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September 1996 - July 2013
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Publications (142)
In this paper, a class of semi-Hamiltonian diagonal systems of hydrodynamic type is constructed using algebraic-geometric methods. For such systems, hydrodynamic integrals and hydrodynamic symmetries are constructed from algebraic-geometric data. Besides, it is described what algebraic-geometric data distinguish in this class Hamiltonian diagonal s...
Алгебро-геометрическими методами построен класс полугамильтоновых диагональных систем гидродинамического типа. Для таких систем по алгебро-геометрическим данным построены гидродинамические интегралы и гидродинамические симметрии. Кроме того, выяснено, какие алгебро-геометрические данные выделяют в этом классе гамильтоновы диагональные системы с гам...
In this paper, a canonical Hamiltonian reduction of the associativity equations with the antidiagonal matrix η i j \eta _{ij} in the case of three primary fields is constructed explicitly, and its Liouville integrability is proved. Moreover, for the considered associativity equations a first integral of the second order is found explicitly and an i...
In this paper, the theory of spaces of diagonal curvature is developed. An efficient necessary condition for metrics of diagonal curvature, namely, the vanishing of the Haantjes tensor for the Ricci affinor, is obtained. Examples are constructed.
В данной статье предложено обобщение алгебро-геометрической конструкции Кричевера построения ортогональных систем координат в плоском пространстве. В теории интегрируемых систем гидродинамического типа фундаментальную роль играют также ортогональные координаты в некоторых специальных неплоских пространствах. Важнейший класс таких пространств задает...
In this work, in the case of three primary fields, a reduction of the associativity equations (the Witten–Dijkgraaf–Verlinde–Verlinde system, see (Witten, 1990, Dijkgraaf et al., 1991, Dubrovin, 1994) with antidiagonal matrix ηij on the set of stationary points of a nondegenerate integral quadratic with respect to the first-order partial derivative...
We study the Hamiltonian geometry of systems of hydrodynamic type that are equivalent to the associativity equations in the case of three primary fields and obtain the complete classification of the associativity equations with respect to the existence of a first-order Dubrovin–Novikov Hamiltonian structure.
This survey is devoted to the theory of pencils of compatible Riemannian and pseudo-Riemannian metrics, related non-linear integrable systems, and applications. Bibliography: 82 titles. © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.
Обзор посвящен теории пучков согласованных римановых и псевдоримановых метрик и связанным с ними интегрируемым нелинейным системам и приложениям. Библиография: 82 названия.
We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlin- ear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in...
In this paper we construct examples of commuting ordinary scalar
differential operators with polynomial coefficients that are related to
a spectral curve of an arbitrary genus g>0 and to an arbitrary rank
r>1 of the vector bundle of common eigenfunctions of the commuting
operators over the spectral curve. This solves completely the well-known
exist...
We construct examples of commuting ordinary scalar differential operators
with polynomial coefficients that are related to a spectral curve of an
arbitrary genus g and to an arbitrary even rank r = 2k, and also to an
arbitrary rank of the form r = 3k, of the vector bundle of common
eigenfunctions of the commuting operators over the spectral curve.
The paper is devoted to complete proofs of theorems on consistency on cubic
lattices for $3 \times 3$ determinants. The discrete nonlinear equations on
$\mathbb{Z}^2$ defined by the condition that the determinants of all $3 \times
3$ matrices of values of the scalar field at the points of the lattice
$\mathbb{Z}^2$ that form elementary $3 \times 3$...
We prove that an arbitrary Poisson structure omega^{ij}(u) and an arbitrary closed 3-form T_{ijk}(u) generate the local Poisson structure A^{ij}(u,u_x) = M^i_s(u,u_x)omega^{sj}(u), where M^i_s(u,u_x)(delta^s_j + omega^{sp}(u)T_{pjk}(u)u^k_x) = delta^i_j, on the corresponding loop space. We obtain also a special graded epsilon-deformation of an arbi...
We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we p...
We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as an N-dimensional k-potential submanifold in ((k + 1) N + p)-dimensional pseudo-Euclidean spaces of certain sign...
We consider a special class of two-dimensional discrete equations defined by relations on elementary NxN squares, N>2, of the square lattice Z^2, and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary NxN squares, N>2, in the cubic lattice Z^3. For an arbitrary N we pro...
We propose a modified condition of consistency on cubic lattices for some
special classes of two-dimensional discrete equations and prove that the
discrete nonlinear equations defined by determinants of matrices of orders N >
2 are consistent on cubic lattices in this sense.
We propose a modified condition of consistency on cubic lattices for some special classes of two-dimensional discrete equations and prove that the discrete nonlinear equations defined by determinants of matrices of orders N > 2 are consistent on cubic lattices in this sense.
We introduce a natural class of potential submanifolds in pseudo-Euclidean spaces and prove that each N-dimensional Frobenius manifold can locally be represented as an N-dimensional potential submanifold in a 2N-dimensional pseudo-Euclidean space. We show that all potential submanifolds bear natural Dubrovin–Frobenius structures on their tangent sp...
respectively. The coefficientsa k (u),b � (u),c k (u),d � (u), the metric gij(u) and the functions h��(u) satisfy a number of relations including the Gauss equations, the Codazzi equations and the Ricci equations for any submanifold. Note that in this paper we consider only the local theory of submanifolds. We shall single out a special class of N-...
We prove a duality principle for a special class of submanifolds in
pseudo-Euclidean spaces. This class of submanifolds with potential of
normals is introduced in this paper. We prove also, for example, that an
arbitrary Frobenius manifold can be realized as a certain flat
submanifold of this very natural class.
We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanifold. We show that all potentia...
The Bogoyavlenskiĭ-Novikov principle concerning the connection between
stationary and nonstationary problems is generalized. It is proved that an
arbitrary evolution system is Hamiltonian on the set of stationary points of its local
integral.
Bibliography: 16 titles.
We give an exposition of some recent crucial achievements in the theory of multidimensional Poisson brackets of hydrodynamic type. In particular, we solve the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensiona...
In this paper the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimens...
We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. It is proved that every such Hamiltonian operator (or the submanifold corresponding to the operator) gives a pe...
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We solve the problem of describing compatible nonlocal Poisson brackets of hydrodynamic type. We prove that for nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type, there exist special local coordinates such that the metrics and the Weingarten operators of both brackets are diagonal. The nonlinear evolution equations desc...
We prove that the equations describing compatible NN metrics of constant Riemannian curvature define a special class of integrable N-parameter deformations of quasi-Frobenius (in general, noncommutative) algebras. We discuss connections with open–closed two-dimensional topological field theories, associativity equations, and Frobenius and quasi-Fro...
The problem of description for compatible nonlocal Poisson brackets of hydrodynamic type is solved. The nonlinear equations describing all compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of the inverse scattering problem is proved. A Lax pair with a spectral parameter is f...
Correction to the paper published in Vol. 133, No. 2, November, 2002.
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We prove that the description of pencils of compatible (N x N)-metrics of constant Riemannian curvature is equivalent to a special class of integrable N-parametric deformations of quasi-Frobenius (in general, noncommutative) algebras.
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The description problem is solved for compatible metrics of constant Riemannian curvature. Nonlinear equations describing all nonsingular pencils of compatible metrics of constant Riemannian curvature are derived and their integrability by the inverse scattering method is proved. In particular, a Lax pair with a spectral parameter is found for thes...
We prove that a local Hamiltonian operator of hydrodynamic type K_1 is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K_2 if and only if the operator K_1 is locally the Lie derivative of the operator K_2 along a vector field in the corresponding domain of local coordinates. This result gives a natural invariant defi...
We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This syste...
We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov-Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Po...
We solve the problem of description for nonsingular pairs of compatible flat metrics in the general N-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equatio...
We consider the problem of classifying all compatible Poisson brackets of hydrodynamic type, one of which is linear in one of the coordinates and the other its derivative with respect to that coordinate.
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We deal with the problem of description of nonsingular pairs of compatible flat metrics for the general $N$-component case. We describe the scheme of the integrating the nonlinear equations describing nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics). It is based on the reducing this problem to a...
In the present paper, the notions of compatible and almost compatible Riemannian and pseudo-Riemannian metrics are introduced. These notions are motivated by the theory of compatible Poisson structures of hydrodynamic type (local and nonlocal) and generalize the notion of flat pencils of metrics, which plays an important role in the theory of integ...
We study the problem of classification or efficient description of compatible Poisson structures of hydrodynamic type, i.e. compatible local first-order homogeneous Poisson brackets in field theory. It is proved that all two-component compatible Poisson structures of hydrodynamic type are explicitly described by solutions of a homogeneous four-comp...
ContentsIntroduction Chapter I. Differential geometry of symplectic structures on loop spaces of smooth manifolds § 1.1. Symplectic and Poisson structures on loop spaces of smooth manifolds. Basic definitions § 1.2. Homogeneous symplectic structures of the first order on loop spaces of pseudo-Riemannian manifolds and two-dimensional non-linear sigm...
We exhibit the bi-Hamiltonian structure of the equations of associativity (Witten-Dijkgraaf-Verlinde-Verlinde-Dubrovin equations) in 2-dimensional topological field theory, which reduce to a single equation of Monge-Ampère type f ttt =f xxt 2 -f xxx f xtt , in the case of three primary fields. The first Hamiltonian structure of this equation is bas...
