High-order conservative scheme for the coupled space fractional nonlinear Schrödinger equations. (English) Zbl 1499.65451
Summary: In this paper, an efficient finite difference scheme is proposed for one dimension and two dimension coupled space fractional nonlinear Schrödinger equations. First, the high-order difference scheme and Crank-Nicolson scheme are used to one dimension coupled space fractional nonlinear Schrödinger equations. second, we show that the high-order conservative difference scheme satisfies the mass and energy conservation laws respectively, and convergence and unconditional stability of the scheme are also proved. Next, we give the high-order conservative scheme for two dimension coupled space fractional nonlinear Schrödinger equations. Finally, some numerical results are reported to verify our theoretical analysis.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
fractional nonlinear Schrödinger equations; conservation law; Crank-Nicolson scheme; convergenceReferences:
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