KdV and BO equations with bore-like data

Abstract

The global well-posedness of the initial-value problem associated to the Korteweg-de Vries equation with bore-like data is studied. In particular, we show that in the Sobolev space $H^s$, $s\ge 2$, the solutions of this problem remain bounded for any time. We also establish similar results for solutions of the initial-value problem associated to the Benjamin-Ono equation with this kind of data.