On the Variational Approach to the Stability of Standing Waves for the Nonlinear Schrödinger Equation

Abstract

We consider the orbital stability of standing waves of the nonlinear Schrdinger equationby the approach that was laid down by Cazenave and Lions in 1992. Our work covers several situations that do not seem to be included in previous treatments, namely,(i) g(x, s) − g(x, 0) → 0 as |x| → ∞ for all s ≥ 0. This includes linear problems.(ii) g(x, s) is a periodic function of x ∈ ℝ(iii) g(x, s) is asymptotically periodic in the sense that g(x, s) − gFurthermore, we focus attention on the form of the set that is shown to be stable and may be bigger than what is usually known as the orbit of the standing wave.

Orbital Stability: Analysis Meets Geometry

“…stability does not hold at σ = 1 + 4 d ). The contribution [CL82] is one of the first rigorous results on orbital stability for nonlinear dispersive equations, and is based on variational arguments using the concentration-compactness principle (see for instance [Zhi01,HS04] for more recent results in this direction). This line of argument is conceptually very different from the energy-momentum approach developed here, so we shall not say more about it.…”

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

“…Since 1 < p < 1 + 4 N and P, V ∈ L ∞ (R N ), it follows from Corollary 6.1.2 of [4] that (GWP at ξ) holds for all ξ ∈ X. By Theorem 6.2, it is enough so show that (M) c is satisfied for the relevant values of c. In case (I) this follows from Proposition 5.1 of [18]. In case (II) we use Proposition 5.3 of [18] if at least one of the inequalities in (8.8) is strict on a set of positive measure.…”

“…By Theorem 6.2, it is enough so show that (M) c is satisfied for the relevant values of c. In case (I) this follows from Proposition 5.1 of [18]. In case (II) we use Proposition 5.3 of [18] if at least one of the inequalities in (8.8) is strict on a set of positive measure. If this is not the case, then P ≡ P ∞ and V ≡ V ∞ and we are in the context of assumption (E) of Section 6 with Υ = {γ z : z ∈ Z N } where γ z (v) = v(· + z) for all v ∈ X.…”

“…If this is not the case, then P ≡ P ∞ and V ≡ V ∞ and we are in the context of assumption (E) of Section 6 with Υ = {γ z : z ∈ Z N } where γ z (v) = v(· + z) for all v ∈ X. Proposition 5.2 of [18] shows that (EM) c and (WS) follows from Corollary 6.3.…”

Existence and stability of TE modes in a stratified non-linear dielectric

“…REMARK 7.4 There are many results establishing the orbital stability of standing wave solutions of more general NLSs in higher space dimensions. However, these results are either of a perturbative nature (Grillakis et al, 1987;Strauss, 1989;Rose & Weinstein, 1988;Weinstein, 1986) and deal the solutions near the bifurcation point or they deal with a weaker notion of orbital stability (Cazenave & Lions (1982), Cazenave (2003), Hajaiej & Stuart (2004)). In a major contribution to the understanding of orbital stability, conditions were established in Grillakis et al (1987) giving a rigorous setting for the Vakhitov-Kolokolov (V-K) criterion (1973) (Kivshar & Sukhorukov, 2001).…”