scholarly journals Exponential time differencing schemes for the 3-coupled nonlinear fractional Schrödinger equation

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Xiao Liang ◽  
Harish Bhatt
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Exponential Time ◽  
Fractional Schrödinger Equation ◽  
Exponential Time Differencing ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

Efficient exponential time differencing methods with Padé approximations for the semilinear space-time-fractional Schrödinger equation

2020 ◽  
pp. 2050428
Author(s):  
Xiao Liang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Time ◽  
Padé Approximations ◽  
Exponential Time ◽  
Fractional Schrödinger Equation ◽  
Exponential Time Differencing ◽  
Pade Approximations ◽  
Semilinear Space ◽  
Fractional Schrodinger Equation

The semilinear space-time-fractional Schrödinger equation is solved numerically using one-step and two-step exponential time differencing methods in time, and a fractional centered difference scheme in space. The two-parametric Mittag–Leffler function arising in the time integral is computed with Padé approximations, which improves the efficiency of the scheme markedly. Numerical experiments for well-known models from literature are performed to show the effectiveness and efficiency of the proposed methods.

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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