The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

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 Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.     

Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations

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 describing all nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived in these special coordinates, and the integrability of these equations is proved using the inverse scattering transform. The Lax pairs with a spectral parameter for these equations are found. We construct various classes of integrable reductions of the derived equations. These classes of reductions are of an independent differential-geometric and applied interest. In particular, if one of the compatible Poisson brackets is local, we obtain integrable reductions of the classical Lamé equations describing all orthogonal curvilinear coordinate systems in a flat space; if one of the compatible brackets is generated by a constant-curvature metric, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in a space of constant curvature.     

Compatible Dubrovin–Novikov Hamiltonian Operators,Lie Derivative,and Integrable Systems of Hydrodynamic Type

We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.     

Compatible Poisson Brackets on Lie Algebras

We discuss the relationship between the representation of an integrable system as an L-A-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.     

Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type

We consider two classes of integrable nonlinear hyperbolic systems on Lie algebras. These systems generalize the principal chiral model. Each system is related to a pair of compatible Lie brackets and has a Lax representation, which is determined by the direct sum decomposition of the Lie algebra of Laurent series into the subalgebra of Taylor series and the complementary subalgebra corresponding to the pair. New examples of compatible Lie brackets are given.     

Classification of Integrable (2+1)-Dimensional Quasilinear Hierarchies

We investigate the (2+1)-dimensional hierarchies associated with the integrable PDEs of the form Δ _tt = F(Δ_xx, Δ_xt, Δ_xy), which generalize the dispersionless KP hierarchy. Integrability is understood as the existence of infinitely many hydrodynamic reductions.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 35–43, July, 2005.     

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.     

The classification of nonsingular multidimensional Dubrovin-Novikov brackets

In this paper, the well-known Dubrovin-Novikov problem posed as long ago as in 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 multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such Poisson brackets are generated by nontrivial canonical special infinite-dimensional Lie algebras. In this paper, we obtain a classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number N of components and for any dimension n by differential-geometric methods. A key role in the solution of this problem is played by the theory of compatible metrics earlier constructed by the present author.     

Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models

We construct hierarchies of commutative Poisson subalgebras for Sklyanin brackets. Each of the subalgebras is generated by a complete set of integrals in involution. Some new integrable systems and schemes for separation of variables for them are elaborated using various well-known representations of the brackets. The constructed models include deformations for the Goryachev–Chaplygin top, the Toda chain, and the Heisenberg model.     

Quasi-Frobenius Algebras and Their Integrable N-Parameter Deformations Generated by Compatible N×N Metrics of Constant Riemannian Curvature

We prove that the equations describing compatible N×N 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-Frobenius manifolds. We conjecture that open–closed two-dimensional topological field theories correspond to a special class of integrable deformations of associative quasi-Frobenius algebras.     

齐型空间上的Poisson算子与广义Carleson测度

该文在齐型空间上定义了一类广义Carleson测度和Poisson型位势算子，该算子具有与Poisson核相似的性质，研究了该算子在Lp、Hp和Tent空间与Lorentz空间上的有界性.