The dispersionless completely integrable heavenly type Hamiltonian flows and their differential-eometric structure

Abstract

There are reviewed modern investigations devoted to studying nonlinear dispersiveless heavenly type integrable evolutions systems on functional spaces within the modern differential-geometric and algebraic tools. Main accent is done on the loop diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, generating dispersionless heavenly type integrable systems. As examples, we analyzed the Einstein–Weyl metric equation, the modified Einstein–Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and modified Plebański heavenly equations, the Husain heavenly equation, the general Monge equation and the classical Korteweg-de Vries dispersive dynamical system. We also investigated geometric structures of a class of spatially one-dimensional completely integrable Chaplygin type hydrodynamic systems, which proved to be deeply connected with differential systems on the complexified torus and the related diffeomorphism group orbits on them.