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Ground states of an  L  2 -subcritical focusing nonlinear Schrödinger (NLS) equation are known to be orbitally stable in the energy classthanks to its variational characterization. In this paper, we will show  L  2 -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  L  2 -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.

Bäcklund Transformation and L 2 -stability of NLS Solitons