

Cited by 4 publications

References 12 publications

A discrete Darboux-Lax scheme for integrable difference equations

A discrete Darboux-Lax scheme for integrable difference equations

Tetrahedron maps, Yang-Baxter maps, and partial linearisations

Direct linearization approach to discrete integrable systems associated with ℤ _𝒩 graded Lax pairs

Contact Info

Product

Resources

About

ZN graded discrete Lax pairs and Yang–Baxter maps

Cited by 4 publications

References 12 publications

A discrete Darboux-Lax scheme for integrable difference equations

A discrete Darboux-Lax scheme for integrable difference equations

Tetrahedron maps, Yang-Baxter maps, and partial linearisations

Direct linearization approach to discrete integrable systems associated with ℤ 𝒩 graded Lax pairs

Contact Info

Product

Resources

About

Direct linearization approach to discrete integrable systems associated with ℤ _𝒩 graded Lax pairs