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In the first part of this paper we prove that the flow associated to adispersive Burgers equation with a non local term  of the form $|D|^{\alpha-1} \partial_x u$, $\alpha \in [1,+\infty[$ isLipschitz 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 theSobolev spaces of functions with $0$ mean value, proving that the resultobtained in [37] is optimal on the torus. The proof relies on aparadifferential generalization of a complex Cole-Hopf gauge transformationintroduced by T.Tao in [43] for the Benjamin-Ono equation.  For this we prove a generalization of the Baker-Campbell-Hausdorff formulafor flows of hyperbolic paradifferential equations and prove the stability ofthe class of paradifferential operators modulo more regular remainders, underconjugation by such flows. For this we prove a new characterization ofparadifferential operators in the spirit of Beals [9].  In the second part of this paper we use a paradifferential version of theprevious method to prove that a re-normalization of the flow of the onedimensional periodic gravity capillary equation is Lipschitz from bounded setsof $H^s$ 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 wavessystem.

Regularity results on the flow maps of periodic dispersive Burgers type equations and the Gravity-Capillary equations