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Variational identities and applications to Hamiltonian structures of soliton equations. (English) Zbl 1238.37020
Summary: This is an introductory report concerning our recent research on Hamiltonian structures. We will discuss variational identities associated with continuous and discrete spectral problems, and their applications to Hamiltonian structures of soliton equations. Our illustrative examples are the AKNS hierarchy and the Volterra lattice hierarchy associated with semisimple Lie algebras, and two hierarchies of their integrable couplings associated with non-semisimple Lie algebras. The resulting Hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies. The presented variational identities can be applied to Hamiltonian structures of other soliton hierarchies.
MSC:
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q51Soliton-like equations