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

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 α + 1 with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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.