We discuss the orbital stability of standing waves of a class of nonlinear Schrödinger equations in one space dimension. The crucial feature for our treatment is the presence of a non-constant linear potential that is even and decreasing away from the origin in space. This enables us to establish the orbital stability of all ground states over the whole range of frequencies for which such solutions exist.
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Charles A. Stuart, Uniqueness and stability of ground states for some nonlinear Schrödinger equations. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 399–414DOI 10.4171/JEMS/60