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 $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the nonlinearity $f(x)$, on the background of an $L_{t,x}^{\infty }$-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 $|f^{\prime \prime }(x)|\lesssim 1$.