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 the extension is not weakly sequentially continuous in $L^2_{ r,0}(\mathbb{ T})$. We then classify the traveling wave solutions for the third order Benjamin-Ono equation in $L^2_{ r,0}(\mathbb{ T})$ and study their orbital stability.
Categories: math.AP