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 ${\partial }_{t}u={\partial }_x\left(-{\partial }_{xx}u-\frac{3}{2}uH{\partial }_{x}u-\frac{3}{2}H(u{\partial }_{x}u)+u^3\right).$ We prove that for any $t\in \mathbb{R}$, the flow map continuously extends to $H_{r,0}^s(\mathbb{T})$ 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 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.