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

Abstract

Abstract We consider the orbital stability of standing waves of the nonlinear Schrdinger equation by 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 N for all s ≥ 0. (iii) g(x, s) is asymptotically periodic in the sense that g(x, s) − g ∞ (x, s) → 0 as |x| → ∞ for some function g ∞ that is periodic with respect to x ∈ ℝ N for all s ≥ 0. Furthermore, 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.