The third order Benjamin-Ono equation on the torus: Well-posedness, traveling waves and stability

Abstract

We consider the third order Benjamin-Ono equation on the torus

We prove that for any $t\in \mathbb{R}$, the flow map continuously extends to $H_{r,0}^s(\mathbb{T})$ if $s\ge 0$, but does not admit a continuous extension to $H_{r,0}^{-s}(\mathbb{T})$ if $0<s<\frac{1}{2}$. Moreover, we show that the extension is weakly sequentially continuous in $H_{r,0}^s(\mathbb{T})$ if $s>0$, but is not weakly sequentially continuous in $L_{r,0}^2(\mathbb{T})$. We then classify the traveling wave solutions for the third order Benjamin-Ono equation in $L_{r,0}^2(\mathbb{T})$ and study their orbital stability.