We study the long-time behavior of solutions
of the nonlinear Schr\"odinger equation in one space dimension
for initial conditions in a small neighborhood of a stable solitary
wave.
The method, initiated in [Buslaev and Perelman, St. Petersburg Math. J.
{\bf 4}(1993), 1111-1142], is based on the spectral
decomposition of solutions on the eigenspaces associated to
the discrete and continuous spectrum of
the linearized operator near the solitary wave.
Under some hypothesis on the structure of the spectrum
of the linearized operator,
we prove that, asymptotically in time, the solution decomposes
into a solitary wave with slightly modified parameters
and a dispersive part described by the free
Schr\"odinger equation. We
explicitly calculate the time behavior of the correction.
This is a joint work with V. Buslaev.
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