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

We consider the third order Benjamin-Ono equation on the torus${\partial }_{t}u={\partial }_x(-{\partial }_{xx}u-\frac{3}{2}uH{\partial }_{x}u-\frac{3}{2}H(u{\partial }_{x}u)+u^3).$ We prove that for any $t\in R$, the flow map continuously extends to $H_{r,0}^s\left(T\right)$ if $s\ge 0$, but does not admit a continuous extension to $H_{r,0}^{-s}\left(T\right)$ if $0<s<\frac{1}{2}$. Moreover, we show that the extension is weakly sequentially continuous in $H_{r,0}^s\left(T\right)$ if $s>0$, but is not weakly sequentially continuous in $L_{r,0}^2\left(T\right)$. We then classify the traveling wave solutions for the third order Benjamin-Ono equation in $L_{r,0}^2\left(T\right)$ and study their orbital stability.