arXiv subjects

Sergei Igonin

Publications and source records attributed to Sergei Igonin.

18 recordsLinked to original sources

arXiv2026

On matrix Lax representations for (1+1)-dimensional evolutionary differential-difference equations

Differential-difference matrix Lax representations (Lax pairs), gauge transformations, and discrete Miura-type transformations (MTs) belong to the main tools in the theory of (nonlinear) integrable differential-difference equations. For a given equation, two matrix Lax representations (MLRs) are said to be gauge equivalent if one of them can be obtained from the other by applying a matrix gauge transformation. Generalizing and extending several previous works on MLRs and MTs, we present new results on the following problems: - When and how can one simplify a given MLR by means of gauge transformations? - How can one use MLRs and gauge transformations for constructing MTs? - A MLR is called fake if it is gauge equivalent to a trivial MLR. How to determine whether a given MLR is not fake? We consider the general (1+1)-dimensional evolutionary differential-difference case when a MLR can depend on any shifts of dependent variables and can be non-autonomous. As applications and illustrations of the presented general theory, we construct several new two-component integrable equations (with new MLRs) connected by new MTs to known integrable equations from the papers [S. Konstantinou-Rizos, A.V. Mikhailov, P. Xenitidis, J. Math. Phys. 2015], [E. Mansfield, G. Mari Beffa, Jing Ping Wang, Found. Comput. Math. 2013]), including non-autonomous examples.

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arXiv2026

The gauge action on semi-discrete Lax representations and its invariants

Semi-discrete (differential-difference) matrix Lax representations (Lax pairs) play an essential role in the theory of integrable differential-difference equations. Fix a (1+1)-dimensional evolutionary differential-difference (semi-discrete) equation and consider matrix Lax representations (MLRs) of this equation. Two MLRs are said to be gauge equivalent if one of them can be obtained from the other by applying a (local) matrix gauge transformation. Gauge transformations (GTs) form an infinite-dimensional group, which acts on the set of MLRs of a given equation. Two MLRs are gauge equivalent iff they belong to the same orbit of this action. When one tries to establish integrability (in the sense of soliton theory) for a given equation, one is interested in MLRs which depend on a parameter (usually called the spectral parameter) such that the parameter cannot be removed by any GT. We introduce and study explicit invariants with respect to the action of GTs on the set of MLRs for a given (1+1)-dimensional evolutionary differential-difference equation with any number of components. Using these invariants, we obtain the following results: - Consider a MLR with a parameter $λ$. If at least one of the invariants computed for this MLR depends nontrivially on $λ$, then the parameter cannot be removed by any GT. - When we have two different MLRs for a given equation, we present necessary conditions for these two MLRs to be gauge equivalent. Our results on semi-discrete MLRs of differential-difference equations are inspired by results of S$.$Yu. Sakovich and M. Marvan on (continuous) zero-curvature representations of partial differential equations. A comparison with some of the results of S$.$Yu. Sakovich and M. Marvan is presented.

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arXiv2025

Matrix Lax pairs under the gauge equivalence relation induced by the gauge group action and Miura-type transformations for lattice equations

In this paper we explore interconnections of differential-difference matrix Lax representations (Lax pairs), gauge transformations, and discrete Miura-type transformations (MTs), which belong to the main tools in the theory of integrable differential-difference (lattice) equations. For a given equation, two matrix Lax representations (MLRs) are said to be gauge equivalent if one of them can be obtained from the other by means of a (local) matrix gauge transformation. Matrix gauge transformations constitute an infinite-dimensional group called the matrix gauge group, which acts naturally on the set of MLRs of a given equation. Two MLRs are gauge equivalent if and only if they belong to the same orbit of the matrix gauge group action. For a wide class of MLRs of (vector) evolutionary differential-difference equations, we present results on the following questions: 1. When and how can one simplify a given MLR by matrix gauge transformations and bring the MLR to a form suitable for constructing MTs? 2. A MLR is called fake if it is gauge equivalent to a trivial MLR. How to determine whether a given MLR is not fake? This allows us to construct new integrable equations (with new MLRs) connected by new MTs to known equations.

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arXiv2024

On matrix Lax representations and constructions of Miura-type transformations for differential-difference equations

This paper is part of a research project on relations between differential-difference matrix Lax representations (MLRs) with the action of gauge transformations and discrete Miura-type transformations (MTs) for (nonlinear) integrable differential-difference equations. The paper addresses the following problem: when and how can one eliminate some shifts of dependent variables from the discrete (difference) part of an MLR by means of gauge transformations? Using results on this problem, we present applications to constructing new MLRs and new integrable equations connected by new MTs to known equations. In particular, we obtain results of this kind for equations connected to the following ones: - The two-component Belov--Chaltikian lattice. - The equation (introduced by G. Mari Beffa and Jing Ping Wang) which describes the evolution induced on invariants by an invariant evolution of planar polygons. - An equation from [S. Igonin, arXiv:2405.08579] related to the Toda lattice in the Flaschka--Manakov coordinates.

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arXiv2024

Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations

Matrix differential-difference Lax pairs play an essential role in the theory of integrable nonlinear differential-difference equations. We present sufficient conditions which allow one to simplify such a Lax pair by matrix gauge transformations. Furthermore, we describe a procedure for such a simplification and present applications of it to constructing new integrable equations connected by (non-invertible) discrete substitutions of Miura type to known equations with Lax pairs. Suppose that one has three (possibly multicomponent) equations $E$, $E_1$, $E_2$, a (Miura-type) discrete substitution from $E_1$ to $E$, and a discrete substitution from $E_2$ to $E_1$. Then $E_1$ and $E_2$ can be called a modified version of $E$ and a doubly modified version of $E$, respectively. We demonstrate how the above-mentioned procedure helps (in the considered examples) to construct modified and doubly modified versions of a given equation possessing a Lax pair satisfying certain conditions. The considered examples include scalar equations of Itoh-Narita-Bogoyavlensky type and $2$-component equations related to the Toda lattice. We present several new integrable equations connected by new discrete substitutions of Miura type to known equations.

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arXiv2022

Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation

We 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.

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arXiv2022

Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability

This paper is devoted to tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. We construct a family of tetrahedron maps on associative rings. We show that matrix tetrahedron maps presented in [arXiv:2110.05998] are a particular case of our construction. This provides an algebraic explanation of the fact that the matrix maps from [arXiv:2110.05998] satisfy the tetrahedron equation. Also, Liouville integrability is established for some of the constructed maps.

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arXiv2020

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. In [arXiv:1303.3575], for any (1+1)-dimensional scalar evolution equation $E$, we defined a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. In [arXiv:1804.04652] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations. The algebras $F(E)$ are defined in terms of generators and relations. In this paper we show that, using the algebras $F(E)$, one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order $5$. Also, we prove a result announced in [arXiv:1303.3575] on the structure of the algebras $F(E)$ for certain classes of equations of orders $3$, $5$, $7$, which include KdV, mKdV, Kaup-Kupershmidt, Sawada-Kotera type equations. Among the obtained algebras for equations considered in this paper and in [arXiv:1804.04652], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus $1$ and $0$. In this approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.

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arXiv2018

Lie algebras responsible for zero-curvature representations of scalar evolution equations

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. For any (1+1)-dimensional scalar evolution equation $E$, we define a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations. As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well. In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs. In this paper we describe general properties of $F(E)$ and present generators and relations for these algebras. In other publications we study the structure of $F(E)$ for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.

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arXiv2018

On Lie algebras responsible for zero-curvature representations and Backlund transformations of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for $(1+1)$-dimensional PDEs can be interpreted as ZCRs. In [arXiv:1303.3575], for any $(1+1)$-dimensional scalar evolution equation $E$, we defined a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. Also, using these algebras, one obtains necessary conditions for existence of a Backlund transformation between two given equations. The algebras $F(E)$ are defined in [arXiv:1303.3575] in terms of generators and relations. In this approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs. In this preprint we prove a number of results on $F(E)$ which were announced in [arXiv:1303.3575]. We present applications of $F(E)$ to the theory of Backlund transformations in more detail and describe the explicit structure (up to non-essential nilpotent ideals) of the algebras $F(E)$ for a number of equations of orders $3$ and $5$.

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arXiv2017

On Lie algebras responsible for zero-curvature representations of multicomponent (1+1)-dimensional evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable $(1+1)$-dimensional PDEs. According to the preprint arXiv:1212.2199, for any given $(1+1)$-dimensional evolution PDE one can define a sequence of Lie algebras $F^p$, $p=0,1,2,3,\dots$, such that representations of these algebras classify all ZCRs of the PDE up to local gauge equivalence. ZCRs depending on derivatives of arbitrary finite order are allowed. Furthermore, these algebras provide necessary conditions for existence of Backlund transformations between two given PDEs. The algebras $F^p$ are defined in arXiv:1212.2199 in terms of generators and relations. In the present paper, we describe some methods to study the structure of the algebras $F^p$ for multicomponent $(1+1)$-dimensional evolution PDEs. Using these methods, we compute the explicit structure (up to non-essential nilpotent ideals) of the Lie algebras $F^p$ for the Landau-Lifshitz, nonlinear Schrodinger equations, and for the $n$-component Landau-Lifshitz system of Golubchik and Sokolov for any $n>3$. In particular, this means that for the $n$-component Landau-Lifshitz system we classify all ZCRs (depending on derivatives of arbitrary finite order), up to local gauge equivalence and up to killing nilpotent ideals in the corresponding Lie algebras. The presented methods to classify ZCRs can be applied also to other $(1+1)$-dimensional evolution PDEs. Furthermore, the obtained results can be used for proving non-existence of Backlund transformations between some PDEs, which will be described in forthcoming publications.

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arXiv2016

Miura-type transformations for lattice equations and Lie group actions associated with Darboux-Lax representations

Miura-type transformations (MTs) are an essential tool in the theory of integrable nonlinear partial differential and difference equations. We present a geometric method to construct MTs for differential-difference (lattice) equations from Darboux-Lax representations (DLRs) of such equations. The method is applicable to parameter-dependent DLRs satisfying certain conditions. We construct MTs and modified lattice equations from invariants of some Lie group actions on manifolds associated with such DLRs. Using this construction, from a given suitable DLR one can obtain many MTs of different orders. The main idea behind this method is closely related to the results of Drinfeld and Sokolov on MTs for the partial differential KdV equation. Considered examples include the Volterra, Narita-Itoh-Bogoyavlensky, Toda, and Adler-Postnikov lattices. Some of the constructed MTs and modified lattice equations seem to be new.

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arXiv2014

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux-Egoroff system

The Darboux-Egoroff system of PDEs with any number $n\ge 3$ of independent variables plays an essential role in the problems of describing $n$-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux-Egoroff system and describe some symmetries generated by these operators. The constructed recursion operators are not pseudodifferential, but are Backlund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux-Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Poboril, and Sergyeyev. In the structure of the obtained third and fifth order symmetries of the Darboux-Egoroff system, one finds the third and fifth order flows of an $(n-1)$-component vector modified KdV hierarchy. The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental-van de Leur twisted loop group action on the space of semisimple Frobenius manifolds. We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.

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arXiv2002

Prolongation structure of the Krichever-Novikov equation

We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on u, u_x, u_{xx}, u_{xxx} for the Krichever-Novikov equation u_t=u_{xxx}-3u_{xx}^2/(2u_x)+p(u)/u_x+au_x in the case when the polynomial p(u)=4u^3-g_2u-g_3 has distinct roots. We prove that there is a universal prolongation algebra isomorphic to the direct sum of a commutative 2-dimensional algebra and a certain subalgebra of the tensor product of sl_2(C) with the algebra of regular functions on an affine elliptic curve. This is achieved by identifying this prolongation algebra with the one for the anisotropic Landau-Lifshitz equation. Using these results, we find for the Krichever-Novikov equation a new zero-curvature representation, which is polynomial in the spectral parameter in contrast to the known elliptic ones.

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arXiv2002

Notes on homogeneous vector bundles over complex flag manifolds

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset Σof simple roots of G, and let E_ϕbe a homogeneous vector bundle over the flag manifold G/P corresponding to a linear representation ϕof P. Using Bott's theorem, we obtain sufficient conditions on ϕin terms of the combinatorial structure of Σfor some cohomology groups of the sheaf of holomorphic sections of E_ϕto be zero. In particular, we define two numbers d(P), l(P) such that for any ϕobtained by natural operations from a representation of dimension less than d(P) the q-th cohomology group of E_ϕis zero for 0<q<l(P). We prove also that in this case the vector bundle E_ϕis rigid.

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arXiv2002

Conservation laws for multidimensional systems and related linear algebra problems

We consider multidimensional systems of PDEs of generalized evolution form with t-derivatives of arbitrary order on the left-hand side and with the right-hand side dependent on lower order t-derivatives and arbitrary space derivatives. For such systems we find an explicit necessary condition for existence of higher conservation laws in terms of the system's symbol. For systems that violate this condition we give an effective upper bound on the order of conservation laws. Using this result, we completely describe conservation laws for viscous transonic equations, for the Brusselator model, and the Belousov-Zhabotinskii system. To achieve this, we solve over an arbitrary field the matrix equations SA=A^tS and SA=-A^tS for a quadratic matrix A and its transpose A^t, which may be of independent interest.

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arXiv2002

Conservation laws of generalized higher Burgers and linear evolution equations

By the Cole-Hopf transformation, with any linear evolution equation in 1+1 dimensions a generalized Burgers equation is associated. We describe local conservation laws of these equations. It turns out that any generalized Burgers equation has only one conservation law, while a linear evolution equation with constant coefficients has an infinite number of (x,t)-independent conservation laws iff the equation involves only odd order terms and, therefore, is bi-Hamiltonian.

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arXiv2000

On one-parametric families of Backlund transformations

In the context of the cohomological deformation theory, infinitesimal description of one-parametric families of Backlund transformations of special type including classical examples is given. It is shown that any family of such a kind evolves in the direction of a nonlocal symmetry shadow.

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