Dispersion relations and wave operators in self-similar quasicontinuous linear chains

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 nonlocal particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasicontinuous linear chain of infinite length with a spatially self-similar distribution of nonlocal interparticle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function that 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.

Figure 1

(Color online) Dispersion relation ${\omega }^2\left(kh\right)$ in arbitrary units for $\delta =\text{log}4/\text{log}5$ of fractal dimension $D\approx 1.14$ and $N=5$.

Figure 2

(Color online) Dispersion relation ${\omega }^2\left(kh\right)$ in arbitrary units for $\delta =0.5$ of fractal dimension $D=1.5$ and $N=2$.

Figure 3

(Color online) Dispersion relation ${\omega }^2\left(kh\right)$ in arbitrary units for $\delta =0.25$ of fractal dimension $D=1.75$ and $N=1.5$.