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

Abstract

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The systems of Gibbons--Tsarev type are the 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 generated by the simplest GT system, which was not considered previously because it is in a sense trivial.

47 pages, no figures

References in corpus (12)

• Essays on the theory of elliptic hypergeometric functions
• Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation
• Classification of integrable hydrodynamic chains and generating functions of conservation laws
• Löwner equations and dispersionless hierarchies
• Classification of integrable Vlasov-type equations
• Systems of Gibbons-Tsarev type and integrable 3-dimensional models
• A family of (2+1)-dimensional hydrodynamic type systems possessing pseudopotential
• Integrable Lagrangians and modular forms
• Second order quasilinear PDEs and conformal structures in projective space
• Integrable pseudopotentials related to elliptic curves
• Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian
• Integrable pseudopotentials related to generalized hypergeometric functions

Cited by in corpus (13)

• Characteristic Lie rings, finitely-generated modules and integrability conditions for 2+1 dimensional lattices
• Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings
• Dispersionless BKP Hierarchy and Quadrant Löwner Equation
• Classification of integrable hydrodynamic chains
• Non-homogeneous systems of hydrodynamic type possessing Lax representations
• Symmetric Matrix Ensemble and Integrable Hydrodynamic Chains
• Deformations of complex structures on Riemann surfaces and integrable structures of Whitham type hierarchies
• Multi-variable reductions of the dispersionless DKP hierarchy
• Characteristic integrals in 3D and linear degeneracy
• Loewner equations and reductions of dispersionless hierarchies
• Hydrodynamic integrability via geometry
• Integrable structures of dispersionless systems and differential geometry
• Algebraic Properties of Quasilinear Two-Dimensional Lattices connected with integrability