Evgeny V. Ferapontov

Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d−1)n arbitrary functions of one variable. The most important examples include the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics and its six-dimensional generalization arising in the context of sdiff(Σ2) self-dual Yang–Mills equations. Given a multidimensional PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions. As an example of this phenomenon we discuss the second commuting flow of the dispersionless KP hierarchy. Considered separately, this is a four-dimensional PDE which does not pass the integrability test. However, the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full (2+1)-dimensional dis...

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 u...

Classification of integrable Hamiltonian hydrodynamic chains associated with Kupershmidt’s brackets

We characterize a class of integrable Hamiltonian hydrodynamic chains, based on the necessary condition for the integrability provided by the vanishing of the Haantjes tensor. We prove that the vanishing of the first few components of the Haantjes tensor is already sufficiently restrictive, and allows a complete description of the corresponding Hamiltonian densities. In each of the cases we were able to explicitly construct a generating function for conservation laws, thus establishing the integrability.

Systems of conservation laws with third-order Hamiltonian structures

We investigate n-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in

On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions

\mathbb {P}^{n+2}

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3; 5)

Pn+2 satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space W of dimension

Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property

n+2