NONLOCAL MATRIX HAMILTONIAN OPERATORS, DIFFERENTIAL GEOMETRY, AND APPLICATIONS

A study is made of a class of nonlocal Hamiltonian operators that arise naturally as second Hamiltonian structures of the nonlinear Schrodinger equation, the Heisenberg magnet, the Landau-Lifshitz equation, etc. A complete description of these operators is obtained, and it reveals intimate connections with classical differential geometry. A new nonlocal Hamiltonian structure of first order is constructed for the partly anisotropic (J1 = J2) Landau-Lifshitz equation (hitherto, only Hamiltonian structures of zeroth and second orders were known for the Landau-Lifshitz equation).