Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, numerical examples, symplectic integrators, nonlinear wave equations, NLS equations (nonlinear Schrödinger equations), multisymplectic integrators, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), geometric integrators, Numerical methods for Hamiltonian systems including symplectic integrators, integrable Hamiltonian semi-discretization, Finite difference methods for initial value and initial-boundary value problems involving PDEs, Hamiltonian systems, conservation laws, nonlinear Schrödinger equation, finite difference method