Comparison of split-step and Hamiltonian integration methods for simulation of the nonlinear Schrödinger type equations

We provide a systematic comparison of two numerical methods to solve the widely used nonlinear Schrödinger equation (NLSE). The first one is the standard second order split-step (SS2) method based on operator splitting approach. The second one is the Hamiltonian integration method (HIM), originally proposed in the paper by Dyachenko et al. (1992) [16]. Extension of the HIM to a widely used generalization of NLSE is developed. HIM allows the exact conservation of the Hamiltonian and wave action at the cost of requiring iterative solution for the implicit scheme. The numerical error for HIM is smaller than the SS2 solution for the same time step for almost all simulations we consider. Conversely, one can take orders of magnitude larger time steps in HIM, compared with SS2, still ensuring numerical stability.