TL;DR
This paper investigates chains with long-range interactions characterized by fractal dispersion laws, deriving pseudodifferential equations and describing their dispersion relations using Weierstrass functions, revealing complex oscillatory behaviors.
Contribution
It introduces a novel class of long-range interacting chains with fractal dispersion laws and derives their governing pseudodifferential equations.
Findings
Dispersion laws are described by Weierstrass and Weierstrass-Mandelbrot functions.
The interactions are defined via exponential functions a(m)=b^m, b=2,3,...
Pseudodifferential equations for chain oscillations are obtained.
Abstract
Chains with long-range interactions are considered. The interactions are defined such that each nth particle interacts only with chain particles with the numbers n+a(m) and n-a(m), where m=1,2,3,... and a(m) is an integer-valued function. Exponential type functions a(m)=b^m, where b=2,3,.., are discussed. The correspondent pseudodifferential equations of chain oscillations are obtained. Dispersion laws of the suggested chains are described by the Weierstrass and Weierstrass-Mandelbrot functions.
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