Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential

2018 ◽  
Vol 6 (9) ◽  
pp. 875 ◽  
Author(s):  
Xiankun Yao ◽  
Xueming Liu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Schrödinger Equation ◽  
Lattice Potential ◽  
Fractional Schrodinger Equation

The Numerical Computation of the Time Fractional Schrödinger Equation on an Unbounded Domain

2018 ◽  
Vol 18 (1) ◽  
pp. 77-94
Author(s):  
Dan Li ◽  
Jiwei Zhang ◽  
Zhimin Zhang
Keyword(s):  
Schrödinger Equation ◽  
Fractional Derivative ◽  
Unbounded Domain ◽  
Caputo Fractional Derivative ◽  
Schrodinger Equation ◽  
Computational Cost ◽  
Main Idea ◽  
Fractional Schrödinger Equation ◽  
Direct Scheme ◽  
Fractional Schrodinger Equation

AbstractA fast and accurate numerical scheme is presented for the computation of the time fractional Schrödinger equation on an unbounded domain. The main idea consists of two parts. First, we use artificial boundary methods to equivalently reformulate the unbounded problem into an initial-boundary value (IBV) problem. Second, we present two numerical schemes for the IBV problem: a direct scheme and a fast scheme. The direct scheme stands for the direct discretization of the Caputo fractional derivative by using the L1-formula. The fast scheme means that the sum-of-exponentials approximation is used to speed up the evaluation of the Caputo fractional derivative. The resulting fast algorithm significantly reduces the storage requirement and the overall computational cost compared to the direct scheme. Furthermore, the corresponding stability analysis and error estimates of two schemes are established, and numerical examples are given to verify the performance of our approach.

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