Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications
In the introduction (Section 1) the authors start with a very short review of methods of investigation of integrability properties of some special classes of infinite-dimensional Hamiltonian systems, expressed by nonlinear evolution equations. To describe these properties different approaches have been proposed and are known now. The authors claim that the approach proposed in the paper does not belong to either of the known types. The main idea, of fairly geometrical nature, is based on the direct investigation of the integrability structures defined on some differentiable manifolds in order to give criteria which guarantee the reducibility of such structures on certain regular submanifolds. By using this method we can recover the integrability structure of more relevant hierarchies of evolution equations discussed in the literature. In order to provide an overview of the contents of the whole paper it seems to be enough to list the titles of Sections 2-11: (2) Poisson- Nijenhuis manifolds; (3) The basic PN manifold; (4) The theory of restriction; (5) Some elements of the geometry of PN manifolds; (6) A reduction method for Nijenhuis manifolds; (7) The integrability structure of the AKNS hierarchy; (8) The recursion operator for the Heisenberg chain; (9) Integrability structure for KN hierarchy; (10) The recursion operator for WKI hierarchy; (11) A further example of reduction of a Nijenhuis structure. The KdV hierarchy.
reduction method for Nijenhuis manifolds
scientific article; zbMATH DE number 3971811
scientific article