A nonlocal finite-dimensional integrable system related to the nonlocal mKdV equation

We propose a hierarchy of the nonlocal mKdV (NmKdV) equation. Based on a constraint, we obtain nonlocal finite-dimensional integrable systems in a Lie–Poisson structure. By a coordinate transformation, the nonlocal Lie–Poisson Hamiltonian systems are reduced to nonlocal canonical Hamiltonian systems in the standard symplectic structure. Moreover, using the nonlocal finite-dimensional integrable systems, we give parametric solutions of the NmKdV equation and the generalized nonlocal nonlinear Schrödinger (NNLS) equation. According to the Hamilton–Jacobi theory, we obtain the action–angle-type coordinates and the inversion problems related to Lie–Poisson Hamiltonian systems.