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.
"Korteweg-de Vries and Benjamin-Ono equations on Zhidkov spaces." Adv. Differential Equations 10 (3) 277 - 308, 2005.