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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:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51 Soliton equations
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