  We prove the local well-posedness for the generalized Korteweg-de Vries
equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the
nonlinearity $f(x)$, on the background of an $L^\infty_{t,x}$-function
$\Psi(t,x)$, with $\Psi(t,x)$ satisfying some suitable conditions. As a
consequence of our estimates, we also obtain the unconditional uniqueness of
the solution in $H^s(\mathbb{R})$. This result not only gives us a framework to
solve the gKdV equation around a Kink, for example, but also around a periodic
solution, that is, to consider localized non-periodic perturbations of a
periodic solution. As a direct corollary, we obtain the unconditional
uniqueness of the gKdV equation in $H^s(\mathbb{R})$ for $s>1/2$. We also prove
global existence in the energy space $H^1(\mathbb{R})$, in the case where the
nonlinearity satisfies that $\vert f''(x)\vert\lesssim 1$.
