  For both localized and periodic initial data, we prove local existence in
classical energy space $H^s, s>\frac{3}{2}$, for a class of dispersive
equations $u_{t}+(n(u))_{x}+Lu_{x}=0$ with nonlinearities of mild regularity.
Our results are valid for symmetric Fourier multiplier operators $L$ whose
symbol is of temperate growth, and $n(\cdot)$ in local Sobolev space
$H^{s+2}_{\mathrm{loc}}(\mathbb{R})$. In particular, the results include
non-smooth and exponentially growing nonlinearities. Our proof is based on a
combination of semi-group methods and a new composition result for Besov
spaces. In particular, we extend a previous result for Nemytskii operators on
Besov spaces on $\mathbb{R}$ to the periodic setting by using the
difference-derivative characterization of Besov spaces.
