Abstract
Motivated by the study of the Cauchy problem with bore-like initial data, we show the "well-posedness" for Korteweg-de Vries and Benjamin-Ono equations with initial data in Zhidkov spaces X s , with respectively s > 1 and s > 5/4. Here, "well-posedness" includes local (global in some cases) existence, uniqueness under a supplementary assumption, and continuity with respect to the initial data.