Abstract: The theory of solitons in space dimension D>1 starts with the proof of existence due to W.Strauss in the late 70's. Subsequently M. Weinstein was able to characterize stable and unstable solitons for nonlinear Schroedinger equations (NLS) using the invariants of motion of the equation. This work was generalized by W.Strauss, J.Shatah and M.Grillakis. In the late 80's A.Soffer and M.Weinstein were the first to prove the asymptotic stability of ground states of special NLS's exploiting some technology on Schroedinger operators originating from work by A.Jensen and T.Kato. The first results for translation invariants NLS's are due to semirigorous work by V.Buslaev and G.Perelman in the early 90s for 1D.

In this talk we give a quick presentation of our contribution to the problem of asymptotic stability of ground states of translation invariant NLS's for D>2 developed in the spring 00.

The approach is based on linearization of the equation around the soliton. One obtains a matrix valued linear operator whose spectrum is contained in the imaginary axis but which is not antisymmetric. Viewed by a finite dimensional perspective this linear operator seems of no use in proving asymptotic stability. Furthermore, the fact that it is not diagonalizable seems to have confused the authours in the 80's.

It turns out however that the linearized operator falls in the scope of the scattering theory developed in the 60's by T.Kato. Scattering theory allows to think that in some sense the spectrum of the linearized operator is negative. Thus one has stability, with many caveats. In fact asymptotic stability is very much a very hard open problem.