Title: Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation,
Abstract: This article represents a first step toward understanding the long time
dynamics of solutions for the Benjamin-Ono equation. While this problem is
known to be both completely integrable and globally well-posed in $L^2$, much
less seems to be known concerning its long time dynamics. Here, we prove that
for small localized data the solutions have (nearly) dispersive dynamics almost
globally in time. An additional objective is to revisit the $L^2$ theory for
the Benjamin-Ono equation and provide a simpler, self-contained approach.