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

TL;DR

This paper proves the existence of small solitary waves in one-dimensional particle lattices with power-law repulsive forces, connecting these waves to fractional KdV and Benjamin-Ono equations for certain force exponents.

Contribution

It establishes the existence of near-sonic solitary waves in particle lattices with power-law forces using fixed-point methods, linking them to fractional KdV and Benjamin-Ono equations.

Findings

Existence of small solitary waves for specific force exponents

Connection between lattice waves and fractional KdV/Benjamin-Ono equations

Use of fixed-point arguments to prove wave existence

Abstract

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.