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Tetsu Mizumachi and Dmitry Pelinovsky

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Backlund transformation and L2-stability of NLS solitons

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International Mathematics Research Notices, Vol. 2012, No. 9, pp. 2034–2067 (2012)

**Abstract:**

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.

**Keywords:**

nonlinear Schrodinger equation, solitons, inverse scattering, Backlund transformation,
orbital stability.