Existence and instability of normalized standing waves for the fractional Schrödinger equations in the L2-supercritical case

2020 ◽  
Vol 61 (7) ◽  
pp. 071511
Author(s):  
Binhua Feng ◽  
Jiajia Ren ◽  
Qingxuan Wang
Keyword(s):  
Standing Waves ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Supercritical Case ◽  
Fractional Schrödinger Equations

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.

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.

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