Dispersion Relations and Wave Operators in Self-Similar Quasi-Continuous Linear Chains

TL;DR

This paper develops a self-similar wave operator for a quasi-continuous chain, revealing fractal dispersion relations and connecting self-similarity with fractional calculus in wave dynamics.

Contribution

It introduces a self-similar Laplacian and wave operator based on non-local interactions, linking self-similarity with fractal dispersion and fractional integrals.

Findings

Dispersion relation exhibits fractal, self-similar features.

Self-similar Laplacian relates to fractional integrals.

Power law frequency dependence in low-frequency regime.

Abstract

We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of non-local particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We also derive a continuum approximation which relates the self-similar Laplacian to fractional integrals and yields in the low-frequency regime a power law frequency-dependence of the oscillator density.