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

We prove the local well-posedness for the generalized Korteweg-de Vries equation in Hs.R/, s > 1=2, under general assumptions on the nonlinearity f .x/, on the background of an L1t;x-function parts per thousand.t; x/, with parts per thousand.t; x/ satisfying some suit-able conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in Hs.R/. This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solu-tion, 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 Hs.R/ for s > 1=2. We also prove global existence in the energy space H1.R/, in the case where the nonlinearity satisfies If 00.x/I . 1.