
Anastasia DoikouHeriot-Watt University · Department of Mathematics
Anastasia Doikou
PhD
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165
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Publications (165)
We study solutions of the parametric set-theoretic reflection equation from an algebraic perspective by employing recently derived generalizations of the familiar shelves and racks, called parametric (p)-shelves and racks. Generic invertible solutions of the set-theoretic reflection equation are also obtained by a suitable parametric twist. The twi...
We produce novel non‐involutive solutions of the Yang–Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces, they are not necessarily involutive. In the case of two‐sided (skew) braces, one can assign such solutions to every element...
The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and lead to solutions of the Yang-Baxter equation. We also recall that non-involutive solutions of the braid equation...
We present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel’d homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the underlying Yang–Baxter algebra for solutions of the braid equation associated to a given quan...
The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step towards this objective is the introduction of certain generalizations of the familiar shelves and racks called parametric (p)-shelves and racks. These objects satisfy a parametric self-distributivity condition and lea...
We present connections between left non-degenerate solutions of set-theoretic Yang-Baxter equation and left shelves using certain maps called Drinfel'd homomorphisms. We further generalise the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the Yang-Baxter algebra for solutions of the braid equation associa...
The workshop was focused on the interplay between set-theoretic solutions to the Yang–Baxter equation and several algebraic structures used to construct and understand new solutions. In this vein, the YBE and properties of these algebraic structures are used as a source of inspiration to study other mathematical problems not directly related to the...
Motivated by recent findings on the derivation of parametric noninvolutive solutions of the Yang–Baxter equation, we reconstruct the underlying algebraic structures, called near braces. Using the notion of the near braces we produce new multi‐parametric, nondegenerate, noninvolutive solutions of the set‐theoretic Yang–Baxter equation. These solutio...
We present resent results regarding involutive, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties necessary for the solution of the set theoretic Yang-Baxter equation. We then consider lambda parametric set-theoretic solutions of...
Motivated by recent findings on the derivation of parametric non-involutive solutions of the Yang-Baxter equation we reconstruct the underlying algebraic structures, called near braces. Using the notion of the near braces we produce new multi-parametric, non-degenerate, non-involutive solutions of the set-theoretic Yang-Baxter equation. These solut...
We review the discrete evolution problem and the corresponding solution as a discrete Dyson series in order to rigorously derive a discrete version of Magnus expansion. We also systematically derive the discrete analogue of the pre-Lie Magnus expansion and show that the elements of the discrete Dyson series are expressed in terms of a tridendriform...
We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang–Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists, we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fac...
We examine links between the theory of braces and set-theoretical solutions of the Yang–Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a nove...
We produce novel non-involutive solutions of the Yang-Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from the standard braces and skew braces, and surprisingly in the case of braces they are not necessarily involutive. In the case of two-sided (skew) braces one can assign such solutions to ev...
We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact...
We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable in the sense that they are realisable as Fred-holm Grassmannian flows. In other words, time-evolutionary solutions to such systems can be constructed from solutions to the corresponding underyling linear partial differential system, by s...
We demonstrate how many classes of Smoluchowski-type coagulation models can be realised as Grassmannian or nonlinear graph flows and are therefore linearisable, and/or integrable in this sense. We first prove that a general Smoluchowski-type equation with a constant frequency kernel, that encompasses a large class of such models, is realisable as a...
We consider involutive, non-degenerate, finite set-theoretic solutions of the Yang–Baxter equation (YBE). Such solutions can be always obtained using certain algebraic structures that generalize nilpotent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent prelimi...
Connections between set-theoretic Yang–Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic R -matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R -matrices being Baxterized solutions...
The time evolution problem for non-self-adjoint second order differential operators is studied by means of the path integral formulation. Explicit computation of the path integral via the use of certain underlying stochastic differential equations, which naturally emerge when computing the path integral, leads to a universal expression for the asso...
We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary...
We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r -matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable sy...
We present a method for linearising classes of matrix-valued nonlinear partial differential equations with local and nonlocal nonlinearities. Indeed we generalise a linearisation procedure originally developed by Pöppe based on solving the corresponding underlying linear partial differential equation to generate an evolutionary Hankel operator for...
We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable sys...
We consider space discretizations of the matrix Zakharov–Shabat AKNS scheme, in particular the discrete matrix non-linear Scrhrödinger (DNLS) model, and the matrix generalization of the Ablowitz–Ladik (AL) model, which is the more widely acknowledged discretization. We focus on the derivation of solutions via local Darboux transforms for both discr...
In this paper, we present a method for linearising certain classes of nonlinear partial differential equations. Originally constructed so as to target PDEs with nonlocal nonlinearities, herein we extend our approach in a non-commutative manner that accommodates local nonlinearities as well, thus enabling us to linearise (matrix) integrable systems....
Connections between set theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We provide examples of set theoretic R-matrices expressed as simple twists of known solutions. Based on these solutions we construct the associated "twisted" co-products. Suitable generalizations regarding the q-deformed case are...
The time evolution problem for non-self adjoint second order differential operators is studied by means of the path integral formulation. Explicit computation of the path integral via the use of certain underlying stochastic differential equations, which naturally emerge when computing the path integral, leads to a universal expression for the asso...
We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a nove...
We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a nove...
We consider space discretizations of the matrix Zakharov-Shabat AKNS scheme, in particular the discrete matrix non-linear Scrhrodinger (DNLS) model, and the matrix generalization of the Ablowitz-Ladik (AL) model, which is the more widely acknowledged discretization. We focus on the derivation of solutions via local Darboux transforms for both discr...
We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable. By this we mean that solutions can be generated by solving a corresponding linear partial differential system together with a linear Fredholm integral equation. The flows of such nonlinear systems are examples of linear flows on Fredhol...
We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable. By this we mean that solutions can be generated by solving a corresponding linear partial differential system together with a linear Fredholm integral equation. The flows of such nonlinear systems are examples of linear flows on Fredhol...
We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic...
We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation...
We focus on the non-linear Schrodinger model and we extend the notion of space-time dualities in the presence of integrable time-like boundary conditions. We identify the associated time-like `conserved' quantities and Lax pairs as well as the corresponding boundary conditions. In particular, we derive the generating function of the space component...
We consider the generalized matrix non-linear Schrödinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS-type hierarc...
We focus on the non-linear Schrödinger model and we extend the notion of space-time dualities in the presence of integrable time-like boundary conditions. We identify the associated time-like “conserved” quantities and Lax pairs as well as the corresponding boundary conditions. In particular, we derive the generating function of the space component...
We review some of the fundamental notions associated to the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that display solitonic solutions and are associated to discrete or continuous integrable...
We consider the generalized matrix non-linear Schrodinger hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS hierarchy. These r...
We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic...
We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogo...
We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto...
We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto...
We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogou...
We consider the deformed harmonic oscillator as a discrete version of the Liouville theory and study this model in the presence of local integrable defects. From this, the time evolution of the defect degrees of freedom are determined, found in the form of the local equations of motion. We also revisit the continuous Liouville theory, deriving its...
We explore the notion of the quantum auxiliary linear problem and the associated problem of quantum Backlund transformations (BT). In this context we systematically construct the analogue of the classical formula that provides the whole hierarchy of the time components of Lax pairs at the quantum level for both closed and open integrable lattice mo...
We show how solutions to a large class of Riccati evolutionary nonlinear partial differential equations can be generated from the corresponding linearized equations. The key is an integral equation analogous to the Marchenko equation, or more generally dressing transformation, in integrable systems. We show explicitly how this can be achieved for s...
We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto...
We consider the discrete and continuous vector non-linear Schrodinger (NLS) model.
We focus on the case where space-like local discontinuities are present, and we are primarily
interested in the time evolution on the defect point. This in turn yields the time part of
a typical Darboux-Backlund transformation. Within this spirit we then explicitly w...
We consider the discrete and continuous vector non-linear Schrodinger (NLS) model. We focus on the case where space-like local discontinuities are present, and we are primarily interested in the time evolution on the defect point. This in turn yields the time part of a typical Darboux-Backlund transformation. Within this spirit we then explicitly w...
We consider the deformed harmonic oscillator as a discrete version of the Liouville theory and study this model in the presence of local integrable defects. From this, the time evolution of the defect degrees of freedom are determined, found in the form of the local equations of motion. We also revisit the continuous Liouville theory, deriving its...
We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the equations of motion on the defect point via the space-like and time-like description. We then exploit the structural similarity of these equations with the discrete and continuous Backlund transformations. And although these equation...
We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the "equations of motion" on the defect point via the space-like and time-like description. We then exploit the structural similarity of these equations with the discrete and continuous Backlund transformations. And although these equati...
We study the bulk and boundary scattering of the sl(N) twisted Yangian spin
chain via the solution of the Bethe ansatz equations in the thermodynamic
limit. Explicit expressions for the scattering amplitudes are obtained and the
factorization of the bulk scattering is shown. The issue of defects in twisted
Yangians is also briefly discussed.
We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schrödinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zero-curvature representation of the NLEE can be realized by two Hamiltonia...
We propose a classical analogue of the vertex algebra in the context of
classical integrable field theories. We use this fundamental notion to describe
the auxiliary function of the linear auxiliary problem as a classical vertex
operator. Then using the underlying algebra satisfied by the auxiliary function
together with the linear auxiliary proble...
Inspired by recent results on the effect of integrable boundary conditions on
the bulk behavior of an integrable system, and in particular on the behavior of
an existing defect we systematically formulate the Lax pairs in the
simultaneous presence of integrable boundaries and defects. The respective
sewing conditions as well as the relevant equatio...
14 pages, Latex. A few comments added. Version to appear in JSTAT
We consider the sl(N) twisted Yangian quantum spin chain. In particular, we
study the bulk and boundary scattering of the model via the solution of the
Bethe ansatz equations in the thermodynamic limit. Local defects are also
implemented in the model and the associated transmission amplitudes are derived
through the relevant Bethe ansatz equations.
A quantum spin chain with non-conventional boundary conditions is studied. The distinct nature of these boundary conditions arises from the conversion of a soliton to an anti-soliton after being reflected to the boundary, hence the appellation soliton non-preserving boundary conditions. We focus on the simplest non-trivial case of this class of mod...
The concept of point-like "jump" defects is investigated in the context of
affine Toda field theories. The Hamiltonian formulation is employed for the
analysis of the problem. The issue is also addressed when integrable boundary
conditions ruled by the classical twisted Yangian are present. In both periodic
and boundary cases explicit expressions o...
Inspired by recent results on the effect of integrable boundary conditions on the bulk behavior of an integrable system, and in particular on the behavior of an existing defect we systematically formulate the Lax pairs in the simultaneous presence of integrable boundaries and defects. The respective sewing conditions as well as the relevant equatio...
Inspired by recent results on the effect of integrable boundary conditions on the bulk behavior of an integrable system, and in particular on the behavior of an existing defect we systematically formulate the Lax pairs in the simultaneous presence of integrable boundaries and defects. The respective sewing conditions as well as the relevant equatio...
Inspired by recent results on the effect of integrable boundary conditions on the bulk behavior of an integrable system, and in particular on the behavior of an existing defect we systematically formulate the Lax pairs in the simultaneous presence of integrable boundaries and defects. The respective sewing conditions as well as the relevant equatio...
Classical integrable impurities associated to high rank (gl_N) algebras are
investigated. A particular prototype i.e. the vector non-linear Schr\"{o}dinger
(NLS) model is chosen as an example. A systematic construction of local
integrals of motion as well as the time components of the corresponding Lax
pairs is presented based on the underlying cla...
Type-I quantum defects are considered in the context of the gl_N spin chain.
The type-I defects are associated to the generalized harmonic oscillator
algebra, and the chosen defect matrix is the one of the vector non-linear
Schrodinger (NLS) model. The transmission matrices relevant to this particular
type of defects are computed via the Bethe ansa...
The and quantum spin chains in the presence of integrable spin
impurities are considered. Within the Bethe ansatz formulation, we
derive the associated transmission amplitudes, and the corresponding
transmission matrices — representations of the underlying
quadratic algebra — that physically describe the interaction
between the various particle-lik...
Type-I quantum impurities are investigated in the context of the integrable
Heisenberg model. This type of defects is associated to the (q)-harmonic
oscillator algebra. The transmission matrices associated to this particular
type of defects are computed via the Bethe ansatz methodology for the XXX
model, as well as for the critical and non-critical...
The \( \mathfrak{g}{{\mathfrak{l}}_{\mathcal{N}}} \) and \( {{\mathfrak{U}}_q}\left( {\mathfrak{g}{{\mathfrak{l}}_{\mathcal{N}}}} \right) \) quantum spin chains in the presence of integrable spin impurities are considered. Within the Bethe ansatz formulation, we derive the associated transmission amplitudes, and the corresponding transmission matri...
The sine-Gordon model in the presence of dynamical integrable defects is
investigated. This is an application of the algebraic formulation introduced
for integrable defects in earlier works. The quantities in involution as well
as the associated Lax pairs are explicitly extracted. Integrability i also
shown using certain sewing constraints, which e...
We consider the Heisenberg spin chain in the presence of integrable spin
defects. Using the Bethe ansatz methodology, we extract the associated
transmission amplitudes, that describe the interaction between the
particle-like excitations displayed by the models and the spin impurity. In the
attractive regime of the XXZ model, we also derive the brea...
Point-like Liouville integrable dynamical defects are introduced in the
context of the Landau-Lifshitz and Principal Chiral (Faddeev-Reshetikhin)
models. Based primarily on the underlying quadratic algebra we identify the
first local integrals of motion, the associated Lax pairs as well as the
relevant sewing conditions around the defect point. The...
Application of our algebraic approach to Liouville integrable defects is
proposed for the sine-Gordon model. Integrability of the model is ensured by
the underlying classical r-matrix algebra. The first local integrals of motion
are identified together with the corresponding Lax pairs. Continuity conditions
imposed on the time components of the ent...
Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integr...
In this paper, solutions of the generic non-compact Weyl equation are
obtained. In particular, by identifying a suitable similarity transformation
and introducing a non-trivial change of variables we are able to implement
azimuthal dependence on the solutions of the diagonal non-compact Weyl
equation. We also discuss some open questions related to...
A systematic approach to Liouville integrable defects is proposed, based on
an underlying Poisson algebraic structure. The non-linear Schrodinger model in
the presence of a single particle-like defect is investigated through this
algebraic approach. Local integrals of motions are constructed as well as the
time components of the corresponding Lax p...
We construct SU(n + 1) Bogomolny-Prasad-Sommerfeld (BPS) spherically symmetric monopoles with minimal symmetry breaking by solving the full Weyl equation. In this context, we explore and discuss the existence of open spin chainlike part within the Weyl equation. For instance, in the SU(3) case the relevant spin chain is the 2-site spin 1/2XXX chain...
We apply the ADHMN construction to obtain the SU(n+1) (for generic
values of n) spherically symmetric BPS monopoles with minimal symmetry
breaking. In particular, the problem simplifies by solving the Weyl
equation, leading to a set of coupled equations, whose solutions are
expressed in terms of the Whittaker functions. Next, this construction
is g...
The discrete non-linear Schrodinger (NLS) model in the presence of an
integrable defect is examined. The problem is viewed from a purely algebraic
point of view, starting from the fundamental algebraic relations that rule the
model. The first charges in involution are explicitly constructed, as well as
the corresponding Lax pairs. These lead to set...
We study the classical generalized gl(n) Landau-Lifshitz (L-L) model with
special boundary conditions that preserve integrability. We explicitly derive
the first non-trivial local integral of motion, which corresponds to the
boundary Hamiltonian for the sl(2) L-L model. Novel expressions of the modified
Lax pairs associated to the integrals of moti...
A non-compact version of the Weyl equation is proposed, based on the infinite dimensional spin zero representation of the
\mathfraks\mathfrakl2 \mathfrak{s}{\mathfrak{l}_2} algebra. Solutions of the aforementioned equation are obtained in terms of the Kummer functions. In this context, we discuss
the ADHMN approach in order to construct the corre...
We present certain classical continuum long wave-length limits of prototype
integrable quantum spin chains, and define the corresponding construction of
classical continuum Lax operators. We also provide two specific examples, i.e.
the isotropic and anisotropic Heisenberg models.
In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equation...
A general framework for obtaining certain types of contracted and centrally extended algebras is reviewed. The whole process relies on the existence of quadratic algebras, which appear in the context of boundary integrable models.
We apply the ADHMN construction to obtain the SU(n+1)(for generic values of n) spherically symmetric BPS monopoles with minimal symmetry breaking. In particular, the problem simplifies by solving the Weyl equation, leading to a set of coupled equations, whose solutions are expressed in terms of the Whittaker functions. Next, this construction is ge...
We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gln magne...