Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability

Abstract

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. Given a PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions (if consistent).

16 pages

References in corpus (12)

• Classification of integrable equations on quad-graphs. The consistency approach
• 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
• The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type
• On Associativity Equations in Dispersionless Integrable Hierarchies
• Towards a theory of differential constraints of a hydrodynamic hierarchy
• Boyer-Finley equation and systems of hydrodynamic type
• Hydrodynamic reductions of the heavenly equation
• On the Whitham Hierarchies: Reductions and Hodograph Solutions
• A hodograph transformation which applies to the heavenly equation

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