# 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…

## 29 Citations

### Non-homogeneous Systems of Hydrodynamic Type Possessing Lax Representations

- MathematicsCommunications in Mathematical Physics
- 2013

We consider 1 + 1-dimensional non-homogeneous systems of hydrodynamic type that possess Lax representations with movable singularities. We present a construction, which provides a wide class of…

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For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if…

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- Mathematics
- 2015

Let Gr(d,n) be the Grassmannian of d ‐dimensional linear subspaces of an n ‐dimensional vector space Vn . A submanifold X⊂Gr(d,n) gives rise to a differential system Σ(X) that governs d ‐dimensional…

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- Mathematics
- 2020

The equations of Löwner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable…

### Classification of integrable hydrodynamic chains

- Mathematics
- 2010

Using the method of hydrodynamic reductions, we find all integrable infinite (1+1)-dimensional hydrodynamic-type chains of shift 1. A class of integrable infinite (2+1)-dimensional hydrodynamic-type…

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- 2018

We consider multi-variable reductions of the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) in the elliptic parametrization. The reduction is given by a system of…

### Integrable structures of dispersionless systems and differential geometry

- MathematicsTheoretical and Mathematical Physics
- 2017

We develop the theory of Whitham-type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application, we construct Gibbons–Tsarev systems…

### Integrable structures of dispersionless systems and differential geometry

- Mathematics
- 2016

We develop the theory of Whitham-type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application, we construct Gibbons–Tsarev systems…

### Loughborough University Institutional Repository Dispersionless integrable systems in 3 D and Einstein-Weyl geometry

- Mathematics
- 2018

For several classes of second-order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if…

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