scholarly journals Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one

2017 ◽  
Vol 37 (8) ◽  
pp. 4309-4328 ◽  
Author(s):  
Daniele Garrisi ◽  
◽  
Vladimir Georgiev ◽  
◽  
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Orbital Stability ◽  
Non Linear ◽  
Dimension One

scholarly journals Orbital stability vs. scattering in the cubic-quintic Schrödinger equation

2020 ◽  
pp. 2150004
Author(s):  
Rémi Carles ◽  
Christof Sparber
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Orbital Stability ◽  
Cubic Nonlinearity ◽  
The Cauchy Problem ◽  
Stability Of Solitary Waves ◽  
Dimension One ◽  
Three Space ◽  
Quintic Nonlinear Schrödinger Equation

We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable.

Existence of a Ground State Solution for a Quasilinear Schrödinger Equation

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Xiang-dong Fang ◽  
Zhi-qing Han
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Quasilinear Schrödinger Equation

AbstractIn this paper we are concerned with the quasilinear Schrödinger equation−Δu + V(x)u − Δ(uwhere N ≥ 3, 4 < p < 4N/(N − 2), and V(x) and q(x) go to some positive limits V

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