The SymmetryGauge command receives a symmetry and returns it rewritten in its most general form, by introducing as many arbitrary functions as the number of independent variables. The dependent variables of the problem must be specified as a function or a list of functions.

When an optional indication of the desired "gauge" is given, as in $\mathrm{\xi }=\mathrm{...}$ or $\mathrm{\eta }=\mathrm{...}$, before returning, the symmetry is "gauged" to satisfy that indication by adjusting the values of the arbitrary functions of the most general form of the symmetry.

The symmetry can be passed as a list of symmetry infinitesimals, or as its equivalent infinitesimal generator operator. Accordingly, the output will be a list or an operator. To override this behavior use the optional argument output = ..., where the right-hand side can be list or operator.

When the input symmetry is passed as an infinitesimal generator operator, or the optional argument output = operator is used, the formulas in the body of the operator returned are not expanded (for details, see the discussion in InfinitesimalGenerator). To receive them in expanded form pass the optional argument expanded.

For a problem with $n$ independent and $m$ dependent variables, a list of infinitesimal components of a symmetry contains $n$ (the $\mathrm{\xi }$) components and $m$ (the $\mathrm{\eta }$) components. To specify the gauge of the symmetry returned by SymmetryGauge, only one at a time of the optional arguments $\mathrm{\xi }=\mathrm{...}$ or $\mathrm{\eta }=\mathrm{...}$, can be specified, and the right-hand side is expected as an algebraic expression or a list of them, representing the values of the $\mathrm{\xi }$ or $\mathrm{\eta }$ components to appear in the returned symmetry.

When the right-hand side of $\mathrm{\xi }=\mathrm{...}$ or $\mathrm{\eta }=\mathrm{...}$ is an algebraic expression, all the corresponding infinitesimals, that is $n$ (for $\mathrm{\xi }=\left[\mathrm{...}\right]$), or $m$ (for $\mathrm{\eta }=\left[\mathrm{...}\right]$), where $n$ and $m$ are, respectively, the number of independent and dependent variables, are returned with that value. When the right-hand side is a list ($\mathrm{\xi }=\left[\mathrm{...}\right]$ or $\mathrm{\eta }=\left[\mathrm{...}\right]$), if one or many of the elements are the word arbitrary, then the corresponding infinitesimal will be returned in the most general form. Also, when the right-hand side is a list, there cannot be more than $n$ (for $\mathrm{\xi }=\left[\mathrm{...}\right]$) or $m$ (for $\mathrm{\eta }=\left[\mathrm{...}\right]$) elements in the list. When the lists are passed with less than $n$ or $m$ elements, the remaining ones are assumed to be arbitrary in that they will appear in the returned symmetry in its most general form, in terms of arbitrary functions.

The prolongation value of the returned symmetry is the prolongation value of the input symmetry unless specified otherwise with the option prolongation = k, where k is a non-negative integer.

The dependency of the arbitrary functions introduced by SymmetryGauge consists of the independent and dependent variables, as well as all their partial derivatives up to order $k$, where $k$ is the prolongation, unless otherwise indicated using the optional argument differentialorder = N, here N is a non-negative integer.

The jet notation used in the output is the one of the input symmetry S unless indicated otherwise using the option jetnotation = ... where the right-hand side is any of jetnumbers', jetODE, jetvariables or jetvariableswithbrackets; for details about the available jet notations see ToJet.

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.

$\mathrm{with}\left(\mathrm{PDEtools}\right)\:$

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

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

$\mathrm{PDE}≔\mathrm{InvariantEquation}\left(S\,u\left(x\,t\right)\,\mathrm{order}=2\,\mathrm{name}=\mathrm{\Lambda }\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}name\text{'}\; against\; keyword\; \text{'}arbitraryfunctionname\text{'}}$

$\mathrm{PDE}≔\mathrm{\Lambda }\left(\frac{\partial }{\partial t}u\left(x\,t\right)\,t{\ⅇ}^{-x}\,u\left(x\,t\right){\ⅇ}^{-x}\,\frac{{\partial }^2}{\partial t\partial x}u\left(x\,t\right)\,\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right){\ⅇ}^{-x}\,\left(\frac{{\partial }^2}{\partial t^2}u\left(x\,t\right)\right){\ⅇ}^x\,\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right){\ⅇ}^{-x}\right)$

$\mathrm{pde}≔\mathrm{isolate}\left(\mathrm{op}\left(-1\,\mathrm{PDE}\right)=\mathrm{subsop}\left(-1=\mathrm{NULL}\,\mathrm{PDE}\right)\,\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\right)\right)$

$\mathrm{pde}≔\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)=\frac{\mathrm{\Lambda }\left(\frac{\partial }{\partial t}u\left(x\,t\right)\,t{\ⅇ}^{-x}\,u\left(x\,t\right){\ⅇ}^{-x}\,\frac{{\partial }^2}{\partial t\partial x}u\left(x\,t\right)\,\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right){\ⅇ}^{-x}\,\left(\frac{{\partial }^2}{\partial t^2}u\left(x\,t\right)\right){\ⅇ}^x\right)}{{\ⅇ}^{-x}}$

$\mathrm{S1}≔\mathrm{SymmetryGauge}\left(S\,u\left(x\,t\right)\right)$

$\mathrm{S1}≔\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_1}\left(x\,t\,u\right)+1\,{\mathrm{\_\ξ}}_t=\mathrm{f\_\_2}\left(x\,t\,u\right)+t\,{\mathrm{\_\η}}_u=\mathrm{f\_\_1}\left(x\,t\,u\right)u_x+\mathrm{f\_\_2}\left(x\,t\,u\right)u_t+u\right]$

$\mathrm{SymmetryTest}\left(\mathrm{S1}\,\mathrm{pde}\right)$

$\left\{0\right\}$

$\mathrm{PDEtools}:-\mathrm{Library}:-\mathrm{ConstructTrivialSymmetry}\left(\left[\mathrm{\_F1}\,\mathrm{\_F2}\right]\,\left[u\right]\,\left[x\,t\right]\,\mathrm{jetnotation}=\mathrm{jetvariables}\right)$

$\left[\mathrm{f\_\_1}\left(x\,t\,u\right)\,\mathrm{f\_\_2}\left(x\,t\,u\right)\,\mathrm{f\_\_1}\left(x\,t\,u\right)u_x+\mathrm{f\_\_2}\left(x\,t\,u\right)u_t\right]$

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

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

$\mathrm{SymmetryGauge}\left(G\,u\left(x\,t\right)\right)$

$f\→\mathrm{add}\left({\mathrm{\ξ}}_{x_j}\left(\frac{\partial }{\partial x_j}f\right)\,j\=1..2\right)\+\mathrm{add}\left({\mathrm{\η}}_{u_m}\left(\frac{\partial }{\partial u_m}f\right)\,m\=1..1\right)$

$\mathrm{SymmetryGauge}\left(G\,u\left(x\,t\right)\,\mathrm{expanded}\right)$

$f\→\left(\mathrm{\_F1}\left(x\,t\,u\right)\+1\right)\left(\frac{\partial }{\partial x}f\right)\+\left(\mathrm{\_F2}\left(x\,t\,u\right)\+t\right)\left(\frac{\partial }{\partial t}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u\right)u_x\+\mathrm{\_F2}\left(x\,t\,u\right)u_t\+u\right)\left(\frac{\partial }{\partial u}f\right)$

$\mathrm{SymmetryGauge}\left(G\,u\left(x\,t\right)\,\mathrm{output}=\mathrm{list}\right)$

$\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_1}\left(x\,t\,u\right)+1\,{\mathrm{\_\ξ}}_t=\mathrm{f\_\_2}\left(x\,t\,u\right)+t\,{\mathrm{\_\η}}_u=\mathrm{f\_\_1}\left(x\,t\,u\right)u_x+\mathrm{f\_\_2}\left(x\,t\,u\right)u_t+u\right]$

$\mathrm{SymmetryGauge}\left(G\,u\left(x\,t\right)\,\mathrm{expanded}\,\mathrm{prolongation}=1\right)$

$f\→\left(\mathrm{\_F1}\left(x\,t\,u\,u_x\,u_t\right)\+1\right)\left(\frac{\partial }{\partial x}f\right)\+\left(\mathrm{\_F2}\left(x\,t\,u\,u_x\,u_t\right)\+t\right)\left(\frac{\partial }{\partial t}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u\,u_x\,u_t\right)u_x\+\mathrm{\_F2}\left(x\,t\,u\,u_x\,u_t\right)u_t\+u\right)\left(\frac{\partial }{\partial u}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u\,u_x\,u_t\right)u_{x\,x}\+\mathrm{\_F2}\left(x\,t\,u\,u_x\,u_t\right)u_{x\,t}\+u_x\right)\left(\frac{\partial }{\partial u_x}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u\,u_x\,u_t\right)u_{x\,t}\+\mathrm{\_F2}\left(x\,t\,u\,u_x\,u_t\right)u_{t\,t}\right)\left(\frac{\partial }{\partial u_t}f\right)$

$\mathrm{SymmetryGauge}\left(G\,u\left(x\,t\right)\,\mathrm{expanded}\,\mathrm{prolongation}=1\,\mathrm{notation}=\mathrm{jetnumbers}\right)$

$f\→\left(\mathrm{\_F1}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)\+1\right)\left(\frac{\partial }{\partial x}f\right)\+\left(\mathrm{\_F2}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)\+t\right)\left(\frac{\partial }{\partial t}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)u_1\+\mathrm{\_F2}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)u_2\+u_{\[\]}\right)\left(\frac{\partial }{\partial u_{\[\]}}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)u_{1\,1}\+\mathrm{\_F2}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)u_{1\,2}\+u_1\right)\left(\frac{\partial }{\partial u_1}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)u_{1\,2}\+\mathrm{\_F2}\left(x\,t\,u_{\[\]}\,u_1\,u_2\right)u_{2\,2}\right)\left(\frac{\partial }{\partial u_2}f\right)$

$\mathrm{SymmetryGauge}\left(G\,u\left(x\,t\right)\,\mathrm{expanded}\,\mathrm{prolongation}=1\,\mathrm{order}=0\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}order\text{'}\; against\; keyword\; \text{'}differentialorder\text{'}}$

$f\→\left(\mathrm{\_F1}\left(x\,t\,u\right)\+1\right)\left(\frac{\partial }{\partial x}f\right)\+\left(\mathrm{\_F2}\left(x\,t\,u\right)\+t\right)\left(\frac{\partial }{\partial t}f\right)\+\left(\mathrm{\_F1}\left(x\,t\,u\right)u_x\+\mathrm{\_F2}\left(x\,t\,u\right)u_t\+u\right)\left(\frac{\partial }{\partial u}f\right)\+\left(u_{x\,x}\mathrm{\_F1}\left(x\,t\,u\right)\+u_{x\,t}\mathrm{\_F2}\left(x\,t\,u\right)\+u_x\right)\left(\frac{\partial }{\partial u_x}f\right)\+\left(u_{x\,t}\mathrm{\_F1}\left(x\,t\,u\right)\+u_{t\,t}\mathrm{\_F2}\left(x\,t\,u\right)\right)\left(\frac{\partial }{\partial u_t}f\right)$

$\mathrm{S0\_xi}≔\mathrm{SymmetryGauge}\left(S\,u\left(x\,t\right)\,\mathrm{\xi }=0\right)$

$\mathrm{S0\_xi}≔\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=-t{u}_t+u-u_x\right]$

$\mathrm{SymmetryTest}\left(\mathrm{S0\_xi}\,\mathrm{pde}\right)$

$\left\{0\right\}$

$\mathrm{S0\_eta}≔\mathrm{SymmetryGauge}\left(S\,u\left(x\,t\right)\,\mathrm{\eta }=0\right)$

$\mathrm{S0\_eta}≔\left[{\mathrm{\_\ξ}}_x=-\frac{\mathrm{f\_\_2}\left(x\,t\,u\right)u_t+u-u_x}{u_x}\,{\mathrm{\_\ξ}}_t=\mathrm{f\_\_2}\left(x\,t\,u\right)+t\,{\mathrm{\_\η}}_u=0\right]$

$\mathrm{SymmetryTest}\left(\mathrm{S0\_eta}\,\mathrm{pde}\right)$

$\left\{0\right\}$

$\mathrm{SymmetryGauge}\left(S\,u\left(x\,t\right)\,\mathrm{\xi }=\left[0\,1\right]\right)$

$\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_t=1\,{\mathrm{\_\η}}_u=-u_x+\left(-t+1\right)u_t+u\right]$

$\mathrm{SymmetryGauge}\left(S\,u\left(x\,t\right)\,\mathrm{\xi }=\left[0\right]\right)$

$\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_t=\mathrm{f\_\_2}\left(x\,t\,u\right)+t\,{\mathrm{\_\η}}_u=\mathrm{f\_\_2}\left(x\,t\,u\right)u_t+u-u_x\right]$

$\mathrm{SymmetryGauge}\left(S\,u\left(x\,t\right)\,\mathrm{\xi }=\left[\mathrm{arbitrary}\,0\right]\right)$

$\left[{\mathrm{\_\ξ}}_x=\mathrm{f\_\_1}\left(x\,t\,u\right)+1\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=\mathrm{f\_\_1}\left(x\,t\,u\right)u_x-t{u}_t+u\right]$

The PDEtools[SymmetryGauge] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.