53 citations · 193 across the 25 of their papers we have counts for
25 papers
Local description of gl-regular Haantjes operators
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We study Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. Our main result is a complete local description of gl-regular Haantjes operators. Additio…
Duality of operator Frobenius algebras and solution of Eisenhart-Stäckel problem in the non-diagonal case
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We study Frobenius algebras of operator fields and introduce a novel notion of duality for them. We show that, under the assumption that the operator fields forming the Frobenius a…
Existence of Riemannian invariants for integrable systems of hydrodynamic type
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We show that for a hyperbolic system of hydrodynamic type admitting n symmetries, there exists a coordinate system in which the generator of the system and all the symmetries are d…
Lax pairs for BKM hierarchy
Andrey Yu. Konyaev, Vladimir S. Matveev
We construct Lax pairs for the recently (2023) introduced integrable PDE systems known as the BKM equations. As many known and previously studied integrable systems are special cas…
Stäckel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We construct integrable Hamiltonian systems such that functionally independent Poisson commuting integrals are quadratic in the momenta. Unlike the classical Stäckel setting, we al…
On operator fields in the upper triangular Toeplitz form
M. M. Chernin, A. Yu. Konyaev
In this work, we solve the fundamental problem of describing the coordinate transformations that preserve the upper triangular Toeplitz form of the given operator field. Surprising…
On the Linearization of Certain Singularities of Nijenhuis Operators
Andrey Yu. Konyaev
We consider a linearization problem for Nijenhuis operators in dimension two around a point of scalar type in analytic category. The problem was almost completely solved in arXiv:1…
Integrability of the magnetic geodesic flow on the sphere with a constant 2-form
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We prove a recent conjecture of Dragovic et al arXiv2504.20515 stating that the magnetic geodesic flow on the standard sphere whose magnetic 2-form is…
On the Jordan-Chevalley decomposition problem for operator fields in small dimensions and Tempesta-Tondo conjecture
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We explore the Jordan-Chevalley decomposition problem for an operator field in small dimensions. In dimensions three and four, we find tensorial conditions for an operator field $L…
Thoughts about potentials with finite-band spectrum and finite-dimensional reductions of integrable systems
Andrey Yu. Konyaev, Vladimir S. Matveev
We repeat, using methods developed for BKM systems, the famous results of S. Novikov (1974), J. Moser (1981, 1982) , and A. Veselov (1980) that relate Schrödinger-Hill operators wi…
Research problems on relations between Nijenhuis geometry and integrable systems
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
The paper surveys open problems and questions related to interplay between the theory of integrable systems with infinitely and finitely many degrees of freedom and Nijenhuis geome…
Finite-dimensional reductions and finite-gap type solutions of multicomponent integrable PDEs
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known…
Nijenhuis operators with a unity and F-manifolds
Evgenii I. Antonov, Andrey Yu. Konyaev
The core object of this paper is a pair , where is a Nijenhuis operator and is a vector field satisfying a specific Lie derivative condition, i.e., $Lie_{e}L=\opera…
When a -tensor generates separation of variables of a certain metric
Andrey Yu. Konyaev, Jonathan M. Kress, Vladimir S. Matveev
By a -tensor field , we construct an explicit system of differential invariants that vanish if and only if there (locally) exists a metric for which generates…
Applications of Nijenhuis Geometry V: geodesically equivalent metrics and finite-dimensional reductions of certain integrable quasilinear systems
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We describe all metrics geodesically compatible with a gl-regular Nijenhuis operator . The set of such metrics is large enough so that a generic local curve is a geodesic fo…
Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl-regular Nijenhuis operators. We prove the splitting Theorem for symmetri…
Orthogonal separation of variables for spaces of constant curvature
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating…
Applications of Nijenhuis Geometry IV: multicomponent KdV and Camassa-Holm equations
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We construct a new series of multicomponent integrable PDE systems that contain as particular example (with appropriately chosen parameters) many famous integrable systems includin…
Geometry of inhomogeneous Poisson brackets, multicomponent Harry Dym hierarchies and multicomponent Hunter-Saxton equations
Andrey Yu. Konyaev
We introduce a natural class of multicomponent local Poisson structures , where is a local Poisson bracket of order one and $\mathcal P_…
Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We consider multicomponent local Poisson structures of the form , under the assumption that the third order term is Darboux-Poisson and…
Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydr…
Nijnehuis Geometry III: gl-regular Nijenhuis operators
Alexey Bolsinov, Andrey Konyaev, Vladimir Matveev
We study Nijenhuis operators, that is, (1,1)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric…
Applications of Nijenhuis geometry: Nondegenerate singular points of Poisson-Nijenhuis structures
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.
Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators
Andrey Yu. Konyaev
A field of endomorphisms is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of called points of scal…