Researches On Split-Step Compact Difference Methods For A Class Of Nonlinear Schr?dinger Equations

In this article,the split-step compact finite difference method is used to study the nonlinear Schrodinger equation involving quintic terms and the coupled Gross-Pitaevskii equationsFirstly,the split-step compact difference method is constructed for the nonlinear Schrodinger equation with quintic terms.The problem is mainly divided into two parts,nonlinear subproblems and linear ones,and then the split equations are solved in sequence.Commonly,the nonlinear subproblems can be solved by integral directly,but it fails when the nonlinear subproblems can not be integrated exactly.So the midpoint rule or the trapezoidal rule is applied approximately,at the same time,the split order is not reduced.For linear ones,the split-step method can also split the high-dimensional problem into one dimensional problems,then the compact difference method can be designed directly.Furthermore,it reduces the computational complexity of high-dimensional problems and simplifies the process of solving high-dimensional problems fundamentally.The scheme is convergent with second-order in time and fourth-order in spaceSecondly,the split-step compact difference method is extended to solve more complex coupled Gross-Pitaevskii equations.However,compared with the construction of the split-step compact finite difference method for the nonlinear Schrodinger equation involving quintic terms,there are two more details to be dealt with.On the one hand,the existence of angular momentum of the coupled Gross-Pitaevskii equations makes it difficult to construct compact difference schemes.To circumvent this problem,an orthogonal rotation transformation is introduced to eliminate the angular momentum directly.In fact,the transformed equations are completely consistent with the coupled nonlinear Schrodinger equations in form.So the split-step compact finite difference method also can be extended to solve the coupled nonlinear Schrodinger equations.On the other hand,the splitted linear parts of the coupled Gross-Pitaevskii equations are still coupled equations.Therefore,by introducing a linear transformation to decouple the system,a compact difference scheme can be constructed directly.Finally,numerical experiments for the nonlinear Schrodinger equation involving quintic terms and the coupled Gross-Pitaevskii equations,as well as the coupled nonlinear Schrodinger equations,are well simulated,conservative properties and convergence rates are demonstrated as well.It is shown from the numerical tests that the present method is efficient and reliable.