On well-posedness and wave operator for the gKdV equation

TL;DR

This paper provides an alternative proof of well-posedness for the gKdV equation in a critical Sobolev space, introduces a new linear estimate, and constructs a wave operator extending previous results.

Contribution

It offers a new proof of well-posedness, establishes a blow-up alternative, and constructs a wave operator in the critical space for the gKdV equation.

Findings

Established local and global well-posedness in the critical Sobolev space.

Developed a new linear estimate for the gKdV equation.

Constructed a wave operator extending previous results.

Abstract

We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k>4$ is an integer number and $\mu=\pm1$. We give an alternative proof of the Kenig, Ponce, and Vega result in \cite{kpv1}, which asserts local and global well-posedness in $\dot{H}^{s_k}(\R)$, with $s_k=(k-4)/2k$. A blow-up alternative in suitable Strichatz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space $\dot{H}^{s_k}(\R)$, extending the results of C\^ote [2].