Stability of standing waves for all frequencies to nonlinear Schrödinger equations with potentials in one dimension
Authors
Noriyoshi Fukaya
Masahiro Ikeda
Hiroaki Kikuchi
Abstract
In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schrödinger equations with potentials. We show that the standing waves are orbitally stable for all frequencies in the $L^{2}$- subcritical and critical cases. Since the presence of potentials breaks the scale invariance of the equations, it is a delicate problem to apply the abstract theory of Grillakis, Shatah, and Strauss (1987) directly without a perturbative argument. For this reason, little is known about the orbital stability of standing waves for \textit{all} frequencies in the non-scale-invariant setting. We overcome this difficulty by employing the approach of Noris, Tavares, and Verzini (2014).
