Given a list with the infinitesimals of a symmetry generator, or the corresponding infinitesimal generator differential operator, SymmetryTransformation computes the actual finite form of that symmetry transformation.

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, SymmetryTransformation 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 that you have written to discard, change or do what you find necessary with the transformation.

In some cases, the Lie group parameter introduced by SymmetryTransformation appears embedded into a subexpression, for example as in ${\ⅇ}^{\mathrm{\_\ε}}$, and only appears through functions of that subexpression. To have these cases returned with $\mathrm{\_\ε}$ instead of - say - ${\ⅇ}^{\mathrm{\_\ε}}$, use the option redefinegroupparameter.

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.

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

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

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

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

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

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

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

$\left\{r=x{\ⅇ}^{\mathrm{\_\ε}}\,s=\mathrm{\_\ε}+t\,v\left(r\,s\right)={\ⅇ}^{\mathrm{\_\ε}}u\left(x\,t\right)\right\}$

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

$\left\{r=x{\ⅇ}^{\mathrm{\_\ε}}\,s=\mathrm{\_\ε}+t\,v={\ⅇ}^{\mathrm{\_\ε}}u\right\}$

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

$\left\{r=x{\ⅇ}^{\mathrm{\_\ε}}\,s=\mathrm{\_\ε}+t\,v\left[\right]={\ⅇ}^{\mathrm{\_\ε}}u\left[\right]\right\}$

$\mathrm{TR},\mathrm{NewVars}≔\mathrm{solve}\left(\,\left\{t\,x\,u\left(x\,t\right)\right\}\right),\mathrm{map}\left(\mathrm{lhs}\,\right)$

$\mathrm{TR},\mathrm{NewVars}≔\left\{t=s-\mathrm{\_\ε}\,x=\frac{r}{{\ⅇ}^{\mathrm{\_\ε}}}\,u\left(x\,t\right)=\frac{v\left(r\,s\right)}{{\ⅇ}^{\mathrm{\_\ε}}}\right\},\left\{r\,s\,v\left(r\,s\right)\right\}$

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

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

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

$f\→r\left(\frac{\partial }{\partial r}f\right)\+\frac{\partial }{\partial s}f\+v\left(\frac{\partial }{\partial v}f\right)$

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

$\left\{\mathrm{\_t1}=x{\ⅇ}^{\mathrm{\_\ε}}\,\mathrm{\_t2}=\mathrm{\_\ε}+t\,\mathrm{\_u1}\left(\mathrm{\_t1}\,\mathrm{\_t2}\right)={\ⅇ}^{\mathrm{\_\ε}}u\left(x\,t\right)\right\}$

$\mathrm{SymmetryTransformation}\left(\left[0\,0\,z\,0\,0\right]\,\left[u\left(x\,y\,z\,t\right)\right]\right)$

$\left\{\mathrm{\_t1}=x\,\mathrm{\_t2}=y\,\mathrm{\_t3}=z{\ⅇ}^{\mathrm{\_\ε}}\,\mathrm{\_t4}=t\,\mathrm{\_u1}\left(\mathrm{\_t1}\,\mathrm{\_t2}\,\mathrm{\_t3}\,\mathrm{\_t4}\right)=u\left(x\,y\,z\,t\right)\right\}$

$\mathrm{SymmetryTransformation}\left(\left[0\,0\,z\,0\,0\right]\,\left[u\left(x\,y\,z\,t\right)\right]\,'\mathrm{redefinegroupparameter}'\right)$

$\left\{\mathrm{\_t1}=x\,\mathrm{\_t2}=y\,\mathrm{\_t3}=z\mathrm{\_\ε}\,\mathrm{\_t4}=t\,\mathrm{\_u1}\left(\mathrm{\_t1}\,\mathrm{\_t2}\,\mathrm{\_t3}\,\mathrm{\_t4}\right)=u\left(x\,y\,z\,t\right)\right\}$