Hamiltonian Systems Of Hydrodynamic Type Research Articles

First integrals of affine connections and Hamiltonian systems of hydrodynamic type

Cambridge Commonwealth, European & International Trust and CAPES Foundation (Grant Proc. BEX 13656/13-9) to F.C.

Dispersive deformations of Hamiltonian systems of hydrodynamic type in 2+1 dimensions

We develop a theory of integrable dispersive deformations of 2+1 dimensional Hamiltonian systems of hydrodynamic type following the scheme proposed by Dubrovin and his collaborators in 1+1 dimensions. Our results show that the multi-dimensional situation is far more rigid, and generic Hamiltonians are not deformable. As an illustration we discuss a particular class of two-component Hamiltonian systems, establishing the triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order.

Lie algebra structures for four-component Hamiltonian hydrodynamic type systems

Four-component Hamiltonian systems of hydrodynamic type induce, through the Haantjes tensor, a Lie algebra structure on tangent planes in the space of dependent variables. We show that this Lie algebra is either reductive or solvable with a nilpotent three-dimensional subalgebra. We demonstrate how the precise Lie algebraic structure is determined by the Hamiltonian structure of the system. An application to perturbations of the Benney system is presented.

Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure, and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.

Hamiltonian Systems of Hydrodynamic Type in 2 + 1 Dimensions

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.

Dupin hypersurfaces and integrable hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants

We establish a close relationship between hamiltonian systems of hydrodynamic type and hypersurfaces of a pseudoeuclidean space. This correspondence provides a complete classification of the 3 × 3 integrable nondiagonalizable hamiltonian systems, based upon the classification of Dupin hypersurfaces in E 4.

Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants

We formulate several conjectures concerning the structure and properties of thenxn integrable nondiagonalizable hamiltonian systems of hydrodynamic type. Forn=3 our results are, in fact, complete: a 3×3 nondiagonalizable hamiltonian system is integrable if and only if it is weakly nonlinear (linearly degenerate).

On the matrix Hopf equation and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants

We propose a complete description of the integrable Hamiltonian 3 x 3 systems of hydrodynamic type, which do not possess Riemann invariants: the system of this class is integrable if and only if it is weakly nonlinear (linearly degenerate). Any 3 x 3 weakly nonlinear nondiagonalizable Hamiltonian system can be transformed to the matrix Hopf equation U t =( U 2) x , where U is a 3 x 3 symmetric matrix subject to the constraints tr U=const, tr U 2=const. The matrix Hopf equation is equivalent to the system of resonant three-wave interaction and hence is integrable via the inverse scattering transform. We formulate several conjectures and unsolved problems concerning the structure and general properties of the integrable Hamiltonian systems of hydrodynamic type.

Hamiltonian systems of hydrodynamic type and constant curvature metrics

We consider the one-dimensional systems of hydrodynamic type. The general relations for the nonlocal Poisson brackets of hydrodynamic type are presented. For a nondegenerate nonlocal Poisson bracket of hydrodynamic type the relations mean exactly that the bracket is generated by a metric of constant Riemannian curvature.

Hamiltonian systems of hydrodynamic type and their realization on hypersurfaces of a pseudo-Euclidean space

A canonical correspondence between Hamiltonian systems of differential equations of hydrodynamic type and hypersurfaces of a pseudo-Euclidean space is constructed. In this correspondence multi-Hamiltonian systems correspond to hypersurfaces admitting nontrivial deformations that preserve the principal directions and principal curvatures. A description of the surfaces inE3 that admit a three-parameter family of such deformations is given.

Integration of Hamiltonian systems of hydrodynamic type with two dependent variables with the aid of the Lie—Bäcklund group

Integration of Hamiltonian systems of hydrodynamic type with two dependent variables with the aid of the Lie—Bäcklund group