Geometric structures on the orbits of loop diffeomorphism groups and related “heavenly-type” Hamiltonian systems. I
Abstract
UDC 517.9
review of differential-geometric and Lie-algebraic approaches to the study of a broad class of nonlinear integrable differential systems of ``heavenly'' type associated with Hamiltonian flows on the spaces conjugate to the loop Lie algebras of vector fields on the tori. These flows are generated by the corresponding orbits of the coadjoint action of the diffeomorphism loop group and satisfy the Lax–Sato-type vector-field compatibility conditions. The corresponding hierarchies of conservation laws and their relationships with Casimir invariants are analyzed. Typical examples of these systems are considered and their complete integrability is established by using the developed Lie-algebraic construction. We describe new generalizations of the integrable dispersion-free systems of ``heavenly'' type for which the corresponding generating elements of orbits have a factorized structure, which allows their extension to the multidimensional case.
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