Tetsu Mizumachi and Dmitry Pelinovsky

Backlund transformation and L2-stability of NLS solitons

International Mathematics Research Notices, Vol. 2012, No. 9, pp. 20342067 (2012)

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.

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