A symplectic non-squeezing theorem for BBM equation

We study the initial value problem for the BBM equation: $$\{\begin{array}{l}{u}_t+u_x+u{u}_x-u_{txx}=0x\in \text{T},t\in \mathbb{R}u(0,x)=u_0(x)\end{array}\mathrm{.}$$ We prove that the BBM equation is globaly well-posed on $H^s(\text{T})$ for $s\ge 0$ and a symplectic non-squeezing theorem on $H^{1\mathrm{/}2}(\text{T})$. That is to say the flow-map $u_0\mapsto u(t)$ that associates to initial data $u_0\in H^{1\mathrm{/}2}(\text{T})$ the solution $u$ cannot send a ball into a symplectic cylinder of smaller width.