scholarly journals Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity

2017 ◽  
Vol 155 ◽  
pp. 52-64 ◽  
Author(s):  
Alex H. Ardila
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Fractional Schrödinger Equation ◽  
Existence And Stability ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

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.

Defect solitons supported by nonlinear fractional Schrödinger equation with a defective lattice

2019 ◽  
Vol 28 (02) ◽  
pp. 1950021
Author(s):  
Yunji Meng ◽  
Renxia Ning ◽  
Kun Ma ◽  
Zheng Jiao ◽  
Haijiang Lv ◽  
...  
Keyword(s):  
Schrödinger Equation ◽  
Low Power ◽  
Schrodinger Equation ◽  
Fractional Schrödinger Equation ◽  
Existence Domain ◽  
Existence And Stability ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Power Region ◽  
Fractional Schrodinger Equation ◽  
Deep Defects

We investigate numerically the existence and stability of defect solitons in nonlinear fractional Schrödinger equation. For positive defects, defect solitons are only existent in the semi-infinite gap and are stable in their whole existence domain irrespective of Lévy index. For moderate deep defects, defect solitons are existent in both the semi-infinite gap and first gap, and their instability domains occur in the low-power region of the semi-infinite gap. While for deep enough defects, stable defect solitons can be found in the second gap. Increasing the strength of defect (or Lévy index) will narrow (or broaden) the existence and stability domains.

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