Birational solutions to the set-theoretical 4-simplex equation

The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov's tetrahedron equation and the Yang--Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a Kadomtsev--Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schr\"odinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.Comment: Revised version, published in Physica

Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang--Baxter equation. Sergeev [S.M. Sergeev 1998 Lett. Math. Phys. 45, 113--119] presented classification results on three-dimensional tetrahedron maps obtained from the local Yang--Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang--Baxter equation for some simple $2\times 2$ matrix-functions, we show that there are (non-unique) solutions to the local Yang--Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang--Baxter equation, we obtain several new birational tetrahedron maps, including maps with matrix Lax representations on arbitrary groups, a $9$-dimensional map associated with a Darboux transformation for the derivative nonlinear Schr\"odinger (NLS) equation, and a $9$-dimensional generalisation of the $3$-dimensional Hirota map.Comment: 18 pages. New results added (section 4), and also the references list was update

A non-commutative extension of the Adler-Yamilov Yang-Baxter map

In this paper, we construct a noncommutative extension of the Adler-Yamilov Yang-Baxter map which is related to the nonlinear Schr�dinger equation. Moreover, we show that this map is partially integrable

Darboux transformations, discrete integrable systems and related Yang-Baxter maps

Darboux transformations constitute a very important tool in the theory of integrable systems. They map trivial solutions of integrable partial differential equations to non-trivial ones and they link the former to discrete integrable systems. On the other hand, they can be used to construct Yang-Baxter maps which can be restricted to completely integrable maps (in the Liouville sense) on invariant leaves. In this thesis we study the Darboux transformations related to particular Lax operators of NLS type which are invariant under the action of the so-called reduction group. Specifically, we study the cases of: 1) the nonlinear Schrödinger equation (with no reduction), 2) the derivative nonlinear Schrödinger equation, where the corresponding Lax operator is invariant under the action of the Z₂-reduction group and 3) a deformation of the derivative nonlinear Schrödinger equation, associated to a Lax operator invariant under the action of the dihedral reduction group. These reduction groups correspond to recent classification results of automorphic Lie algebras. We derive Darboux matrices for all the above cases and we use them to construct novel discrete integrable systems together with their Lax representations. For these systems of difference equations, we discuss the initial value problem and, moreover, we consider their integrable reductions. Furthermore, the derivation of the Darboux matrices gives rise to many interesting objects, such as Bäcklund transformations for the corresponding partial differential equations as well as symmetries and conservation laws of their associated systems of difference equations. Moreover, we employ these Darboux matrices to construct six-dimensional Yang-Baxter maps for all the afore-mentioned cases. These maps can be restricted to four-dimensional Yang-Baxter maps on invariant leaves, which are completely integrable; we also consider their vector generalisations. Finally, we consider the Grassmann extensions of the Yang-Baxter maps corresponding to the nonlinear Schrödinger equation and the derivative nonlinear Schrödinger equation. These constitute the first examples of Yang-Baxter maps with noncommutative variables in the literature

Anticommutative extension of the Adler map

We construct a noncommutative (Grassmann) extension of the well-known Adler Yang–Baxter map. It satisfies the Yang–Baxter equation, it is reversible and birational. Our extension preserves all the properties of the original map except the involutivity