Department of Mathematical Sciences Loughborough University Tel: +44 (0) 1509 22 2861 Fax: +44 (0) 1509 22 3969

Eugene Ferapontov

Differential geometry and integrability of multi-dimensional dispersionless systems

In a series of recent publications with Dr K Khusnutdinova (Loughborough University) we have proposed a new original approach to the integrability of multi-dimensional quasilinear systems. Known as the `method of hydrodynamic reductions', this approach consists of seeking special multi-phase solutions which can be interpreted as nonlinear interactions of planar simple waves. This is done by `decoupling' a given (2+1)-dimensional equation into a pair of n-component commuting systems of hydrodynamic type. To be precise, a (2+1)-dimensional quasilinear system is said to be integrable if, for any n, it possesses an infinity of hydrodynamic reductions parametrized by n arbitrary functions of a single variable. The crucial observation was that this definition provides an efficient classification criterion. Based on this approach, we have classified integrable equations within various particularly interesting classes including

- multi-dimensional systems of hydrodynamic type,

- hydrodynamic chains,

- equations of the dispersionless Hirota type,

- second order quasilinear PDEs,

etc. This research revealed remarkable connections of dispersionless integrable systems with classical differential geometry, in particular, pencils of matrices with the zero Haantjes tensor, hypersurfaces of the Lagrangian Grassmannian, Gl(2, R)-structures and flat conformal structures in projective space.

I collaborate with Dr B Doubrov, Dr K Khusnutdinova, Prof C Klein, Dr A Moro, Dr V Novikov, Dr A Odesskii, Dr M Pavlov, Prof S Tsarev. This project was recently supported by an EPSRC research grant.

I supervise the following PhD students working on various aspects of multi-dimensional integrability: David Marshall, Lenos Hadjikos, Pavel Burovskiy, Nikola Stoilov

Related publications:

E.V. Ferapontov, L. Hadjikos and K.R. Khusnutdinova,
Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian,
arXiv: 0705.1774, 2007, submitted.
Paper: PDF

E.V. Ferapontov and D.G. Marshall,
Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor,
Math. Ann. 339, no. 1 (2007) 61-99.
Paper: PDF

E.V. Ferapontov, K.R. Khusnutdinova, D.G. Marshall and M.V. Pavlov,
Classification of integrable Hamiltonian hydrodynamic chains associated with Kupershmidt brackets,
J. Math. Phys. 47 (2006) 103507 1-13.
Paper: PDF

E.V. Ferapontov and K.R. Khusnutdinova,
The Haantjes tensor and double waves in multi-dimensional systems of hydrodynamic type: a necessary condition for integrability,
Proc. Royal Soc. A 462 (2006) 1197-1219.
Paper: PDF

E.V. Ferapontov, K.R. Khusnutdinova and S.P. Tsarev,
On a class of three-dimensional integrable Lagrangians,
Comm. Math. Phys. 261 (2006) 225-243.
Paper: PDF

E.V. Ferapontov, K.R. Khusnutdinova and M.V. Pavlov,
Classification of integrable (2+1)-dimensional quasilinear hierarchies,
Theor. Math. Phys. 144 (2005) 35-43.
Paper: PDF

E.V. Ferapontov, K.R. Khusnutdinova,
Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability,
J. Math. Phys. 45 (2004) 2365-2377.
Paper: PDF

E.V. Ferapontov, K.R. Khusnutdinova,
The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type,
J. Phys. A: Math. Gen. 37 (2004) 2949-2963.
Paper: PDF

E.V. Ferapontov, K.R. Khusnutdinova,
On the integrability of (2+1)-dimensional quasilinear systems,
Comm. Math. Phys. 248 (2004) 187-206.
Paper: PDF

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