  In the first part of this paper we prove that the flow associated to a
dispersive Burgers equation with a non local term
  of the form $|D|^{\alpha-1} \partial_x u$, $\alpha \in [1,+\infty[$ is
Lipschitz from bounded sets of $H^s_0(\mathbb{T};\mathbb{R})$ to
$C^0([0,T],H^{s-(2-\alpha)^+}_0(\mathbb{T};\mathbb{R}))$ for $T>0$ and
$s>\lceil \frac{\alpha}{\alpha-1}\rceil-\frac{1}{2}$, where $H^s_0$ are the
Sobolev spaces of functions with $0$ mean value, proving that the result
obtained in [37] is optimal on the torus. The proof relies on a
paradifferential generalization of a complex Cole-Hopf gauge transformation
introduced by T.Tao in [43] for the Benjamin-Ono equation.
  For this we prove a generalization of the Baker-Campbell-Hausdorff formula
for flows of hyperbolic paradifferential equations and prove the stability of
the class of paradifferential operators modulo more regular remainders, under
conjugation by such flows. For this we prove a new characterization of
paradifferential operators in the spirit of Beals [9].
  In the second part of this paper we use a paradifferential version of the
previous method to prove that a re-normalization of the flow of the one
dimensional periodic gravity capillary equation is Lipschitz from bounded sets
of $H^s$ to $C^0([0,T],H^{s-\frac{1}{2}})$ for $T>0$ and $s>3+\frac{1}{2}$.
This proves that the result obtained in [37] is optimal for the water waves
system.
