Papers by Artur Sergyeyev
Boletín de la Sociedad Matemática Mexicana, 2025
Upon having presented a bird's eye view of history of integrable systems, we give a brief review ... more 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, in terms used in physics, classical integrable 4D field theories, in general non-relativistic and non-Lagrangian). Namely, we review a recent construction for a large new class of (3+1)-dimensional integrable systems with Lax pairs involving contact vector fields. This class contains two infinite families of such systems, thus establishing that there is significantly more integrable (3+1)-dimensional systems than it was believed for a long time. In fact, the construction under study yields (3+1)-dimensional integrable generalizations of many well-known dispersionless integrable (2+1)-dimensional systems like the dispersionless KP equation, as well as a first example of a (3+1)-dimensional integrable system with an algebraic, rather than rational, nonisospectral Lax pair.
To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik-Veselov-Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.
Acta Applicandae Mathematicae, 2022
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We present a novel construction of recursion operators f... more Free to read at https://rdcu.be/cVZCJ
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, employs an ansatz for the sought-for recursion operator of the equation under study based on the Lax pair for the latter.
Annales Henri Poincaré, 2019
We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an in... more 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 an infinite-dimensional differential covering over the Przanowski equation.
Applied Mathematics Letters, 2019
We present a first example of an integrable (3+1)-dimensional dispersionless system with nonisosp... more 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 appeared before. The Lax pair in question is of the type recently introduced in [A. Sergyeyev, Lett. Math. Phys. 108 (2018), no. 2, 359-376, arXiv:1401.2122 ].
Nonlinear Dynamics, 2018
Published version is free to read at http://rdcu.be/A7w6 The search for new integrable (3+1)-dim... more Published version is free to read at http://rdcu.be/A7w6 The search for new integrable (3+1)-dimensional partial differential systems is among the most important challenges in the modern integrability theory. It turns out that such a system can be associated to any pair of rational functions of one variable in general position, as established below using contact Lax pairs introduced in A. Sergyeyev, Lett. Math. Phys. 108 (2018), no. 2, 359-376.
Reports on Mathematical Physics, 1998
Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant va... more Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant variables) on specific finite-dimensional subspaces of the space of generalized symmetries of some system of partial differential equations, we explicitly determine the dependance of coefficients of generalized symmetries from these subspaces on the above-mentioned variables. We establish the connection of our results with the theory of quasiexactly solvable models. Some generalizations of the approach proposed also are discussed.
Phys Lett a, 2011
We consider the generalized Stäckel systems, the broadest class of integrable Hamiltonian systems... more 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 systems under study are related through appropriately chosen reciprocal transformations and how the respective constants of motion are related through generalized Stäckel transforms.
Concise Encyclopedia of Supersymmetry, 2000
Reports on Mathematical Physics, 2009
Using a (1, 1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the num... more 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 the seed systems. The reciprocal transformations constructed from these conservation laws yield a considerably larger class of hydrodynamic-type systems from the seed systems, and we ...
Physics Letters A, 2013
The Stäckel separability of a Hamiltonian system is well known to ensure existence of a complete ... more The Stäckel 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. We consider a class of Stäckel separable systems where the entries of the Stäckel matrix are monomials in the separation variables. We show that the only systems in this class for which the integrals of motion arising from the Stäckel construction keep commuting after quantization are, up to natural equivalence transformations, the so-called Benenti systems. Moreover, it turns out that the latter are the only quantum separable systems in the class under study.
Journal of Geometry and Physics, 2014
We consider the four-dimensional integrable Martínez Alonso-Shabat equation, and present three in... more We consider the four-dimensional integrable Martínez 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.
Letters in Mathematical Physics, 2018
We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless syst... more We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields.
In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.
Journal of Mathematical Analysis and Applications, 2017
The article is now free to read and download on the publisher website at https://doi.org/10.1016/... more The article is now free to read and download on the publisher website at https://doi.org/10.1016/j.jmaa.2017.04.050 under Elsevier open archive terms and conditions.
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 the general heavenly equation, which describes a class of anti-self-dual solutions of the vacuum Einstein equations, and a six-dimensional equation resulting from a system of Ferapontov and Khusnutdinova.
Concise Encyclopedia of Supersymmetry, 2003
Concise Encyclopedia of Supersymmetry, 2003
Reports on Mathematical Physics, 2002
We present sufficient conditions ensuring the locality of hierarchies of symmetries generated by ... more We present sufficient conditions ensuring the locality of hierarchies of symmetries generated by repeated commutation of master symmetry with a 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 symmetry is a nontrivial master symmetry and hence the system in question has infinitely many symmetries and is integrable.
Physics Letters A, 2011
Abstract We consider the generalized Stäckel systems, the broadest class of integrable Hamiltonia... more Abstract 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 systems under study are related through appropriately chosen reciprocal transformations and how the respective constants of motion are related through ...
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Papers by Artur Sergyeyev
To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik-Veselov-Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.
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, employs an ansatz for the sought-for recursion operator of the equation under study based on the Lax pair for the latter.
In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.
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 the general heavenly equation, which describes a class of anti-self-dual solutions of the vacuum Einstein equations, and a six-dimensional equation resulting from a system of Ferapontov and Khusnutdinova.
To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik-Veselov-Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.
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, employs an ansatz for the sought-for recursion operator of the equation under study based on the Lax pair for the latter.
In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.
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 the general heavenly equation, which describes a class of anti-self-dual solutions of the vacuum Einstein equations, and a six-dimensional equation resulting from a system of Ferapontov and Khusnutdinova.
https://www-math.ias.tokushima-u.ac.jp/~yasumoto/gsis20221126/
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 independent variables that are integrable in the sense of soliton theory. It turns out that such systems are far more numerous than it was believed, and we provide an effective explicit construction, involving contact vector fields, for a large class of systems in question along with their Lax pairs. As a byproduct, we present a first example of an integrable partial differential system in four independent variables with a nonisospectral Lax pair which is algebraic, rather than rational, in the spectral parameter.
Using this result we prove, under some natural assumptions, the Maltsev–Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.