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Alexander V. Mikhailov

Publications and source records attributed to Alexander V. Mikhailov.

At least 19 recordsLinked to original sources
arXiv2026

Global Results on the Classification of Two-Component Integrable Evolutionary Systems

We derive necessary and sufficient integrability conditions for two-component polynomial evolutionary systems of odd order in $(1+1)$ dimensions. Integrable systems are members of infinite hierarchies of commuting symmetries, which are characterised by their spectral invariants. We prove that there are precisely $24$ possible spectral classes of integrable hierarchies. As an application, we obtain a complete classification of integrable homogeneous hierarchies whose lowest-order equations are of order 3 and 5. The resulting classification naturally splits into two classes. The C--integrable systems are reduced, by means of differential substitutions, to linear--triangular form, while the S--integrable systems are related, through linear changes of variables and differential substitutions, to canonical Drinfeld--Sokolov KdV-type systems associated with affine Lie algebras of rank two.

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arXiv2025

Algebraic quantisation approach to integrable differential-difference equations

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 hierarchy admits a bi-quantum structure whose classical (commutative) limit recovers a well-known Poisson pencil. In addition, we discover a non-standard quantisation that has no commutative counterpart. In both cases we present the quantum systems in the Heisenberg form. The generality of the method is illustrated through a wide range of integrable lattices, including the modified Volterra, Bogoyavlensky, Ablowitz-Ladik, relativistic Toda, Merola-Ragnisco-Tu, Adler-Yamilov, Chen-Lee-Liu, Belov-Chaltikian, and Blaszak-Marciniak systems. For each of them, we construct explicit quantisation ideals and present the first few commuting quantum Hamiltonians.

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arXiv2024

Commutative Poisson algebras from deformations of noncommutative algebras

It is well-known that a formal deformation of a commutative algebra ${\mathcal A}$ leads to a Poisson bracket on ${\mathcal A}$ and that the classical limit of a derivation on the deformation leads to a derivation on ${\mathcal A}$, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalisation of it for formal deformations of an arbitrary noncommutative algebra ${\mathcal A}$. The deformation leads in this case to a Poisson algebra structure on $Π({\mathcal A}):=Z({\mathcal A})\times({\mathcal A}/Z({\mathcal A}))$ and to the structure of a $Π({\mathcal A})$-Poisson module on ${\mathcal A}$. The limiting derivations are then still derivations of ${\mathcal A}$, but with the Hamiltonian belong to $Π({\mathcal A})$, rather than to ${\mathcal A}$. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantised Grassmann algebra.

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arXiv2023

Hamiltonians for the quantised Volterra hierarchy

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 the whole hierarchy of its symmetries admits a quantisation dependent on a parameter $ω$. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admits an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter $ω$. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.

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arXiv2023

Quantisations of the Volterra hierarchy

In this paper we explore a recently emerged approach to the problem of quantisation based on the notion of quantisation ideals. We explicitly prove that the nonabelian Volterra together with the whole hierarchy of its symmetries admit a deformation quantisation. We show that all odd-degree symmetries of the Volterra hierarchy admit also a non-deformation quantisation. We discuss the quantisation problem for periodic Volterra hierarchy including their quantum Hamiltonians, central elements of the quantised algebras, and demonstrate super-integrability of the quantum systems obtained. We show that the Volterra system with period $3$ admits a bi-quantum structure, which can be regarded as a quantum deformation of its classical bi-Hamiltonian structure.

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arXiv2020

Automorphic Lie algebras and corresponding integrable systems

We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are infinite dimensional and almost graded. We formulate the concept of a graded isomorphism and classify $sl(2,C)$ based automorphic Lie algebras corresponding to all finite reduction groups. We show that hierarchies of integrable systems, their Lax representations and master symmetries can be naturally formulated in terms of automorphic Lie algebras.

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arXiv2020

Quantisation ideals of nonabelian integrable systems

We consider dynamical systems on the space of functions taking values in a free associative algebra. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commuting symmetries. In this paper we propose a new approach to the problem of quantisation of dynamical systems, introduce the concept of quantisation ideals and provide meaningful examples.

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arXiv2018

PreHamiltonian and Hamiltonian operators for differential-difference equations

In this paper we are developing a theory of rational (pseudo) difference Hamiltonian operators, focusing in particular on its algebraic aspects. We show that a pseudo--difference Hamiltonian operator can be represented as a ratio $AB^{-1}$ of two difference operators with coefficients from a difference field $\mathcal{F}$ where $A$ is preHamiltonian. A difference operator $A$ is called preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket of evolutionary vector fields on $\mathcal{F}$. We show that a skew-symmetric difference operator is Hamiltonian if and only if it is preHamiltonian and satisfies simply verifiable conditions on its coefficients. We show that if $H$ is a rational Hamiltonian operator, then to find a second Hamiltonian operator $K$ compatible with $H$ is the same as to find a preHamiltonian pair $A$ and $B$ such that $AB^{-1}H$ is skew-symmetric. We apply our theory to non-trivial multi-Hamiltonian structures of Narita-Itoh-Bogoyavlensky and Adler-Postnikov equations.

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arXiv2018

Rational recursion operators for integrable differential-difference equations

In this paper we introduce preHamiltonian pairs of difference operators and study their connections with Nijenhuis operators and the existence of weakly non-local inverse recursion operators for differential-difference equations. We begin with a rigorous setup of the problem in terms of the skew field $Q$ of rational (pseudo--difference) operators over a difference field $F$ with a zero characteristic subfield of constants $k\subset F$ and the principal ideal ring $M_n(Q)$ of matrix rational (pseudo-difference) operators. In particular, we give a criteria for a rational operator to be weakly non--local. A difference operator $H$ is called preHamiltonian, if its image is a Lie $k$-subalgebra with respect the the Lie bracket on $F$. Two preHamiltonian operators form a preHamiltonian pair if any $k$-linear combination of them is preHamiltonian. Then we show that a preHamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a preHamiltonian pair. This provides a systematical method to check whether a rational operator is Nijenhuis. As an application, we construct a preHamiltonian pair and thus a Nijenhuis recursion operator for the differential-difference equation recently discovered by Adler \& Postnikov. The Nijenhuis operator obtained is not weakly non-local. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well known examples including the Toda, the Ablowitz-Ladik and the Kaup-Newell differential-difference equations.

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arXiv2016

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.

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arXiv2016

Dressing method for the vector sine-Gordon equation and its soliton interactions

In this paper, we develop the dressing method to study the exact solutions for the vector sine-Gordon equation. The explicit formulas for one kink and one breather are derived. The method can be used to construct multi-soliton solutions. Two soliton interactions are also studied. The formulas for position shift of the kink and position and phase shifts of the breather are given. These quantities only depend on the pole positions of the dressing matrices.

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arXiv2016

Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system

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 $k$ and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmanians ${\rm Gr}_{\mathbb{R}}(k,N)$ and ${\rm Gr}_{\mathbb{C}}(k,N)$.

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arXiv2016

Grassmann extensions of Yang-Baxter maps

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 on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative Nonlinear Schrodinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the Yang-Baxter property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional Yang-Baxter maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations.

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arXiv2015

Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere

We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations we derive new vector Yang-Baxter map and integrable discrete vector sine-Gordon equation on a sphere.

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arXiv2014

Integrable Discretisations for a Class of Nonlinear Schrodinger Equations on Grassmann Algebras

Integrable discretisations for a class of coupled (super) nonlinear Schrodinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initialboundary problems are formulated.

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arXiv2014

Darboux transformation with Dihedral reduction group

We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Backlund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix we obtain local conservation laws of the system.

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