scholarly journals Variational properties and orbital stability of standing waves for NLS equation on a star graph

2014 ◽  
Vol 257 (10) ◽  
pp. 3738-3777 ◽  
Author(s):  
Riccardo Adami ◽  
Claudio Cacciapuoti ◽  
Domenico Finco ◽  
Diego Noja
Keyword(s):  
Standing Waves ◽  
Orbital Stability ◽  
Star Graph ◽  
Nls Equation

scholarly journals Constrained energy minimization and orbital stability for the NLS equation on a star graph

2014 ◽  
Vol 31 (6) ◽  
pp. 1289-1310 ◽  
Author(s):  
Riccardo Adami ◽  
Claudio Cacciapuoti ◽  
Domenico Finco ◽  
Diego Noja
Keyword(s):  
Energy Minimization ◽  
Orbital Stability ◽  
Star Graph ◽  
Nls Equation

scholarly journals Nonlinear Schrödinger equation on graphs: recent results and open problems

2014 ◽  
Vol 372 (2007) ◽  
pp. 20130002 ◽  
Author(s):  
Diego Noja
Keyword(s):  
Schrodinger Equation ◽  
Standing Waves ◽  
Special Consideration ◽  
Star Graph ◽  
Nls Equation ◽  
Open Problems ◽  
Hamiltonian Equations ◽  
Nonlinear Schrödinger ◽  
Physical Context ◽  
Solitary Solutions

In this paper, an introduction to the new subject of nonlinear dispersive Hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of the nonlinear Schrödinger (NLS) equation. Special consideration is given to the existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning the NLS equation on a star graph: the standing waves of the NLS equation on a graph with a δ interaction at the vertex, and the scattering of fast solitons through a Y-junction in the cubic case. The emphasis is on a description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.

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.

scholarly journals A Note on Sign-Changing Solutions to the NLS on the Double-Bridge Graph

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 161
Author(s):  
Diego Noja ◽  
Sergio Rolando ◽  
Simone Secchi
Keyword(s):  
Boundary Conditions ◽  
Standing Waves ◽  
Nls Equation ◽  
Sign Changing Solutions

We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on the circle, by proving some existing results of sign-changing solutions non-periodic on the circle.

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