71 research outputs found
Darboux transformations, discrete integrable systems and related Yang-Baxter maps
No full textDarboux 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
Entwining Yang–Baxter maps related to NLS type equations
No full textWe construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense
Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
No full textWe present several algebraic and differential-geometric constructions of
tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov
tetrahedron equation. In particular, we obtain a family of new (nonlinear)
polynomial tetrahedron maps on the space of square matrices of arbitrary size,
using a matrix refactorisation equation, which does not coincide with the
standard local Yang--Baxter equation. Liouville integrability is established
for some of these maps.
Also, we show how to derive linear tetrahedron maps as linear approximations
of nonlinear ones, using Lax representations and the differentials of nonlinear
tetrahedron maps on manifolds. We apply this construction to two nonlinear
maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton
solutions of vector KP equations and a tetrahedron map obtained in
[arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a
Darboux matrix associated with a Lax operator for the NLS equation. We derive
parametric families of new linear tetrahedron maps (with nonlinear dependence
on parameters), which are linear approximations for these nonlinear ones.
Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron
map from Sergeev's classification [arXiv:solv-int/9709006]. These matrix
generalisations can be regarded as tetrahedron maps in noncommutative
variables.
Besides, several tetrahedron maps on arbitrary groups are constructed.Comment: 21 pages; v3: minor corrections in Example 5.12, references update
A discrete Darboux-Lax scheme for integrable difference equations
No full textWe propose a discrete Darboux-Lax scheme for deriving auto-Backlund 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 Schroedinger (NLS) equation [21]. In particular, we construct an auto-Backlund 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
Conference of Mathematical Physics Kezenoi-Am 2016
No full textThis volume, whose contributors include leading researchers in their field, covers a wide range of topics surrounding Integrable Systems, from theoretical developments to applications. Comprising a unique collection of research articles and surveys, the book aims to serve as a bridge between the various areas of Mathematics related to Integrable Systems and Mathematical Physics. Recommended for postgraduate students and early career researchers who aim to acquire knowledge in this area in preparation for further research, this book is also suitable for established researchers aiming to get up to speed with recent developments in the area, and may very well be used as a guide for further study
'Les Perses de l'Occident' de Sotiris Skipis: Une invasion barbare en Europe dans les années 1920
No full textThe study examines the contribution of Sotiris Skipis (1881-1952) to theatrical art, an aspect which has been neglected by contemporary research. It focuses on an almost unknown three-act drama of Skipis, which the present research retrieved from the rare collections of Greek libraries: Les Perses de l' Occident (The Persians of the West). The author published the work in Paris in 1917, before the bloodshed of World War I was beginning to wane, to openly condemn pan-Germanism and racism, even before the latter term prevailed. The paper explores the ideological, aesthetic, and dramaturgical substratum of the text reflecting the spread of racial theories in Europe is explored. It also interprets the reasons that did not allow for the stage realization of the play on Parisian stages at the time of publication, despite the warm praise from Parisian scholarship, and that it took more than a century to pull the play out of oblivion
A noncommutative discrete potential KdV lift
No full textIn this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared in the work of Kouloukas and Papageorgiou [J. Phys. A: Math. Theor. 42, 404012 (2009)] and can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension satisfies the Yang-Baxter equation, and it admits a 3 × 3 Lax matrix. Moreover, we show that it can be squeezed down to a novel system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, we consider commutative analogs of the constructed Yang-Baxter map and its associated quad-graph system, and we discuss their integrability
Integrable extensions of the Adler map via Grassmann algebras
No full textWe study certain extensions of the Adler map on Grassmann algebras Γ(n) of order n. We consider a known Grassmann-extended Adler map and under the assumption that n =1, obtain a commutative extension of the Adler map in six dimensions. We show that the map satisfies the Yang–Baxter equation, admits three invariants, and is Liouville integrable. We solve the map explicitly by regarding it as a discrete dynamical system.</p
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