Self similar functions

Question

I'm trying to undestand the self-similarity as an invariance of a function under certain transformation. For example I think $$f(\lambda x)=\lambda^\epsilon f(x)$$ could be understood as a self-similarity like follows: a stretch of the $x$ variable is equivalent to a scale factor $\lambda^\epsilon$ over the whole profile $f(x)$. However it is not clear to me, when multiple variable are involved, if self-similarity could "couple" them together. For example $$u(x,t)=At^{-\alpha}f(\xi) \qquad \xi=xt^{-\beta}$$ is a self-similar function? In this case it is not true $$u(\lambda x,t)=\lambda^\gamma u(x,t)$$ nor $$u(x,\lambda t)=\lambda^\delta u(x,t)$$ but $$u(x,\lambda t)=\lambda^{-\beta}u(\frac{x}{\lambda^\beta},t)$$ still holds. So a stretch over the time is equivalent to a stretch of the profile AND of the $x$ variable.

Is there a general rule to check if a function is self similar? I looked up online but nothing sufficiently clear.

Answer 1

Continuing Dave's answer (and using his notation), a practical way to check for this kind of self-similarity is to draw a log-log plot of $u(x,t)$ as a function of $x$ for different values of $t$. If $u$ is self-similar, the different curves would all all have the same shape and would be related by translations.

To see this, define $X \equiv \log x$ and $T \equiv \log t$. With these variables, $$ u(x,t) = u(e^X, e^T) = e^{-\alpha T} f(e^{T-X/\beta}) \equiv e^{F(X-\beta T)-\alpha T}, $$ where the last equality defines the function $F$. Therefore, $$ U(X,T) \equiv \log u(e^X,e^T) = F(X-\beta T) - \alpha T. $$ This way you can in fact identify more general scaling functions of the form $u(x,t) = a(t) f\big(x b(t)\big)$ for functions $a(t)$ and $b(t)$ that are not necessarily power-laws.

Answer 2

What's going on is that the definition of the scaling variable $\xi=x^{-1/\beta} t$ (note: I prefer to parameterize it this way) defines how to resale the the two variables in a way that generates a scale transformation:

$$ u(\lambda^\beta x, \lambda t) = A \lambda^{-\alpha} t^{-\alpha} f( x^{-1/\beta} t ) = \lambda^{-\alpha} u(x,t) $$

by using different, but related, scale factors for $x,t$ they drop out of the $f(.)$ factor. One way to think of this is that you have a set of "families" of self similar systems, one for each $\xi_0$ value, and in terms of the $x,t$ domain, they lie along particular curves of the form $t = \xi_0 x^{1/\beta}$.

If you were just given a two parameter function function (rather than assuming that this form holds), the process for checking would be to first identify $\xi=x^{-1/\beta} t$, re-express the equation in terms of it, and see if everything can be expressed in terms of it, except for an overall $t^{-\alpha}$ factor.