Nonlocal matrix hamiltonian operators, differential geometry, and applications

A study is made of a class of nonlocal Hamiltonian operators that arise naturally as second Hamittonian structures of the nonlinear Schrödinger 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 ($J_1=J_2$) Landau–Lifshitz equation (hitherto, only Hamiltonian structures of zeroth and second orders were known for the Landau–Lifshitz equation).