Fourier spectral method with an adaptive time strategy for nonlinear fractional Schrödinger equation

2019 ◽  
Vol 36 (4) ◽  
pp. 823-838
Author(s):  
Haidong Qu ◽  
Zihang She
Keyword(s):  
Schrödinger Equation ◽  
Spectral Method ◽  
Schrodinger Equation ◽  
Fourier Spectral Method ◽  
Fractional Schrödinger Equation ◽  
Adaptive Time ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

scholarly journals A Conservative Crank-Nicolson Fourier Spectral Method for the Space Fractional Schrödinger Equation with Wave Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Lei Zhang ◽  
Rui Yang ◽  
Li Zhang ◽  
Lisha Wang
Keyword(s):  
Schrödinger Equation ◽  
Spectral Method ◽  
Schrodinger Equation ◽  
Fourier Spectral Method ◽  
Fractional Schrödinger Equation ◽  
Wave Operators ◽  
Conservative Difference ◽  
Fourier Collocation ◽  
Fractional Schrodinger Equation ◽  
Crank Nicolson

In this paper, the Crank-Nicolson Fourier spectral method is proposed for solving the space fractional Schrödinger equation with wave operators. The equation is treated with the conserved Crank-Nicolson Fourier Galerkin method and the conserved Crank-Nicolson Fourier collocation method, respectively. In addition, the ability of the constructed numerical method to maintain the conservation of mass and energy is studied in detail. Meanwhile, the convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Finally, the numerical experiments verify the properties of the conservative difference scheme and demonstrate the correctness of theoretical results.

Orbital stability of standing waves for nonlinear fractional Schrödinger equation with unbounded potential

2019 ◽  
Vol 12 (03) ◽  
pp. 1950043
Author(s):  
Xiaohua Liu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Orbital Stability ◽  
Fractional Schrödinger Equation ◽  
Unbounded Potential ◽  
Nonlinear Fractional Schrödinger Equation ◽  
The Stability ◽  
First Time ◽  
Fractional Schrodinger Equation

In this paper, the orbital stability of standing waves for nonlinear fractional Schrödinger equation is considered. By constructing the constrained functional extreme-value problem, the existence of standing waves is studied. With the help of the orbital stability theories presented by Grillakis, Shatah and Strauss, the orbital stability of standing waves is determined by the sign of a discriminant. To our knowledge, it is the first time that the abstract orbital stability theories presented by Grillakis, Shatah and Strauss are applied to study the stability of solutions for fractional evolution equation.

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