72 citations · 136 across the 10 of their papers we have counts for
11 papers
Finite dimensional reductions of integrable differential-difference equations
Alexander Mikhailov
Integrable partial differential equations, such as the Korteweg--De Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give…
Integrable Volterra hierarchies over nonabelian algebras
J. P. Wang, S. Carpentier, A. V. Mikhailov
Known integrable systems with noncommutative dependent variables are typically formulated over free associative algebras, quantum algebras, or Grassmann algebras. For differential-…
Algebraic quantisation approach to integrable differential-difference equations
Sylvain Carpentier, Alexander V. Mikhailov, Jing Ping Wang
We develop an algebraic quantisation approach, based on quantisation ideals, and apply it to integrable non-Abelian differential--difference equations. We show that the Toda hierar…
Discrete integrable principal chiral field model and its involutive reduction
Jan L. Cieśliński, Alexander V. Mikhailov, Maciej Nieszporski +1
We discuss an integrable discretization of the principal chiral field models equations and its involutive reduction. We present a Darboux transformation and general construction of…
Commutative Poisson algebras from deformations of noncommutative algebras
Alexander V. Mikhailov, Pol Vanhaecke
It is well-known that a formal deformation of a commutative algebra leads to a Poisson bracket on and that the classical limit of a derivation on the…
Hamiltonians for the quantised Volterra hierarchy
Sylvain Carpentier, Alexander V. Mikhailov, Jing Ping Wang
This paper builds upon our recent work, published in Lett. Math. Phys., 112: 94, 2022, where we established that the integrable Volterra lattice on a free associative algebra and t…
Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system
Rhys Bury, Alexander V. Mikhailov, Jing Ping Wang
In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N. We derive soliton solutions of arbitrary rank and…
Anticommutative extension of the Adler map
Sotiris Konstantinou-Rizos, Alexander V. Mikhailov
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 p…
Grassmann extensions of Yang-Baxter maps
Georgi G. Grahovski, Sotiris Konstantinou-Rizos, Alexander V. Mikhailov
In this paper we show that there are explicit Yang-Baxter maps with Darboux-Lax representation between Grassmann extensions of algebraic varieties. Motivated by some recent results…
Darboux transformations and Recursion operators for differential--difference equations
Farbod Khanizadeh, Alexander V. Mikhailov, Jing Ping Wang
In this paper we review two concepts directly related to the Lax representations: Darboux transformations and Recursion operators for integrable systems. We then present an extensi…
Reductions of integrable equations on A.III-type symmetric spaces
V S Gerdjikov, A V Mikhailov, T I Valchev
We study a class of integrable non-linear differential equations related to the A.III-type symmetric spaces. These spaces are realized as factor groups of the form SU(N)/S(U(N-k) x…