A multisymplectic Fourier pseudo-spectral scheme,which exactly preserves the discrete multisymplectic conservation law,is presented to solve the Klein-Gordon-Schrdinger equations.The scheme is of spectral accuracy in space and of second order in time.The scheme preserves the discrete multisymplectic conservation law and the charge conservation law.Moreover,the residuals of some other conservation laws are derived for the geometric numerical integrator.Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme,and demonstrate the correctness of the theoretical analysis.
In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.
The local one-dimensional multisymplectic scheme(LOD-MS)is developed for the three-dimensional(3D)Gross-Pitaevskii(GP)equation in Bose-Einstein condensates.The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional(LOD)method.The 3D GP equation is split into three linear LOD Schrodinger equations and an exactly solvable nonlinear Hamiltonian ODE.The three linear LOD Schrodinger equations are multisymplectic which can be approximated by multisymplectic integrator(MI).The conservative properties of the proposed scheme are investigated.It is masspreserving.Surprisingly,the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable.This is impossible for conventional MIs in nonlinear Hamiltonian context.The numerical results show that the LOD-MS can simulate the original problems very well.They are consistent with the numerical analysis.
