  We consider the generalized Korteweg-de Vries equation
  \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in
[0,T)\times \mathbb{R}, (1)
  with general $C^3$ nonlinearity $f$. Under an explicit condition on $f$ and
$c>0$, there exists a solution in the energy space $H^1$ of (1) of the type
$u(t,x)=Q_c(x-x_0-ct)$, called soliton.
  In this paper, under general assumptions on $f$ and $Q_c$, we prove that the
family of soliton solutions around $Q_c$ is asymptotically stable in some local
sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\geq 0$), then
$u(t)$ locally converges in the energy space to some $Q_{c_+}$ as $t\to
+\infty$. Note in particular that we do not assume the stability of $Q_{c}$.
This result is based on a rigidity property of equation (1) around $Q_{c}$ in
the energy space whose proof relies on the introduction of a dual problem.
These results extend the main results in previous works devoted to the pure
power case.
