The Spectral Approximation For Coupled Nonlinear Schr?dinger Equations

The Coupled Nonlinear Schr?dinger(CNLS)equations is well known as its wide applications in the areas of engineering and science.In this paper,we con-sidered the numerical solution of CNLS equations which include the first-order differential term in space,firstly optimized the collocations chosen by orthogo-nal spline collocation method.It s called optimized orthogonal spline collocation method.And we proved the conservation properties by using the energy method.Then we strictly proved its second-order accuracy in the time direction and high-order accuracy in the spatial direction.By optimizing,the time complexity is reduced to the 1/2 of the orthogonal spline collocation method,which greatly in-creases the efficiency of the operation.Besides that,a full implicit second-order Legendre Spectrum is presented to get the spectral accuracy.Strict proof were also given to prove the conservation law and accuracy of its numerical solution by the discrete energy method,and then we gave an estimate of the convergence order.Finally,we demonstrate the effectiveness of optimized orthogonal spline colloca-tion method and the high-order accuracy of Legendre Spectrum method by the numerical experiments.