On the generalized Korteweg-de Vries-type equations

Abstract

We show well-posedness results for the generalized Korteweg-de Vries equation with nonlinear term $F(u)\partial_xu$. We assume $F(u)$ is a $C^4$ function and $F(0)=0$. Using a version of the chain rule for fractional derivatives and some estimates on the evolution group, we prove existence, uniqueness and regularity properties of the solution of the equation when the space of the initial data is $H^s(\mathbb{R}), \, s>1/2$. The theorem we prove is sharp. We obtain all the above results also for a mixed KdV and Schrödinger type equation proposed as a model for the propagation of a signal in an optic fiber.