  We study the paralinearised weakly dispersive Burgers type equation:
$$\partial_t u+\partial_x [T_u u]-T_{\frac{\partial_x u}{2}}u+\partial_x
|D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ which contains the main non linear
"worst interaction" terms, i.e low-high interaction terms, of the usual weakly
dispersive Burgers type equation: $$\partial_t u+u\partial_x u+\partial_x
|D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ with $u_0 \in H^s(\mathbb D)$, where
$\mathbb D=\mathbb T \text{ or } \mathbb R$.
  Through a paradifferential complex Cole-Hopf type gauge transform we
introduce for the study of the flow map regularity of Gravity-Capillary
equation, we prove a new a priori estimate in $H^s(\mathbb D)$ under the
control of $\left\Vert(1+\left\Vert u\right\Vert_{L^\infty_x})\left\Vert u
\right\Vert_{W^{2-\alpha,\infty}_x}\right\Vert_{L^1_t}$, improving upon the
usual hyperbolic control $\left\Vert \partial_x
u\right\Vert_{L^1_tL^\infty_x}$. Thus we eliminate the "standard" wave breaking
scenario in case of blow up as conjectured by J. C. Saut and C. Klein in their
numerical study of the dispersive Burgers equation.
  For $\alpha\in ]2,3[$ we show that we can completely conjugate the
paralinearised dispersive Burgers equation to a semi-linear equation of the
form: $$\partial_tu+ \partial_x |D|^{\alpha-1}u=R_\infty(u),\ \alpha \in
]2,3[,$$ where $R_\infty$ is a regularizing operator under the control of
$\left\Vert u\right\Vert_{L^\infty_t C^{2-\alpha}_*}$.
