scholarly journals Existence of Standing Waves for a Generalized Davey-Stewartson System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Xiaoxiao Hu ◽  
Xiao-xun Zhou ◽  
Wu Tunhua ◽  
Min-Bo Yang
Keyword(s):  
Schrödinger Equation ◽  
Variational Methods ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Existence Results ◽  
Equation Problem

The purpose of this paper is to investigate the existence of standing waves for a generalized Davey-Stewartson system. By reducing the system to a single Schrödinger equation problem, we are able to establish some existence results for the system by variational methods.

scholarly journals Orbital stability of standing waves for a class of Schrödinger equations with unbounded potential

2006 ◽  
Vol 2006 ◽  
pp. 1-7
Author(s):  
Guanggan Chen ◽  
Jian Zhang ◽  
Yunyun Wei
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Variational Calculus ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Unbounded Potential

This paper is concerned with the nonlinear Schrödinger equation with an unbounded potential iϕt=−Δϕ+V(x)ϕ−μ|ϕ|p−1ϕ−λ|ϕ|q−1ϕ, x∈ℝN, t≥0, where μ>0, λ>0, and 1<p<q<1+4/N. The potential V(x) is bounded from below and satisfies V(x)→∞ as |x|→∞. From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.

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.

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