  Ground states of a L2-subcritical focusing nonlinear Schrodinger (NLS)
equation are known to be orbitally stable in the energy class H1 thanks to its
variational characterization. In this paper, we will show L2-orbital stability
of 1-solitons to a one-dimensional cubic NLS equation for any initial data
which are close to 1-solitons in L2. Moreover, we prove that if the initial
data are in H3 in addition to being small in L2, then the solution remains in
an L2-neighborhood of a specific 1-soliton solution for all the time. The proof
relies on the Backlund transformation between zero and soliton solutions of
this integrable equation.
