Given a list with the infinitesimals S of a generator of symmetry transformations leaving invariant a PDE system (PDESYS), or the corresponding infinitesimal generator differential operator, the SimilarityTransformation command computes a transformation that reduces by one the number of independent variables of PDESYS. The output consists of a sequence of two sets respectively containing the transformation and inverse transformation equations.

These similarity transformations are special cases of group invariant transformations able to reduce the number of independent variables by many in one go, computed with the InvariantTransformation command.

The process of computing similarity transformations implies computing the invariants associated to the given infinitesimals. The typical formulation of these transformations in textbooks, however, sometimes avoids those wordings and instead presents these transformations as the introduction of variables $\mathrm{\psi }$ and ${\mathrm{\phi }}_k$, where $k=n+m-1$ and $n$ and $m$ are, respectively, the number of independent and dependent variables of the problem, such that the infinitesimals of the symmetry generator used to construct the transformation assume the form $\left[\mathrm{...}{\mathrm{\xi }}_r=1\,\mathrm{...}\,{\mathrm{\eta }}_1=0\,\mathrm{...}\,{\mathrm{\eta }}_m=0\right]$, where $0\le r$ $\le n$. Hence, by applying this similarity transformation to any PDE invariant under S you obtain a PDE not depending on the rth independent variable to which corresponds ${\mathrm{\xi }}_r=1$ - see the examples below.

When there is only one dependent variable, DepVars and NewVars can be a function; otherwise they must be a list of functions representing dependent variables. If NewVars are not given, SimilarityTransformation will generate a list of globals to represent them.

You can optionally specify a simplifier, to be used instead of the default which is simplify/size, as well as requesting the output to be in jet notation by respectively using the optional arguments simplifier = ... and jetnotation. Note that the option simplifier = ... can be used not just to "simplify" the output but also to post-process this output in the way you want, for instance using a procedure written by you, to discard, change or do what you find necessary with the transformation.

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{SimilarityTransformation}\,\mathrm{ChangeSymmetry}\,\mathrm{InfinitesimalGenerator}\right)$

$\left[\mathrm{SimilarityTransformation}\,\mathrm{ChangeSymmetry}\,\mathrm{InfinitesimalGenerator}\right]$

$S≔\left[\mathrm{\_\ξ}\left[x\right]=x\,\mathrm{\_\ξ}\left[t\right]=1\,\mathrm{\_\η}\left[u\right]=u\right]$

$S≔\left[{\mathrm{\_\ξ}}_x=x\,{\mathrm{\_\ξ}}_t=1\,{\mathrm{\_\η}}_u=u\right]$

$G≔\mathrm{InfinitesimalGenerator}\left(S\,u\left(x\,t\right)\right)$

$G≔f\→x\left(\frac{\partial }{\partial x}f\right)\+\frac{\partial }{\partial t}f\+u\left(\frac{\partial }{\partial u}f\right)$

$\mathrm{ITR},\mathrm{TR}≔\mathrm{SimilarityTransformation}\left(S\,u\left(x\,t\right)\,v\left(r\,s\right)\right)$

$\mathrm{ITR},\mathrm{TR}≔\left\{r=-\ln \left(x\right)+t\,s=\ln \left(x\right)\,v\left(r\right)=\frac{u\left(x\,t\right)}{x}\right\},\left\{t=r+s\,x={\ⅇ}^s\,u\left(x\,t\right)=v\left(r\right){\ⅇ}^s\right\}$

$\mathrm{SimilarityTransformation}\left(S\,u\left(x\,t\right)\,v\left(r\,s\right)\,\mathrm{jetnotation}\right)$

$\left\{r=-\ln \left(x\right)+t\,s=\ln \left(x\right)\,v=\frac{u}{x}\right\},\left\{t=r+s\,u=v{\ⅇ}^s\,x={\ⅇ}^s\right\}$

$\mathrm{SimilarityTransformation}\left(S\,u\left(x\,t\right)\,v\left(r\,s\right)\,\mathrm{jetnotation}=\mathrm{jetnumbers}\right)$

$\mathrm{jet\; notation:\; [1\; =\; x,\; 2\; =\; t,\; 3\; =\; r]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left\{r=-\ln \left(x\right)+t\,s=\ln \left(x\right)\,v\left[\right]=\frac{u\left[\right]}{x}\right\},\left\{t=r+s\,x={\ⅇ}^s\,u\left[\right]=v\left[\right]{\ⅇ}^s\right\}$

$\mathrm{NewVars}≔\mathrm{map}\left(\mathrm{lhs}\,\mathrm{ITR}\right)$

$\mathrm{NewVars}≔\left\{r\,s\,v\left(r\right)\right\}$

$\mathrm{ChangeSymmetry}\left(\mathrm{TR}\,S\,u\left(x\,t\right)\,\mathrm{NewVars}\right)$

$\left[{\mathrm{\_\ξ}}_r=0\,{\mathrm{\_\ξ}}_s=1\,{\mathrm{\_\η}}_v=0\right]$

$\mathrm{InfinitesimalGenerator}\left(\,v\left(r\,s\right)\right)$

$f\→\frac{\partial }{\partial s}f$

$\mathrm{SimilarityTransformation}\left(S\,u\left(x\,t\right)\right)$

$\left\{\mathrm{\_\ψ}=\ln \left(x\right)\,{\mathrm{\_\φ}}_1=-\ln \left(x\right)+t\,{\mathrm{\_\φ}}_2\left({\mathrm{\_\φ}}_1\right)=\frac{u\left(x\,t\right)}{x}\right\},\left\{t={\mathrm{\_\φ}}_1+\mathrm{\_\ψ}\,x={\ⅇ}^{\mathrm{\_\ψ}}\,u\left(x\,t\right)={\mathrm{\_\φ}}_2\left({\mathrm{\_\φ}}_1\right){\ⅇ}^{\mathrm{\_\ψ}}\right\}$