Local well-posedness for the gKdV equation on the background of a bounded function

Abstract

We prove the local well-posedness for the generalized Korteweg-de Vries equation in H(R), s > 1/2, under general assumptions on the nonlinearity f(x), on the background of an Lt,x-function Ψ(t, x), with Ψ(t, x) satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in H(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(R) for s > 1/2. We also prove global existence in the energy space H(R), in the case where the nonlinearity satisfies that |f (x)| . 1.