Consider the nonlinear Schrödinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.
Kenji NAKANISHI. "Global dynamics below excited solitons for the nonlinear Schrödinger equation with a potential." J. Math. Soc. Japan 69 (4) 1353 - 1401, October, 2017. https://doi.org/10.2969/jmsj/06941353