scholarly journals Existence and Stability of Standing Waves for Nonlinear Fractional Schrödinger Equations with Hartree Type and Power Type Nonlinearities

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Na Zhang ◽  
Jie Xin
Keyword(s):  
Minimization Problem ◽  
Standing Waves ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Power Type ◽  
Initial Value ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Existence And Stability ◽  
Constrained Minimization Problem

We consider the standing wave solutions for nonlinear fractional Schrödinger equations with focusing Hartree type and power type nonlinearities. We first establish the constrained minimization problem via applying variational method. Under certain conditions, we then show the existence of standing waves. Finally, we prove that the set of minimizers for the initial value problem of this minimization problem is stable.

scholarly journals On the Existence and Stability of Standing Waves for 2-Coupled Nonlinear Fractional Schrödinger System

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Xiuyan Sha ◽  
Huanmin Ge ◽  
Jie Xin
Keyword(s):  
Minimization Problem ◽  
Standing Waves ◽  
Constrained Minimization ◽  
Stable Set ◽  
Initial Value ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Existence And Stability ◽  
Constrained Minimization Problem ◽  
Schrödinger System

We study a system of 2-coupled nonlinear fractional Schrödinger equations. Firstly, we construct constrained minimization problem to the system. Next, we prove the existence of standing waves for the system by using the concentration-compactness and commutator estimates method. Lastly, we also consider the set of minimizers of the constrained minimization problem. We prove that it is a stable set for initial value of the problem; that is, a solution to the system with initial value which is near the set will remain near it for all time.

Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity

2016 ◽  
Vol 15 (05) ◽  
pp. 699-729 ◽  
Author(s):  
Yonggeun Cho ◽  
Mouhamed M. Fall ◽  
Hichem Hajaiej ◽  
Peter A. Markowich ◽  
Saber Trabelsi
Keyword(s):  
Cauchy Problem ◽  
Standing Waves ◽  
Orbital Stability ◽  
Schrodinger Equations ◽  
Well Posedness ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Existence And Stability

This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method.

Sign in / Sign up

Export Citation Format

Share Document