  We prove a full asymptotic stability result for solitary wave solutions of
the mKdV equation. We consider small perturbations of solitary waves with
polynomial decay at infinity and prove that solutions of the Cauchy problem
evolving from such data tend uniformly, on the real line, to another solitary
wave as time goes to infinity. We describe precisely the asymptotics of the
perturbation behind the solitary wave showing that it satisfies a nonlinearly
modified scattering behavior. This latter part of our result relies on a
precise study of the asymptotic behavior of small solutions of the mKdV
equation.
