E. V. Ferapontov

Compatible Poisson brackets of hydrodynamic type

Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular: (a) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor which is slightly different from those existing in the literature; (b) the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case; (c) the connection of this problem with the class of surfaces in Euclidean space which possess non-trivial deformations preserving the Weingarten operator.

Integrable Schrödinger operators with magnetic fields: Factorization method on curved surfaces

The factorization method for Schrodinger operators with magnetic fields on a two-dimensional surface M2 with nontrivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as the Dirac magnetic monopole and the Landau problem. The global geometric aspects and related spectral properties of the operators from the factorization chains are discussed in detail. We also consider the Laplace transformations on a curved surface and extend the class of Schrodinger operators with two integrable levels introduced in the flat case by S. P. Novikov and one of the authors.

Quasiclassical limit of coupled KdV equations: Riemann invariants and multi-Hamiltonian structure

We consider the quasiclassical limit of the first nontrivial flow in coupled KdV hierarchy. Written down in Riemann invariants, it assumes the extremely simple form R+i=∑k=1nRk + 2RiRxi, i=1…,n. Up to certain natural equivalence this turns out to be the only hydrodynamic system with n + 1 compatible purely local Hamiltonian structures.

Hydrodynamic reductions of the heavenly equation

We demonstrate that Plebanskis first heavenly equation decouples in infinitely many ways into a triple of commuting (1 + 1)-dimensional systems of hydrodynamic type which satisfy the Egorov property. Solving these systems by the generalized hodograph method, one can construct exact solutions of the heavenly equation parametrized by arbitrary functions of one variable. We discuss explicit examples of hydrodynamic reductions associated with the equations of one-dimensional nonlinear elasticity, linearly degenerate systems and the equations of associativity.

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

Abstract 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 .

On the integrability of symplectic Monge–Ampère equations

Abstract Let u be a function of n independent variables x 1 , … , x n , and let U = ( u i j ) be the Hessian matrix of u . The symplectic Monge–Ampere equation is defined as a linear relation among all possible minors of U . Particular examples include the equation det U = 1 governing improper affine spheres and the so-called heavenly equation, u 13 u 24 − u 23 u 14 = 1 , describing self-dual Ricci-flat 4 -manifolds. In this paper we classify integrable symplectic Monge–Ampere equations in four dimensions (for n = 3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of ‘maximally singular’ hyperplane sections of the Plucker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F ( u i j ) = 0 in more than three dimensions is necessarily of the symplectic Monge–Ampere type.

Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems

Abstract Isoparametric hypersurface M n ⊂ S n + 1 can be defined as an intersection of the unit sphere r 2 = ( u 1 ) 2 + … + ( u n + 2 ) 2 = 1 with the level set F ( u ) = const of a homogeneous polynomial F of degree g , satisfying Cartan-Munzner equations (▽ F ) 2 = g 2 r 2 g − 2 , Δ F = cr g − 2 c = const . We introduce a hamiltonian system of hydrodynamic type u i t = 1 g δ ij d dx ∂F ∂u j , with the hamiltonian operator δ ij d dx and the hamiltonian density F (u) g . Under the additional assumption of the homogeneity of the hypersurface M n , the restriction of this system to M n proves to be nondiagonalizable, but integrable and can be transformed to an appropriate integrable reduction of the N -wave system. Possible generalizations to isoparametric submanifolds (finite or infinite dimensional) are also briefly indicated.

Separable Hamiltonians and integrable systems of hydrodynamic type

Abstract We exhibit a surprising relationship between separable Hamiltonians and integrable, linearly degenerate systems of hydrodynamic type. This gives a new way of obtaining the general solution of the latter. Our formulae also lead to interesting canonical transformations between large classes of Stackel systems.

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

Abstract 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 Ut=(U2)x, where U is a 3 x 3 symmetric matrix subject to the constraints tr U=const, trU2=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.

Boyer–Finley equation and systems of hydrodynamic type

We reduce the Boyer–Finley equation to a family of compatible systems of hydrodynamic type, with characteristic speeds expressed in terms of spaces of rational functions. The systems of hydrodynamic type are then solved by the generalized hodograph method, providing solutions of the Boyer–Finley equation containing functional parameters.