A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations (TFSEs) subject to ...initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre Gauss–Lobatto collocation (SL-GL-C) method for spatial discretization; an expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the TFSE is investigated. In addition, the Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions. Thereby, the expansion coefficients are then determined by reducing the TFSE with its nonlocal conditions to a system of fractional differential equations (SFDEs) for these coefficients. The second step is to propose a shifted Legendre Gauss–Radau collocation (SL-GR-C) scheme, for temporal discretization, to reduce such system into a system of algebraic equations which is far easier to be solved. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to solve the two-dimensional TFSE. Numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms. By selecting relatively limited Legendre Gauss–Lobatto and Gauss–Radau collocation nodes, we are able to get very accurate approximations, demonstrating the utility and high accuracy of the new approach over other numerical methods.
In this paper, a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional Schrödinger equations. We prove that the scheme preserves both the ...mass and energy, which are defined by virtue of some recursion relationships. Using the Sobolev inequalities and then employing the mathematical induction, the discrete scheme is proved to be unconditionally convergent in the sense of L2-norm and Hα/2-norm, which means that there are no any constraints on the grid ratios. Then, the prior bound of the discrete solution in L2-norm and L∞-norm are also obtained. Moreover, we propose an iterative algorithm, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners. This method can reduce the memory requirement of the proposed linearized finite element scheme from O(M2) to O(M) and the computational complexity from O(M3) to O(MlogM) in each iterative step, where M is the number of grid nodes. Finally, numerical results are carried out to verify the correction of the theoretical analysis, simulate the collision of two solitary waves, and show the utility of the fast numerical solution techniques.
•A new linearized conservative FEM for the strongly coupled nonlinear fractional Schrödinger equations is proposed.•The mass and energy conservation results are obtained for the proposed discretized scheme.•The scheme is proved to be unconditionally convergent in both the L2-norm and fractional norm.•To reduce the memory requirement, a super-fast method is used for the iterative algorithm of the proposed linearized FEM.
•Improved L1-Galerkin spectral methods for coupled nonlinear time-space fractional Schrödinger equations are proposed.•Sharp convergence rates reflecting the regularity of solution are obtained over ...uniform and nonuniform time-steps.•The well-posedness of the numerical solution is proved.•Two numerical examples with smooth and non-smooth solutions are presented to support our theoretical contributions.•The effects of fractional-order parameters on the pattern formations of coupled Schrödinger equations are studied.
Recently there has been a growing interest in designing efficient numerical methods for the solution of fractional differential equations. The solutions of such equations in general exhibit a weak singularity near the initial time. In this paper, we propose finite difference/spectral methods to solve the coupled nonlinear time-space fractional Schrödinger equations with non-smooth solutions in the time direction. The proposed methods combine the strength of the L1 scheme on both uniform and non-uniform grids and the Galerkin-Legendre method. The well-posedness of the numerical solution is proved. The convergence analysis shows clearly how the grading of the mesh and the regularity of the solution affect the order of convergence of the L1 scheme, so one can choose an optimal mesh grading which can be seen numerically by considering some numerical experiments.
In this paper we study the concentration phenomenon of solutions for the nonlinear fractional Schrödinger equationε2s(−Δ)su+V(x)u=K(x)|u|p−1u,x∈RN, where ε is a positive parameter, s∈(0,1), N≥2 and ...1<p<N+2sN−2s, V(x) and K(x) are positive smooth functions. Let Γ(x)=V(x)p+1p−1−N2sK(x)−2p−1. Under certain assumptions on V(x) and K(x), we show existence and multiplicity of solutions which concentrate near some critical points of Γ(x) by a perturbative variational method.
Firstly, we construct a fourth-order numerical differential formula for approximating Riesz derivative. Substituting it into the damped nonlinear space fractional Schrödinger equation, the original ...equation becomes a matrix form of nonlinear ordinary differential equation system. Then, by means of implicit integrating factor method and Padé approximation, we obtain a new numerical scheme whose convergence order is higher than the existing algorithms. Finally, a numerical example is given to verify the effectiveness of the numerical algorithm and the correctness of the theoretical analysis.
In this paper, a linearly implicit conservative difference scheme for the coupled nonlinear Schrödinger equations with space fractional derivative is proposed. This scheme conserves the mass and ...energy in the discrete level and only needs to solve a linear system at each step. The existence and uniqueness of the difference solution are proved. The stability and convergence of the scheme are discussed, and it is shown to be convergent of order O(τ2+h2) in the discrete l2 norm with the time step τ and mesh size h. When the fractional order is two, all those results are in accord with the difference scheme proposed for the classical non-fractional coupled nonlinear Schrödinger equations. Some numerical examples are also reported.
In this paper, we present a fully discrete and structure-preserving scheme for the nonlinear fractional Schrödinger equations. The key is to introduce a scalar auxiliary variable and rewrite the ...equations as a new family of systems. The new systems are approximated by using the implicit midpoint rule, the repeated trapezoidal rule and the fractional centered difference method. It is shown the fully discrete scheme is mass- and energy-conserved. This is sharp contrast to the former result, i.e., the fully discrete scheme is only mass-conserved for the original equations.
In this paper, we investigate the dynamics and stability of multi-peak solitons from the coupled nonlinear Schrödinger equation with the fractional dimension based on Lévy random flights. By ...implementing linear stability analysis and direct simulations, we demonstrate regions where the single and multi-peak modes are stable. Analysis of perturbed coupled solitons confirms the stability of higher-order modes compared to lower-order modes under the same configurations. The stability diagrams show that the force coupling, Lévy index, power, and the nonlinear intensity significantly influence the stability of high-order modes. Our findings indicate that stability is favored in self-defocusing systems with high Lévy indices under weak coupling conditions, with higher-order states exhibiting smaller stability regions.
•Stable excited modes in coupled fractional nonlinear equations are found.•The linear stability and numerical simulations confirm the stability of the modes.•The stability of the modes is affected by the Levy index, coupling, and power.•Stability is favored in self-defocusing systems with weak coupling.
Fractional quantum couplers Zeng, Liangwei; Zeng, Jianhua
Chaos, solitons and fractals,
November 2020, 2020-11-00, Letnik:
140
Journal Article
Recenzirano
•Fractional Schrödinger equation is an important extension of Schrödinger equation.•Two-component fractional Schrödinger equation is investigated.•Quantum couplers with fractional-order diffraction ...can support asymmetric solitons.•The existence, stability and evolution of the obtained solitons are studied.
Fractional quantum coupler, a new type of quantum couplers that is composed of arrays of two coupled waveguides or a dual-core waveguide with intermodal coupling, within which the light waves diffraction is of the fractional-order differentiation, is put forward in the territory of fractional quantum mechanics. The modelling equations of such fractional couplers are derived in the framework of coupled nonlinear fractional Schrödinger equations with the space derivative of fractional order denoted by Lévy index α, and localized wave solutions as spatial optical solitons of these equations are constructed and their nonlinear propagation properties are discussed. Linear perturbation method based on linear stability analysis, and direct simulations are conducted to identify the stability and instability regions of the predicted solitons.
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