Canonically conjugate variables for the Volterra lattice with periodic boundary conditions (Q1282514)

The author considers the Volterra lattice \[ \dot c_i= c_i(c_{i+1}- c_i), \] where \(i\in \mathbb{R}\) and \(c_i(t)>0\), with the periodic boundary conditions \(c_{T+i}= c_i\), \(T\in\mathbb{N}\). This dynamical system is known to be Hamiltonian with respect to two compatible Poisson brackets (quadratic and cubic). For each of the two brackets, a set of canonically conjugate variables is found by using the spectral theory of the Jacobi operator. The problem of constructing canonically conjugate variables under quasiperiodic boundary conditions is not treated in the present paper and remains open.