We consider 1-dimensional non-linear Schrödinger equations around a travelling wave. We prove its asymptotic stability for general non-linearities, under the hypotheses that the orbital stability condition of Grillakis-Shatah-Strauss is satisfied and that the linearized operator does not have a resonance and only has 0 as an eigenvalue. As a by-product of our approach, we show long-range scattering for the radiation remainder. Our proof combines for the first time modulation techniques and the study of space-time resonances. We rely on the use of the distorted Fourier transform, akin to the work of Buslaev and Perelman, and of Krieger and Schlag, and on precise computations and estimates of space-time resonances to handle its interaction with the soliton.
This is joint work with Pierre Germain.
This video was produced by the SITE Research Center at New York University, as part of their talk series.
