  In the first part of this paper we prove that the flow associated to the
Burgers equation with a non local term of the form $\partial_x |D|^{\alpha-1}
u$ fails to be uniformly continuous from bounded sets of $H^s({\mathbb D})$ to
$C^0([0,T],H^s({\mathbb D}))$ for $T>0$, $s>\frac{1}{2}+2$, $0\leq \alpha <2$,
${\mathbb D}={\mathbb R} \ \text{or} \ {\mathbb T} $. Furthermore we show that
the flow cannot be $C^1$ from bounded sets of $H^s({\mathbb D})$ to
$C^0([0,T],H^{s-1+(\alpha-1)^+ +\epsilon}({\mathbb D}))$ for $\epsilon>0$. We
generalize this result to a large class of nonlinear transport-dispersive
equations in any dimension, that in particular contains the Whitham equation
and the paralinearization of the water waves system with and without surface
tension. The current result is optimal in the sense that for $\alpha=2$ and
${\mathbb D}={\mathbb T}$ the flow associated to the Benjamin-Ono equation is
Lipschitz on function with $0$ mean value $H^s_0$.
  In the second part of this paper we apply this method to deduce the
quasi-linearity of the water waves system, which is the main result of this
paper.
