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We consider the third order Benjamin-Ono equation on the torus $\partial_t u=\partial_x \left( -\partial_{xx}u-\frac{3}{2}u H\partial_x u -\frac{3}{2}H(u\partial_x u) + u^3 \right).$ We prove that for any$t\in\mathbb{R}$, the flow map continuously extends to $H^s_{r,0}(\mathbb{T})$if $s\geq 0$, but does not admit a continuous extension to$H^{-s}_{r,0}(\mathbb{T})$ if $0<s<\frac{1}{2}$. Moreover, we show that theextension is not weakly sequentially continuous in $L^2_{r,0}(\mathbb{T})$. Wethen classify the traveling wave solutions for the third order Benjamin-Onoequation in $L^2_{r,0}(\mathbb{T})$ and study their orbital stability.

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