A nonlocal finite-dimensional integrable system related to the nonlocal nonlinear Schrödinger equation hierarchy

Based on the Lenard gradient sequence, a hierarchy of the nonlocal nonlinear Schrodinger (NNLS) equations is obtained. Using the Lax representation, the nonlocal finite-dimensional integrable system with Lie-Poisson structure is presented. Then, under coordinate transformation, the nonlocal finite-dimensional integrable system with Lie-Poisson structure can be expressed as the canonical Hamiltonian system of the standard symplectic structures. Moreover, the parametric representation of the NNLS equation and the nonlocal complex modified Kortewegde Vries (NcmKdV) equation are constructed. Finally, according to the Hamilton-Jacobi theory, the action-angle variables are built and the inversion problem related to the Lie-Poisson Hamiltonian systems is discussed.