2017/05/31 by Stephen C. Anco, Michelle Przedborski
Mathematics · Physics and Astronomy · #Advanced Fiber Laser Technologies #Classical mechanics #Materials science #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Optics #Physics #Quantum mechanics #Wavelength #math-ph #math.MP
paper · pdf · doi:10.1103/physreve.98.042208
published as Phys. Rev. E 98, 042208 (2018) · 43 pages; 6 figures. This version is a major rewrite, with over 50% new material
arxiv created 2018/06/30 · openalex publication_date 2018/10/16 · arxiv updated 2018/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Properties of solitary waves in precompressed Hertzian chains of particles are studied in the long-wavelength limit using a well-known continuum model. Several main results are obtained by parametrizing the solitary waves in terms of their wave speed and their asymptotic amplitude. First, the asymptotic amplitude is shown to be directly related to the continuum sound speed, and the ratio of asymptotic amplitude to peak amplitude is shown to describe the degree of dynamical nonlinearity in the underlying discrete system. Second, an algebraic relation is derived that determines the dynamical nonlinearity ratio in terms of the ratio of the solitary wave speed to the sound speed. In particular, highly supersonic solitary waves correspond to highly nonlinear propagating pulses in weakly compressed systems, and slightly supersonic solitary waves correspond to weakly nonlinear propagating pulses in strongly compressed systems. Third, explicit formulas for the physical height, width, impulse, and energy of the solitary waves are obtained in both the strongly nonlinear regime and the weakly nonlinear regime. Asymptotic expansions are used to show that in the strongly nonlinear regime, solitary waves are well approximated by Nesterenko's compacton (having the same wave speed), while in the weakly nonlinear regime, solitary waves coincide with solitons of the Korteweg--de Vries equation, with the same wave speed. All of these results are illustrated by means of exact solitary wave solutions, including the physically important case that models a chain of spherical particles.