scholarly journals Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

2008 ◽  
Vol 76 (1) ◽  
pp. 329-399 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

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 THE NONLINEAR SCHRÖDINGER EQUATION WITH POTENTIAL

2001 ◽  
Vol 13 (12) ◽  
pp. 1529-1546 ◽  
Author(s):  
CARLOS CID ◽  
PATRICIO FELMER
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Local Minimum ◽  
Standing Waves ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

We prove existence and orbital stability of standing waves for the nonlinear Schrödinger equation [Formula: see text] concentrating near a possibly degenerate local minimum of the potential V, when the Plank's constant ℏ is small enough. Our method applies to general nonlinearities, including f(s)=sp - 1 with p ∈ (1,1 + 4/N), but does not require uniqueness nor non-degeneracy of the limiting equation.

scholarly journals Existence and orbital stability of standing waves to a nonlinear Schrödinger equation with inverse square potential on the half-line

2021 ◽  
Vol 28 (5) ◽  
Author(s):  
Elek Csobo
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Wave Solution ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Half Line ◽  
Minimizing Sequences ◽  
Nonlinear Schrödinger

AbstractIn our work, we establish the existence of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We apply a profile decomposition argument to overcome the difficulty arising from the non-compactness of the setting. We obtain convergent minimizing sequences by comparing the problem to the problem at “infinity” (i.e., the equation without inverse square potential). Finally, we establish orbital stability/instability of the standing wave solution for mass subcritical and supercritical nonlinearities respectively.

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