Chains with Fractal Dispersion Law

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.

9 pages, LaTeX

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• Dispersion Relations and Wave Operators in Self-Similar Quasi-Continuous Linear Chains
• Lattice fractional Laplacian and its continuum limit kernel on the finite cyclic chain
• Fractional Diffusion Equations for Lattice and Continuum: Grunwald-Letnikov Differences and Derivatives Approach