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

TL;DR

This paper establishes local well-posedness and unconditional uniqueness for the gKdV equation in certain Sobolev spaces, accommodating background functions and nonlinearities, with implications for solutions around Kinks and periodic waves.

Contribution

It extends well-posedness results for the gKdV equation to include background functions and nonlinearities, providing a framework for solutions near Kinks and periodic solutions.

Findings

Proves local well-posedness for gKdV in H^s with s>1/2.

Establishes unconditional uniqueness in H^s for s>1/2.

Demonstrates global existence in H^1 when the nonlinearity's second derivative is bounded.

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 $L^\infty_{t,x}$-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 $\vert f''(x)\vert\lesssim 1$.