  We prove the existence of small solitary waves for one-dimensional lattices
of particles that each repel every other particle with a force that decays as a
power of distance. For force exponents $\alpha+1$ with $\frac43<\alpha<3$, we
employ fixed-point arguments to find near-sonic solitary waves having scaled
velocity profiles close to non-degenerate solitary-wave profiles of fractional
KdV or generalized Benjamin-Ono equations. These equations were recently found
to approximately govern unidirectional long-wave motions in these lattices.
