# Symmetry in Self-Similarity in Space and Time—Short Time Transients and Power-Law Spatial Asymptotes

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## Abstract

**:**

## 1. Introduction

## 2. Symmetry Properties in the Family of Scaling Functions for the 1D Diffusion Process

#### 2.1. Scaling Functions

#### 2.2. Symmetry in the Family of Scaling Functions

## 3. Self-Similarity in Space and Time with Scaling Function Having Algebraic Tail

#### 3.1. Late-Stage Self-Similarity in Space and Time of Gel Network

#### 3.2. Self-Similarity in Space and Time in the Short-Time Limit of 1D Diffusion

#### 3.3. Short-Time Self-Similarity in Space and Time of Capillary-Driven Thin-Film Equation

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Family of Scaling Functions for the Self-Similarity Solution of 1D Diffusion Equation

p | ${\mathbf{\Phi}}_{\mathit{p},\mathit{e}}\left(\mathit{s}\right)$ | ${\mathbf{\Phi}}_{\mathit{p},\mathit{o}}\left(\mathit{s}\right)$ |
---|---|---|

−3 | ${\Phi}_{-3,e}\left(s\right)=\frac{{s}^{2}}{2}+1$ | ${\Phi}_{-3,o}\left(s\right)=\frac{\sqrt{\pi}}{2}(\frac{{s}^{2}}{2}+1)\text{}\mathrm{erf}\left(\frac{s}{2}\right)+\frac{s}{2}{e}^{-\frac{{s}^{2}}{4}}$ |

−2 | ${\Phi}_{-2,e}\left(s\right)=\frac{\sqrt{\pi}}{2}s\text{}\mathrm{erf}\left(\frac{s}{2}\right)+{e}^{-{s}^{2}/4}$ | ${\Phi}_{-2,o}\left(s\right)=s$ |

−1 | ${\Phi}_{-1,e}\left(s\right)=1$ | ${\Phi}_{-1,o}=\sqrt{\pi}\mathrm{erf}\left(\frac{s}{2}\right)$ |

0 | ${\Phi}_{0,e}\left(s\right)={e}^{-{s}^{2}/4}$ (*) | ${\Phi}_{0,o}\left(s\right)={e}^{-{s}^{2}/4}{\int}_{0}^{s}{e}^{{w}^{2}/4}dw$ |

1 | ${\Phi}_{1,e}\left(s\right)\propto {\Phi}_{0,o}^{\prime}\left(s\right)$ | ${\Phi}_{1,o}\left(s\right)\propto {\Phi}_{0,e}^{\prime}\left(s\right)$ (*) |

2 | ${\Phi}_{2,e}\left(s\right)\propto {\Phi}_{0,e}^{\u2033}\left(s\right)$ (*) | ${\Phi}_{2,o}\left(s\right)\propto {\Phi}_{0,o}^{\u2033}\left(s\right)$ |

**Figure A1.**Examples of the scaling functions satisfying Equation (2) with power-law asymptotes. ${\varphi}_{0,o}\left(s\right)={e}^{-{s}^{2}/4}{\int}_{0}^{s}{e}^{{w}^{2}/4}dw$ (solid curve), ${\varphi}_{1,e}\left(s\right)={\varphi}_{0,o}^{\prime}\left(s\right)$ (coarse dashed curve), and ${\varphi}_{2,o}\left(s\right)={\varphi}_{0,o}^{\u2033}\left(s\right)$ (fine dashed curve). For $|s|\gg 1$, these functions decays as $\sim {|s|}^{-(p+1)}$ with $p=0,1,$ and 2, respectively.

## Appendix B. Spacetime Self-Similarity Solution of Equation (7)

## References

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**Figure 1.**Diagram showing the relations among the solutions of ${\mathcal{L}}_{p}$ and those of ${\mathcal{L}}_{q}^{\u2020}.$ See the main text.

**Figure 2.**Radial displacement of the gel. The radial displacement, ${U}_{\mathrm{rad}}(r,t)={U}_{1}^{\left(r\right)}(r,t)\equiv {t}^{-1}{\Psi}_{1}(r/\sqrt{t}),$ is shown vs. r for the various time, $t=1$ (blue), 4 (magenta), and 9 (yellow). The vertical scale has been arbitrarily chosen and ${R}_{0}$ is supposed to be infinitesimal. For $r\gg \sqrt{t}$, the displacement, ${U}_{1}^{\left(r\right)}(r,t)$, retains a time-independent tail, $\sim {r}^{-2}$. While the finiteness of ${R}_{0}$ should modify the radial displacements at small scale, $r\sim {R}_{0}$, it does not influence the fate of the algebraic tail.

**Figure 3.**(

**a**,

**b**) Initial conditions (solid curves) as well as the short-time evolution at $t=0.02$ and $t=0.04$ (dashed curves in (

**a**)) and at $t=0.005,0.01,$ and $0.02$ (dashed curves in (

**b**)), respectively, are shown versus the unscaled variables. The initial condition has $u(0,0)=1$ and the support, $[-1,1].$ (

**c**,

**d**) The scaling function and the short-time evolution are represented using the scaled variables: (

**c**) replots the evolution in (

**a**) using $\left[u\right(x,t)-u(0,0\left)\right]/t$ and $s=x/\sqrt{t}$ and compares with ${\varphi}_{-3}\left(s\right)=2+{s}^{2},$ while (

**d**) replots the evolution in (

**b**) using $[u(x,t)-u(0,0)]/\sqrt{t}$ with s and compares with ${\varphi}_{-2}\left(s\right)={e}^{-{s}^{2}/4}+\left(\sqrt{\pi}s/2\right)\mathrm{erf}(s/2)$.

**Figure 4.**The early stage profile at $t=0.04$ calculated from the truncated parabolic initial condition (solid curve) is compared with the short-time self-similarity prediction (lower dashed curve) and with the late-stage one (upper dashed curve).

**Figure 5.**Short-time self-similarity evolution with algebraic asymptote of capillary-driven thin-film equation. From the initial profile $u(x,0)=1-|x|$ (this cannot be shown on this plane), the solution $u(x,t)$ for Equation (8) at $t=1$ (solid curve), $t={2}^{4}$ (long dashed curve) and $t={2}^{8}$ (short dashed curve) are presented using the scaled variables. The scaling function, which is numerically obtained by solving Equation (9) with $\beta =-2$, is almost indistinguishable with the earliest profile (solid curve).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sekimoto, K.; Fujita, T.
Symmetry in Self-Similarity in Space and Time—Short Time Transients and Power-Law Spatial Asymptotes. *Symmetry* **2019**, *11*, 1489.
https://doi.org/10.3390/sym11121489

**AMA Style**

Sekimoto K, Fujita T.
Symmetry in Self-Similarity in Space and Time—Short Time Transients and Power-Law Spatial Asymptotes. *Symmetry*. 2019; 11(12):1489.
https://doi.org/10.3390/sym11121489

**Chicago/Turabian Style**

Sekimoto, Ken, and Takahiko Fujita.
2019. "Symmetry in Self-Similarity in Space and Time—Short Time Transients and Power-Law Spatial Asymptotes" *Symmetry* 11, no. 12: 1489.
https://doi.org/10.3390/sym11121489