  We consider the Cauchy problem for one-dimensional dispersive equations with
a general nonlinearity in the periodic setting. Our main hypotheses are both
that the dispersive operator behaves for high frequencies as a Fourier
multiplier by $ i |\xi|^\alpha \xi $, with $ 1\le \alpha \le 2 $, and that the
nonlinear term is of the form $ \partial_x f(u) $ where $ f $ is the sum of an
entire series with infinite radius of convergence. Under these conditions, we
prove the unconditional local well-posedness of the Cauchy problem in
$H^{s}(\mathbb{T})$ for $ s\ge 1-\frac{\alpha}{2(\alpha+1)}$. This leads to
some global existence results above the energy space $ H^{\alpha/2}(\mathbb{T})
$, for $ \alpha \in [\sqrt{2},2]$.
