Comparative numerical study of integrable and nonintegrable discrete models of nonlinear Schrödinger equations

Abstract

. In this paper, we study efficiency of numerical simulation for integrable and nonintegrable discrete Non-linear Schr ¨ odinger equations (NLSE). We first discretize the NLSE into two classical spatial models, nonintegrable direct discrete model and integrable Ablowitz-Ladik model. By some simple transformations and Doubarx transformation, we obtain two integrable models from Ablowitz-Ladik model. Then, five different kinds of schemes can be applied to simulate four models in bright and dark cases for comparing the performance in preserving the conserved quantities’ approximations of NLSE. The numerical experiments indicate that Gauss symplectic method is more efficient than nonsymplectic schemes and splitting schemes when simulating the same model. Both intergrable models and nonintergrable model have their own advantages in preserving the conserved quanti-ties’ approximations. For the three integrable models, Ablowitz-Ladik Model and the model which has a general symplectic structure have similar simulation effects, and the model owing a cononical symplectic structure has low efficiency because the complicated Doubarx transformations make the model difficult to solve. Moreover, symplectic scheme and symmetric scheme have overwhelming superiorities over nonsymplectic schemes in preserving the invariants of Hamiltonian system.