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.
Michelitsch
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TM
Institut Jean le Rond d'Alembert, CNRS UMR 7190, Université Pierre et Marie Curie, Paris 6, 4, Place Jussieu 75252 Paris Cedex 05, France. michel@lmm.jussieu.fr
Maugin
G A
GA
Nicolleau
F C G A
FC
Nowakowski
A F
AF
Derogar
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