Artur Sergyeyev

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Artur verified their affiliation via an institutional email.

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• DSc. (Czech Academy of Sciences)
• Professor (Full) at Silesian University in Opava

The search for partial differential systems that are integrable in the sense of soliton theory is well known to be an important problem of modern mathematical physics. Alas, this search quickly becomes very difficult with the increase in the number D of independent variables. In this paper we deal with the first case where the said difficulty mani...

Published version of the article is free to read (even without valid subscription) at http://rdcu.be/tNXs -------------------------- ------------------------------------ Given a Poisson structure (or, equivalently, a Hamiltonian operator) P, we show that its Lie derivative L_τ(P) along a vector field τ defines another Poisson structure, which is a...

We present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a Lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect to the latter. New examples of recursion operators obtained using our technique include inter alia those for t...

We present a novel construction of recursion operators for integrable second-order multidimensional PDEs admitting isospectral scalar Lax pairs with Lax operators being first-order scalar differential operators linear in the spectral parameter. Our approach, illustrated by several examples and applicable to many other PDEs of the kind in question,...

Upon having presented a bird's eye view of history of integrable systems, we give a brief review of certain recent advances in the longstanding problem of search for partial differential systems in four independent variables that are integrable in the sense of soliton theory (such systems are known as integrable (3+1)-dimensional systems, or as cla...

Upon having presented a bird's eye view of history of integrable systems, we give a brief review of certain earlier advances (arXiv:1401.2122 & arXiv:1812.02263) in the longstanding problem of search for partial differential systems in four independent variables, often referred to as (3+1)-dimensional or 4D systems, that are integrable in the sense...

We introduce an integrable two-component extension of the general heavenly equation and prove that the solutions of this extension are in one-to-one correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we demonstrate that if the metrics in question are hyper-para-Kähler, then our system reduces to the general heavenly equati...

We give a complete description of inequivalent nontrivial local conservation laws of all orders for a natural generalization of the dissipative Westervelt equation and, in particular, show that the equation under study admits an infinite number of inequivalent nontrivial local conservation laws for the case of more than two independent variables.

Nonlinear partial differential systems in four independent variables that are integrable in the sense of soliton theory were for a long time believed to be quite scarce. In this talk we dispel this misconception and present an effective explicit construction for a large class of such systems with nonisospectral Lax pairs involving contact vector fi...

We present a new multidimensional integrable system, whose solutions are in one-to-one correspondence with four-dimensional hyper-para-Hermitian metrics. For hyper-para-Kähler metrics our system reduces to the general heavenly equation.

Contact geometry is well known to play an important and manifold role in the geometric approach to the study of partial differential systems. In this talk we showcase a novel application of three-dimensional contact geometry, where it helps answering a longstanding question of just how exceptional are partial differential systems in four independen...

We give an explicit effective construction for a large new class of partial differential systems in four independent variables that are integrable in the sense of soliton theory, thus showing inter alia that there is significantly more of such systems than it appeared before. This is achieved by employing contact vector fields in dimension three in...

Upon a brief review of the theory of integrable systems in general, we describe new classes of integrable systems in four independent variables resulting from the construction given in the paper A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108(2018) 359–376 a.k.a. https://arxiv.org/abs/1401.2122

Talk at Ostrava Mathematical Seminar 2021

We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein–Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating...

We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein-Gordon equation we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating s...

We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a by product, we obtai...

We present the results of study of a nonlinear evolutionary PDE (more precisely, a one-parameter family of PDEs) associated with the chain of pre-stressed granules. The PDE under study supports solitary waves of compression and rarefaction (bright and dark compactons) and can be written in Hamiltonian form. We investigate {\em inter alia} integrabi...

Two exact solutions for $n=0$ and $n=1$ of the Palatini-modified Lane-Emden equation are found. We have employed these solutions to describe a Palatini-Newtonian neutron star and compared the result with the pure Newtonian counterpart. It turned out that for the negative parameter of the Starobinsky model the star is heavier and larger.

We present a first example of an integrable (3+1)-dimensional dispersionless system with nonisospectral Lax pair involving algebraic, rather than rational, dependence on the spectral parameter, thus showing that the class of integrable (3+1)-dimensional dispersionless systems with nonisospectral Lax pairs is significantly more diverse than it appea...

We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain...

We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain...

We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related $R$-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear...

We present a first example of an integrable (3+1)-dimensional dispersionless system with nonisospectral Lax pair involving algebraic, rather than rational, dependence on the spectral parameter, thus showing that the class of integrable (3+1)-dimensional dispersionless systems with nonisospectral Lax pairs is significantly more diverse than it appea...

Talk at 32nd International Colloquium on Group Theoretical Methods in Physics, Prague 2018, revised version; the key message is that there is significantly more (3+1)-dimensional integrable systems than it appeared before -- in addition to a few previously known isolated examples like the (anti)self-dual Yang--Mills equations, there is a large new...

The search for new integrable (3+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3+1)$$\end{document}-dimensional partial differential systems is among the most impo...

Motivated by the theory of Painlevé equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable in the sense defined below. We establish sufficient conditions under which Hamiltonian vector fields forming a finite-dimensional Lie algebra can be deformed to time-dependent Frobenius integrable Hamil...

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector field...

We present the results of study of a nonlinear evolutionary PDE (more precisely, a one-parameter family of PDEs) associated with the chain of pre-stressed granules. The PDE in question supports solitary waves of compression and rarefaction (bright and dark compactons) and can be written in Hamiltonian form. We investigate {\em inter alia} integrabi...

We perform extended group analysis for the system of differential equations modeling an isothermal no-slip drift flux. The maximal Lie invariance algebra of this system is proved to be infinite-dimensional. We also find the complete point symmetry group of this system using the megaideal-based version of the algebraic method. Optimal lists of one-...

We perform extended group analysis for a system of differential equations modeling an isothermal no-slip drift flux. The maximal Lie invariance algebra of this system is proved to be infinite-dimensional. We also find the complete point symmetry group of this system, including discrete symmetries, using the megaideal-based version of the algebraic...

In the present paper we introduce a multi-dimensional version of the R-matrix approach to the construction of integrable hierarchies. Applying this method to the case of the Lie algebra of functions with respect to the contact bracket, we construct integrable hierarchies of (3+1)-dimensional dispersionless systems of the type recently introduced by...

We construct an infinite hierarchy of nonlocal conservation laws for the ABC equation Autuxy+Buxuty+Cuyutx=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A u_t\,u_{xy...

We construct an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation $A u_t\,u_{xy}+B u_x\,u_{ty}+C u_y\,u_{tx} = 0$, where $A,B,C$ are constants and $A+B+C\neq 0$, using a novel nonisospectral Lax pair. As a byproduct, we present new coverings for the ABC equation. The method of proof of nontriviality of the conservation laws un...

We give a complete description of generalized symmetries and local conservation laws for the fifth-order Karczewska--Rozmej--Rutkowski--Infeld equation describing shallow water waves in a channel with variable depth. In particular, we show that this equation has no genuinely generalized symmetries and thus is not symmetry integrable.

We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the shape operator).

We consider a hydrodynamic-type system of balance equations which is closed by the dynamic equation of state taking into account the effects of spatial nonlocality. Symmetry and local conservation laws of this system are studied. A system of ODEs being obtained via the group theory reduction of the initial system of PDEs is investigated. The reduce...

This paper is devoted to description of the relationship among oriented associativity equations, symmetry consistent conjugate curvilinear coordinate nets, and the widest associated class of semi- Hamiltonian hydrodynamic-type systems.

We consider the four-dimensional integrable Martinez Alonso--Shabat equation, and present three integrable three-dimensional reductions thereof. One of these reductions, the basic Veronese web equation, provides a new example of an integrable three-dimensional PDE. We also construct an infinite hierarchy of commuting nonlocal symmetries (and not ju...

The Stackel separability of a Hamiltonian system is well known to ensure existence of a complete set of Poisson commuting integrals of motion quadratic in the momenta. In the present paper we consider a class of Stackel separable systems where the entries of the Stackel matrix are monomials in the separation variables. We show that the only systems...

The article is in Highlights of 2013, see http://iopscience.iop.org/journal/1751-8121/page/Highlights-of-2013 The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surfac...

We present a new approach to construction of recursion operators for multidimensional integrable systems which have a Lax-type representation in terms of a pair of commuting vector fields. It is illustrated by the examples of the Manakov–Santini system which is a hyperbolic system in N dependent and (N + 4) independent variables, where N is an arbi...

We consider the generalized Stäckel systems, the broadest class of integrable Hamiltonian systems that admit separation of variables and possess separation relations affine in the Hamiltonians. For these systems we construct in a systematic fashion hierarchies of basic separable potentials. Moreover, we show how the equations of motion for the syst...

In the present paper we extend the multiparameter coupling constant metamorphosis, also known as the generalized St\"ackel transform, from Hamiltonian dynamical systems to general finite-dimensional dynamical systems and ODEs. This transform interchanges the values of integrals of motion with the parameters these integrals depend on but leaves the...

We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg–de Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all...

We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all...

We construct infinite hierarchies of nonlocal higher symmetries for the oriented associativity equations using solutions of associated vector and scalar spectral problems. The symmetries in question generalize those found by Chen, Kontsevich and Schwarz (Nucl. Phys. B 730 352–63) for the WDVV equations. As a byproduct, we obtain a Darboux-type tran...

Using a (1, 1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for t...

We present changes of variables that transform new integrable hierarchies found by Szablikowski and B{\l}aszak using the $R$-matrix deformation technique [J. Math. Phys. 47 (2006), paper 043505, nlin.SI/0501044] into known Harry-Dym-type and mKdV-type hierarchies.

We introduce the cotangent universal hierarchy that extends the universal hierarchy from [L. Martínez Alonso, A.B. Shabat, Phys. Lett. A 300 (1) (2002) 58, nlin.SI/0202008; A.B. Shabat, Theor. Math. Phys. 136 (2003) 1066; L. Martínez Alonso, A.B. Shabat, J. Nonlinear Math. Phys. 10 (2) (2003) 229, nlin.SI/0310036; L. Martínez Alonso, A.B. Shabat, T...

We suggest a deterministic delay difference model for the time series of the closing stock price and the intrinsic value of the stock. The most important new feature of this model is the equation describing the evolution of the intrinsic value. We present a general solution for the model in question and study the stability of the stationary points....

Published version is free to read at http://rdcu.be/tN0H We consider the Burgers-type system studied by Foursov, $$\begin{array}{@{}rcl@{}}w_{t}&=&w_{xx}+8ww_{x}+(2-4\alpha)zz_{x},\\[6pt]z_{t}&=&(1-2\alpha)z_{xx}-4\alpha zw_{x}+(4-8\alpha)wz_{x}-(4+8\alpha)w^{2}z+(-2+4\alpha)z^{3},\end{array}$$ for which no recursion operator or master symmetry...

Using a (1,1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for th...

We present a multiparameter generalization of the Stackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stackel transform preserves the Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations...

We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in arXiv:hep-th/0612029 and show that these operators, along wi...

We show that under certain technical assumptions any weakly nonlocal Hamiltonian structure compatible with a given nondegenerate weakly nonlocal symplectic structure $J$ can be written as the Lie derivative of $J^{−1}$ along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary...

We establish quantum and classical exact solvability for two large classes of maximally superintegrable Benenti systems in $n$ dimensions with arbitrarily large $n$. Namely, we solve the Hamilton--Jacobi and Schr\"odinger equations for the systems in question. The results obtained are illustrated for a model with the cubic potential.

We show that under certain technical assumptions any weakly nonlocal Hamiltonian structure compatible with a given nondegenerate weakly nonlocal symplectic structure $J$ can be written as the Lie derivative of $J^{-1}$ along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary...

We consider the St\"ackel transform, also known as the coupling-constant metamorphosis, which under certain conditions turns a Hamiltonian dynamical system into another such system and preserves the Liouville integrability. We show that the corresponding transformation for the equations of motion is nothing but the reciprocal transformation of a sp...

In this brief note we present a zero-curvature representation for one of the new integrable system found by Mikhailov, Novikov and Wang in nlin.SI/0601046. ----------------------------------------------------------------------------------- Note: You can read the published full text of the paper free of charge here: http://rdcu.be/tNZ7

We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.

We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.

We found a new symplectic structure and a recursion operator for the Sasa--Satsuma equation widely used in nonlinear optics, $$ p_t=p_{xxx}+6 p q p_x+3 p (p q)_x,\quad q_t=q_{xxx}+6 p q q_x+3 q (p q)_x, $$ along with an integro-differential substitution linking this system to a third-order generalized symmetry of the complex sine-Gordon II system $...

We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanli) et al (2004 Acta Appl. Math. 83 85-94) is bi-Hamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators found by us. Using this we prove that the system i...

For a class of Hamiltonian systems naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion. Comment: 5 pages, LaTeX 2e, to appear in J. Phys. A: Math. Gen

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries....

You can read the published full text of the paper for free at http://rdcu.be/tNZZ ----------------------------------------------------------------------------------- We generalize earlier results of Fokas and Liu and find all locally analytic (1 + 1)-dimensional evolution equations of order n that admit an N-shock-type solution with N ≤ n + 1. Fo...

Using the methods of the theory of formal symmetries, we obtain new easily verifiable sufficient conditions for a recursion operator to produce a hierarchy of local generalized symmetries. An important advantage of our approach is that under certain mild assumptions it allows to bypass the cumbersome check of hereditarity of the recursion operator...

We derive a formula for the leading terms of formal conservation laws (and hence for the leading terms of higher order cosymmetries as well) for a large class of odd order evolution systems in (1+1)-dimensions. This result yields a simple proof of locality for hierarchies of symmetries generated using master symmetries and recursion operators of su...

In the present paper we prove the integrability (in the sense of existence of formal symmetry of infinite rank) for a class of block-triangular inhomogeneous extensions of (1+1)-dimensional integrable evolution systems. An important consequence of this result is the existence of formal symmetry of infinite rank for "almost integrable" systems, rece...

We present a new general construction of a recursion operator from the zero-curvature representation. Using it, we find a recursion operator for the stationary Nizhnik-Veselov-Novikov equation and present a few low-order symmetries generated with the help of this operator.

We consider the Calogero-Degasperis-Ibragimov-Shabat (CDIS) equation and find the complete set of its nonlocal symmetries depending on the local variables and on the integral of the only local conserved density of the equation in question. The Lie algebra of these symmetries turns out to be a central extension of that of local generalized symmetrie...

This paper presents a first example of parasupersymmetric relativistic quantum-mechanical model with non-oscillator-like interaction: the Coulomb problem for the modified Stueckelberg equation, describing a relativistic massive spin-1 particle in the electromagnetic field of a point charge.

Given a generalized (Lie–Bäcklund) vector field satisfying certain nondegeneracy assumptions, we explicitly describe all (1 + 1)-dimensional evolution systems that admit this vector field as a generalized conditional symmetry. The connection with the theory of symmetries of systems of ODEs and with the theory of invariant modules is discussed.

We present sufficient conditions ensuring the locality of hierarchies of time-independent symmetries generated by master symmetries from a local seed symmetry. These conditions are applicable to a large class of (1+1)-dimensional evolution systems. Our results can also be used for proving that the time-independent part of a suitable linear-in-time...

Given a generalized (Lie--Backlund) vector field, we classify (under certain nondegeneracy assumptions) all (1+1)-dimensional evolution systems which admit this vector field as a conditional symmetry. The connection with the theory of symmetries of systems of ODEs and with the theory of invariant modules is discussed.

We find the complete set of local generalized symmetries (including x, t-dependent ones) for the Calogero–Degasperis–Ibragimov–Shabat (CDIS) equation, and investigate the properties of these symmetries.

In the present paper we prove the integrability (in the sense of existence of formal symmetry of infinite rank) for a class of (1+1)-dimensional evolution systems generalizing the triangular systems considered by Bakirov, Beukers, Sanders and Wang, and by Sanders and van der Kamp. The important consequence of this result is the existence of formal...

PDF IS AVAILABLE FREE OF CHARGE AT https://dml.cz/handle/10338.dmlcz/701699 The sufficient conditions of time independence and commutativity for local and nonlocal homogeneous symmetries of a large class of (1+1)-dimensional evolution systems are obtained. In contrast with the majority of known results, the verification of our conditions does not...

All local generalized symmetries (including x,t-dependent ones) of the Bakirov system are found. In particular, it is shown that its only non-Lie-point local generalized symmetry is the sixth order one found by Bakirov. This result generalizes a similar result of Beukers, Sanders and Wang on x,t-independent symmetries and completes the refutation o...

We present easily verifiable sufficient conditions of time-independence and commutativity for local and nonlocal symmetries for a large class of homogeneous (1+1)-dimensional evolution systems. In contrast with the majority of known results, the verification of our conditions does not require the existence of master symmetry or hereditary recursion...

We consider the recursion operators with nonlocal terms of special form for evolution systems in (1+1) dimensions, and extend them to well-defined operators on the space of nonlocal symmetries associated with the so-called universal Abelian coverings over these systems. The extended recursion operators are shown to leave this space invariant. These...

We consider scalar (1 + 1)-dimensional evolution equation of order n≥2, which possesses time-independent formal symmetry (i.e. it is integrable in the sense of symmetry approach), shared by all local generalized time-independent symmetries of this equation. We show that if such equation possesses the nontrivial canonical conserved density ρ_m, m∈{−...

We present the explicit formulae, describing the structure of symmetries and formal symmetries of any scalar (1+1)-dimensional evolution equation. Using these results, the formulae for the leading terms of commutators of two symmetries and two formal symmetries are found. The generalization of these results to the case of system of evolution equati...

The $x$-dependence of the symmetries of (1+1)-dimensional scalar translationally invariant evolution equations is described. The sufficient condition of (quasi)polynomiality in time $t$ of the symmetries of evolution equations with constant separant is found. The general form of time dependence of the symmetries of KdV-like non-linearizable evoluti...

Using the adjoint action of infinitesimal translations (with respect to some (in)dependent variables) on specific finite-dimensional subspaces of the space of generalized symmetries of a system of partial differential equations, we explicitly determine the dependence of coefficients of generalized symmetries from these subspaces on the variables me...

In this paper we found explicit eigenvalues and eigenstates of discrete spectrum for the modified Stueckelberg equation (written in second order formalism) for the case of external Coulomb field. We considered the case of arbitrary gyromagnetic ratio.

This paper presents a first example of parasupersymmetric relativistic quantum-mechanical model with non-oscillator-like interaction: the Coulomb problem for the modified Stueckelberg equation, describing a relativistic massive spin-1 particle in the electromagnetic field of a point charge.