Hamiltonian Operators Of Hydrodynamic Type Research Articles

Degenerate first-order Hamiltonian operators of hydrodynamic type in 2D

First-order Hamiltonian operators of hydrodynamic type were introduced by Drubrovin and Novikov in 1983. In 2D, they are generated by a pair of contravariant metrics g, and a pair of differential-geometric objects b, . If the determinant of the pencil vanishes for all λ, the operator is called degenerate. In this paper we provide a complete classification of degenerate two- and three-component Hamiltonian operators. Moreover, we study the integrability, by the method of hydrodynamic reductions, of 2+1 Hamiltonian systems arising from the structures we classified.

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.

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems

Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our results to linearly degenerate semi-Hamiltonian systems in Riemann invariants, a typical example being Rti=(∑m=1nRm−Ri)Rxi, i=1,2,…,n. Since all such systems are linearizable by appropriate (generalized) reciprocal transformations, our formulas provide an infinity of mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by n arbitrary functions of one variable.

A Darboux theorem for Hamiltonian operators in the formal calculus of variations

We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

A Darboux theorem for Hamiltonian operators in the formal calculus of variations

We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

Surfaces with flat normal bundle: an explicit construction

An explicit construction of surfaces with flat normal bundle in the Euclidean space E n (unit hypersphere S n ) in terms of solutions of certain linear system is proposed. In the case of E 3 our formulae can be viewed as a direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geometry or the Lelieuvre representation of surfaces in affine 3-space. Parametrization of Ribaucour congruences of spheres by three solutions of the linear system is obtained. In view of the classical Lie correspondence between Ribaucour congruences of spheres and surfaces with flat normal bundle in the Lie quadric in P 5, this gives an explicit representation of surfaces with flat normal bundle in the 4-dimensional space form of the Lorentzian signature. Direct projective analog of this construction is the known parametrization of W-congruences by three solutions of the Moutard equation. Under the Plücker embedding W-congruences give rise to surfaces with flat normal bundle in the Plücker quadric. Integrable evolutions of surfaces with flat normal bundle and parallels with the theory of nonlocal Hamiltonian operators of hydrodynamic type are discussed in the conclusion.

Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type

Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type