  We consider the generalized Korteweg-de Vries equation in the supercritical
case, and we are interested in solutions which converge to a soliton in large
time in H^1. In the subcritical case, such solutions are forced to be exactly
solitons by variational characterization, but no such result exists in the
supercritical case. In this paper, we first construct a "special solution" in
this case by a compactness argument, i.e. a solution which converges to a
soliton without being a soliton. Secondly, using a description of the spectrum
of the linearized operator around a soliton due to Pego and Weinstein, we
construct a one parameter family of special solutions which characterizes all
such special solutions.
