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

Abstract

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 involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the `dispersionless Lax pair'. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.

15 pages

References in corpus (12)

• On integrability of (2+1)-dimensional quasilinear systems
• Integrable hydrodynamic chains
• Einstein-Weyl geometry, the dKP equation and twistor theory
• A class of Einstein--Weyl spaces associated to an integrable system of hydrodynamic type
• Energy-dependent potentials revisited: A universal hierarchy of hydrodynamic type
• Dispersionless scalar integrable hierarchies, Whitham hierarchy and the quasi-classical dbar-dressing method
• Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type
• Boyer-Finley equation and systems of hydrodynamic type
• Three Dimensional Distorted Black Holes
• dbar-equations, integrable deformations of quasiconformal mappings and Whitham hierarchy
• Hydrodynamic reductions of the heavenly equation
• Quasi-classical $\dbar$-method: Generating equations for dispersionless integrable hierarchies

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