Symplectic integrators for discrete nonlinear Schrödinger systems

Symplectic methods for integrating canonical and non-canonical Hamiltonian systems are examined. A general form for higher order symplectic schemes is developed for non-canonical Hamiltonian systems using generating functions and is directly applied to the Ablowitz–Ladik discrete nonlinear Schrödinger system. The implicit midpoint scheme, which is symplectic for canonical systems, is applied to a standard Hamiltonian discretization. The symplectic integrators are compared with an explicit Runge–Kutta scheme of the same order. The relative performance of the integrators as the dimension of the system is varied is discussed.