Existence of solitary waves in particle lattices with power-law forces

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 $\frac{4}{3}<\alpha <3$, 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.