scholarly journals Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

2012 ◽  
Vol 142 (6) ◽  
pp. 1237-1262 ◽  
Author(s):  
Patricio Felmer ◽  
Alexander Quaas ◽  
Jinggang Tan
Keyword(s):  
Schrödinger Equation ◽  
Positive Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Symmetry Properties ◽  
Existence Of Positive Solutions ◽  
Image Position

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

Painlevé analysis of the damped, driven nonlinear Schrödinger equation

1988 ◽  
Vol 109 (1-2) ◽  
pp. 109-126 ◽  
Author(s):  
Peter A. Clarkson
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Analytic Functions ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Completely Integrable ◽  
Real Analytic Functions ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equationwhere a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only ifwhere α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

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