Integrable (2+1)-dimensional systems of hydrodynamic type

@article{Odesskii2010IntegrableS,
  title={Integrable (2+1)-dimensional systems of hydrodynamic type},
  author={Alexander Odesskii and Vladimir Vyacheslavovich Sokolov},
  journal={Theoretical and Mathematical Physics},
  year={2010},
  volume={163},
  pages={549-586}
}
We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The Gibbons-Tsarev (GT) systems are most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g = 0 and g = 1 and also a new GT system corresponding to algebraic curves of genus g = 2. We construct a wide class of integrable models… 

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References

SHOWING 1-10 OF 51 REFERENCES

A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential

Abstract.We construct a family of integrable hydrodynamic type systems with three independent and n ≥ 2 dependent variables in terms of solutions of a linear system of PDEs with rational

On the Integrability of (2+1)-Dimensional Quasilinear Systems

A (2+1)-dimensional quasilinear system is said to be ‘integrable’ if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants.

On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions

We investigate (d + 1)-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case d ≥ 3 we formulate a conjecture that any such system with an

On (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential with movable singularities

We consider a class of hydrodynamic type systems that have three independent and N ⩾ 2 dependent variables and possess a pseudopotential. It turns out that systems having a pseudopotential with

Towards classification of -dimensional integrable equations. Integrability conditions I

In this paper we attempt to extend the symmetry approach (well developed in the case of (1 + 1)-dimensional equations) to the (2 + 1)-dimensional case. Presence of nonlocal terms in symmetries and

Systems of Gibbons-Tsarev type and integrable 3-dimensional models

We review the role of Gibbons-Tsarev-type systems in classification of integrable multi-dimensional hydrodynamic-type systems. Our main observation is an universality of Gibbons-Tsarev-type systems.

Classifying Integrable Egoroff Hydrodynamic Chains

We introduce the notion of Egoroff hydrodynamic chains. We show how they are related to integrable (2+1)-dimensional equations of hydrodynamic type. We classify these equations in the simplest case.

A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

We introduce a hierarchy of integrable partial differential equations in 2+1 dimensions arising from the commutation of one-parameter families of vector fields, and we construct the formal solution

Integrable elliptic pseudopotentials

We construct integrable pseudopotentials with an arbitrary number of fields in terms of an elliptic generalization of hypergeometric functions in several variables. These pseudopotentials are

Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

We investigate integrable second-order equations of the formwhich typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples include
...