  We construct solitary waves for the fractional Korteweg-De Vries type
equation $u_t + (\Lambda^{-s}u + u^2)_x = 0$, where $\Lambda^{-s}$ denotes the
Bessel potential operator $(1 + |D|^2)^{-\frac{s}{2}}$ for $s \in (0,\infty)$.
The approach is to parameterise the known periodic solution curves through the
relative wave height. Using a priori estimates, we show that the periodic waves
locally uniformly converge to waves with negative tails, which are transformed
to the desired branch of solutions. The obtained branch reaches a highest wave,
the behavior of which varies with $s$. The work is a generalisation of recent
work by Ehrnstr\"om-Nik-Walker, and is as far as we know the first simultaneous
construction of small, intermediate and highest solitary waves for the complete
family of (inhomogeneous) fractional KdV equations with negative-order
dispersive operators. The obtained waves display exponential decay rate as $|x|
\to \infty$.
