On well-posedness and wave operator for the gKdV equation

We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+μ\partial_x(u^{k+1})=0$, where $k>4$ is an integer number and $μ=\pm1$. 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 Côte [2].

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