Symmetry in Self-Similarity in Space and Time---Short Time Transients and Power-Law Spatial~Asymptote

TL;DR

This paper analyzes the symmetry properties of self-similar solutions in space and time, focusing on early-stage behaviors and algebraically decaying scaling functions, especially in diffusion processes, revealing internal structures and examples of power-law tails.

Contribution

It provides a symmetry analysis of early-stage self-similarity with algebraic decay, linking polynomial and rapidly decaying functions, and offers new examples of early-stage self-similar solutions.

Findings

Rapidly decaying functions are generated by polynomial scaling functions.

Early-stage self-similarity can have power-law tails.

The analysis reveals internal structures of self-similar solutions.

Abstract

The self-similarity in space and time (hereafter self-similarity), either deterministic or statistical, is characterized by similarity exponents and a function of scaled variable, called the scaling function. In the present paper, we address mainly the self-similarity in the limit of early stage, as~opposed to the latter one, and also consider the scaling functions that decay or grow algebraically, as~opposed to the rapidly decaying functions such as Gaussian or error function. In particular, in~the case of simple diffusion, our symmetry analysis shows a mathematical mechanism by which the rapidly decaying scaling functions are generated by other polynomial scaling functions. While~the former is adapted to the self-similarity in the late-stage processes, the latter is adapted to the early stages. This paper sheds some light on the internal structure of the family of self-similarities generated by a simple diffusion equation. Then, we present an example of self-similarity for the late stage whose scaling function has power-law tail, and also several cases of self-similarity for the early stages. These examples show the utility of self-similarity to a wider range of phenomena other than the late stage behaviors with rapidly decaying scaling function