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PDEtools

 SymmetryTransformation
 computes the finite form of the (symmetry) transformation leaving invariant any PDE system admitting a given symmetry

 Calling Sequence SymmetryTransformation(S, DepVars, NewVars, 'options'='value')

Parameters

 S - a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator DepVars - a function or a list of functions indicating the dependent variables of the problem NewVars - optional - a function or a list of functions representing the new dependent variables jetnotation = ... - (optional) can be true (same as jetvariables), false (default), jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available simplifier = ... - optional - indicates the simplifier to be used instead of the default simplify/size redefinegroupparameter - optional - to simplify a subexpression that involves the Lie group parameter replacing it by another group parameter

Description

 • Given a list with the infinitesimals of a symmetry 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.

Examples

 > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{SymmetryTransformation},\mathrm{ChangeSymmetry},\mathrm{InfinitesimalGenerator}\right)$
 $\left[{\mathrm{SymmetryTransformation}}{,}{\mathrm{ChangeSymmetry}}{,}{\mathrm{InfinitesimalGenerator}}\right]$ (1)

Consider a PDE problem with two independent variables and one dependent variable, $u\left(x,t\right)$, and consider the list of infinitesimals of a symmetry group

 > $S≔\left[\mathrm{_ξ}\left[x\right]=x,\mathrm{_ξ}\left[t\right]=1,\mathrm{_η}\left[u\right]=u\right]$
 ${S}{≔}\left[{{\mathrm{_ξ}}}_{{x}}{=}{x}{,}{{\mathrm{_ξ}}}_{{t}}{=}{1}{,}{{\mathrm{_η}}}_{{u}}{=}{u}\right]$ (2)

In the input above you can also enter the symmetry $S$ without infinitesimals' labels, as in $\left[x,1,u\right]$. The corresponding infinitesimal generator is

 > $G≔\mathrm{InfinitesimalGenerator}\left(S,u\left(x,t\right)\right)$
 ${G}{≔}{f}{→}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){+}\frac{{\partial }}{{\partial }{t}}{}{f}{+}{u}{}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right)$ (3)

A $\mathrm{PDESYS}$ is invariant under the symmetry transformation generated by $G$ in that $G\left(\mathrm{PDESYS}\right)=0$, where, in this formula, $G$ represents the prolongation necessary to act on $\mathrm{PDESYS}$ (see InfinitesimalGenerator).

The actual form of this finite, one-parameter, symmetry transformation relating the original variables $\left\{t,x,u\left(x,t\right)\right\}$ to new variables, $\left\{r,s,v\left(r,s\right)\right\}$, that leaves invariant any PDE system admitting the symmetry represented by $G$ above is obtained via

 > $\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right),v\left(r,s\right)\right)$
 $\left\{{r}{=}{x}{}{{ⅇ}}^{{\mathrm{_ε}}}{,}{s}{=}{\mathrm{_ε}}{+}{t}{,}{v}{}\left({r}{,}{s}\right){=}{{ⅇ}}^{{\mathrm{_ε}}}{}{u}{}\left({x}{,}{t}\right)\right\}$ (4)

where $\mathrm{_ε}$ is a (Lie group) transformation parameter. To express this transformation using jetnotation use

 > $\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right),v\left(r,s\right),\mathrm{jetnotation}\right)$
 $\left\{{r}{=}{x}{}{{ⅇ}}^{{\mathrm{_ε}}}{,}{s}{=}{\mathrm{_ε}}{+}{t}{,}{v}{=}{{ⅇ}}^{{\mathrm{_ε}}}{}{u}\right\}$ (5)
 > $\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right),v\left(r,s\right),\mathrm{jetnotation}=\mathrm{jetnumbers}\right)$
 $\left\{{r}{=}{x}{}{{ⅇ}}^{{\mathrm{_ε}}}{,}{s}{=}{\mathrm{_ε}}{+}{t}{,}{v}\left[\right]{=}{{ⅇ}}^{{\mathrm{_ε}}}{}{u}\left[\right]\right\}$ (6)

That this transformation leaves invariant any PDE system invariant under $G$ above is visible in the fact that it also leaves invariant the infinitesimals $S$; to verify this you can use ChangeSymmetry

 > $\mathrm{TR},\mathrm{NewVars}≔\mathrm{solve}\left(,\left\{t,x,u\left(x,t\right)\right\}\right),\mathrm{map}\left(\mathrm{lhs},\right)$
 ${\mathrm{TR}}{,}{\mathrm{NewVars}}{≔}\left\{{t}{=}{s}{-}{\mathrm{_ε}}{,}{x}{=}\frac{{r}}{{{ⅇ}}^{{\mathrm{_ε}}}}{,}{u}{}\left({x}{,}{t}\right){=}\frac{{v}{}\left({r}{,}{s}\right)}{{{ⅇ}}^{{\mathrm{_ε}}}}\right\}{,}\left\{{r}{,}{s}{,}{v}{}\left({r}{,}{s}\right)\right\}$ (7)
 > $\mathrm{ChangeSymmetry}\left(\mathrm{TR},S,u\left(x,t\right),\mathrm{NewVars}\right)$
 $\left[{{\mathrm{_ξ}}}_{{r}}{=}{r}{,}{{\mathrm{_ξ}}}_{{s}}{=}{1}{,}{{\mathrm{_η}}}_{{v}}{=}{v}\right]$ (8)

which is the same as $S$ (but written in terms of $v\left(r,s\right)$ instead of $u\left(x,t\right)$). So to this list of infinitesimals corresponds, written in terms of $v\left(r,s\right)$, this infinitesimal generator

 > $\mathrm{InfinitesimalGenerator}\left(,v\left(r,s\right)\right)$
 ${f}{→}{r}{}\left(\frac{{\partial }}{{\partial }{r}}{}{f}\right){+}\frac{{\partial }}{{\partial }{s}}{}{f}{+}{v}{}\left(\frac{{\partial }}{{\partial }{v}}{}{f}\right)$ (9)

which is also equal to $G$, only written in terms of $v\left(r,s\right)$.

If the new variables, $v\left(r,s\right)$,  are not indicated, variables prefixed by the underscore _ to represent the new variables are introduced

 > $\mathrm{SymmetryTransformation}\left(S,u\left(x,t\right)\right)$
 $\left\{{\mathrm{_t1}}{=}{x}{}{{ⅇ}}^{{\mathrm{_ε}}}{,}{\mathrm{_t2}}{=}{\mathrm{_ε}}{+}{t}{,}{\mathrm{_u1}}{}\left({\mathrm{_t1}}{,}{\mathrm{_t2}}\right){=}{{ⅇ}}^{{\mathrm{_ε}}}{}{u}{}\left({x}{,}{t}\right)\right\}$ (10)

An example where the Lie group parameter $\mathrm{_ε}$ appears only through the subexpression ${ⅇ}^{\mathrm{_ε}}$

 > $\mathrm{SymmetryTransformation}\left(\left[0,0,z,0,0\right],\left[u\left(x,y,z,t\right)\right]\right)$
 $\left\{{\mathrm{_t1}}{=}{x}{,}{\mathrm{_t2}}{=}{y}{,}{\mathrm{_t3}}{=}{z}{}{{ⅇ}}^{{\mathrm{_ε}}}{,}{\mathrm{_t4}}{=}{t}{,}{\mathrm{_u1}}{}\left({\mathrm{_t1}}{,}{\mathrm{_t2}}{,}{\mathrm{_t3}}{,}{\mathrm{_t4}}\right){=}{u}{}\left({x}{,}{y}{,}{z}{,}{t}\right)\right\}$ (11)

A symmetry transformation with the parameter redefined

 > $\mathrm{SymmetryTransformation}\left(\left[0,0,z,0,0\right],\left[u\left(x,y,z,t\right)\right],'\mathrm{redefinegroupparameter}'\right)$
 $\left\{{\mathrm{_t1}}{=}{x}{,}{\mathrm{_t2}}{=}{y}{,}{\mathrm{_t3}}{=}{z}{}{\mathrm{_ε}}{,}{\mathrm{_t4}}{=}{t}{,}{\mathrm{_u1}}{}\left({\mathrm{_t1}}{,}{\mathrm{_t2}}{,}{\mathrm{_t3}}{,}{\mathrm{_t4}}\right){=}{u}{}\left({x}{,}{y}{,}{z}{,}{t}\right)\right\}$ (12)

 See Also