A conservative Fourier pseudospectral algorithm for the nonlinear Schrödinger equation

In this paper, we derive a new method for a nonlinear Schrödinger system by using the square of the first-order Fourier spectral differentiation matrix D₁ instead of the traditional second-order Fourier spectral differentiation matrix D₂ to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in ‖·‖₂ norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.