# Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

@article{Ferapontov2005DifferentialgeometricAT, title={Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor}, author={Eugene V. Ferapontov and D. G. Marshall}, journal={Mathematische Annalen}, year={2005}, volume={339}, pages={61-99} }

The integrability of an m-component system of hydrodynamic type, ut = V(u)ux, by the generalized hodograph method requires the diagonalizability of the m × m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ × ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation… Expand

#### 53 Citations

Differential geometry of hydrodynamic Vlasov equations

- Mathematics, Physics
- 2007

Abstract We consider hydrodynamic chains in (1+1) dimensions which are Hamiltonian with respect to the Kupershmidt–Manin Poisson bracket. These systems can be derived from single (2+1) equations,… Expand

The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

- Mathematics
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2006

An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type,is developed. We prove that the existence of special solutions known as ‘double… Expand

On classical integrability of the hydrodynamics of quantum integrable systems

- Physics, Mathematics
- 2017

Recently, a hydrodynamic description of local equilibrium dynamics in quantum integrable systems was discovered. In the diffusionless limit, this is equivalent to a certain 'Bethe–Boltzmann' kinetic… Expand

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

- Mathematics, Physics
- 2006

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… Expand

Integrability of the Manakov--Santini hierarchy

- Physics, Mathematics
- 2009

The first example of the so-called "coupled" integrable hydrodynamic chain is presented. Infinitely many commuting flows are derived. Compatibility conditions of the first two of them lead to the… Expand

Double waves in multi-dimensional systems of hydrodynamic type: the necessary condition for integrability

- Physics, Mathematics
- 2005

The invariant differential-geometric approach to the integrability of (2+1)- dimensional systems of hydrodynamic type is developed. It is argued that the existence of special solutions known as… Expand

Classification of conservative hydrodynamic chains. Vlasov type kinetic equation, Riemann mapping and the method of symmetric hydrodynamic reductions

- Physics, Mathematics
- 2009

A complete classification of integrable conservative hydrodynamic chains is presented. These hydrodynamic chains are written via special coordinates -- moments, such that right hand sides of these… Expand

Haantjes Manifolds of Classical Integrable Systems

- Mathematics
- 2014

A general theory of classical integrable systems is proposed, based on the geometry of the Haantjes tensor. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathcal{H}$ manifold),… Expand

Hydrodynamic chains and the classification of their Poisson brackets

- Mathematics, Physics
- 2006

Infinite component Poisson brackets of the Dubrovin-Novikov type [Sov. Math. Dokl. 27, 665–669 (1983)] are considered. The corresponding Jacobi identity is significantly simplified in the Liouville… Expand

A New family of higher-order Generalized Haantjes Tensors, Nilpotency and Integrability

- Mathematics
- 2018

We propose a new infinite class of generalized binary tensor fields, whose first representative of is the known Frolicher--Nijenhuis bracket. This new family of tensors reduces to the generalized… Expand

#### References

SHOWING 1-10 OF 93 REFERENCES

Differential geometry of hydrodynamic Vlasov equations

- Mathematics, Physics
- 2007

Abstract We consider hydrodynamic chains in (1+1) dimensions which are Hamiltonian with respect to the Kupershmidt–Manin Poisson bracket. These systems can be derived from single (2+1) equations,… Expand

The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

- Mathematics
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2006

An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type,is developed. We prove that the existence of special solutions known as ‘double… Expand

Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability

- Mathematics, Physics
- 2004

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… Expand

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

- Mathematics, Physics
- 2006

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… Expand

Double waves in multi-dimensional systems of hydrodynamic type: the necessary condition for integrability

- Physics, Mathematics
- 2005

The invariant differential-geometric approach to the integrability of (2+1)- dimensional systems of hydrodynamic type is developed. It is argued that the existence of special solutions known as… Expand

The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

- Mathematics, Physics
- 2004

We obtain the necessary and sufficient conditions for a two-component (2 + 1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in… Expand

On a Class of Three-Dimensional Integrable Lagrangians

- Mathematics, Physics
- 2004

AbstractWe characterize non-degenerate Lagrangians of the form
such that the corresponding Euler-Lagrange equations are integrable by the method of hydrodynamic reductions. The integrability… Expand

On Hamiltonian perturbations of hyperbolic systems of conservation laws

- Mathematics, Physics
- 2004

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is… Expand

THE GEOMETRY OF HAMILTONIAN SYSTEMS OF HYDRODYNAMIC TYPE. THE GENERALIZED HODOGRAPH METHOD

- Mathematics
- 1991

It is proved that there exists an infinite involutive family of integrals of hydrodynamic type for diagonal Hamiltonian systems of quasilinear equations; the completeness of the family is also… Expand

The ∂-approach to the dispersionless KP hierarchy

- Mathematics
- 2001

The dispersionless limit of the scalar nonlocal ∂-problem is derived. It is given by a special class of nonlinear first-order equations. A quasiclassical version of the ∂-dressing method is… Expand