Bäcklund Transformation and L2-stability of NLS Solitons

Ground states of an L2-subcritical focusing nonlinear Schrödinger (NLS) equation are known to be orbitally stable in the energy class thanks to its variational characterization. In this paper, we will show L2-stability of 1-solitons to a one-dimensional cubic NLS equation in the sense that for any initial data which are sufficiently close to a 1-soliton in , the solution remains in an L2-neighborhood of a nearby 1-soliton for all the time. The proof relies on the Bäcklund transformation between zero and soliton solutions of this integrable equation.