Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability
Seminar Room 1, Newton Institute
A (d+1)-dimensional dispersionless PDE is said to be integrable if it possesses infinitely many n-component hydrodynamic reductions parametrized by (d-1)n arbitrary functions of one variable. Among the most important examples one should primarily mention the three-dimensional dKP and the Boyer-Finley equations, as well as the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics. It was observed that the integrability in the sense of hydrodynamic reductions is equivalent to the existence of a scalar pseudopotential playing the role of dispersionless Lax pair. Lax pairs of this type constitute a basis of the dispersionless d-bar and twistor approaches to multi-dimensional equations.