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

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