Given a list with the infinitesimals of a symmetry generator, or the corresponding infinitesimal generator operator, or a list of n of these lists or operators possibly representing an n-dimensional symmetry group, CharacteristicQInvariants returns a sequence of mathematical expressions ${\mathrm{\Phi }}_k$ simultaneously invariant under all the $G_j$ symmetry generators ($1\le j\le n$) related to the given $n$ infinitesimals. That is, the ${\mathrm{\Phi }}_k$ are functions of the independent and dependent variables of the problem (DepVars), not involving their partial derivatives, and satisfy $G_j\left({\mathrm{\Phi }}_k\right)=0$ for all $j$ and $k$, where the number $k$ depends on the problem. The objects returned are thus differential invariants of order zero.

The same objects can be computed also with the Invariants command of PDEtools. Their computation using CharacteristicQInvariants however is simpler: the invariants are computed directly from the CharacteristicQ associated to each symmetry, thus requiring solving PDE systems where each PDE involved depends on one variable less than the ones used by Invariants, obtained directly from the InfinitesimalGenerator.

Note: In the multidimensional case, depending on the infinitesimals given, the problem may have no solution.

Optionally, a simplifier can be specified to be used instead of the default which is simplify/size. For that purpose, use the optional argument simplifier = .... By default, the output of CharacteristicQInvariants is in jet notation. You can change this by passing the optional argument jetnotation = false.

The invariants returned by CharacteristicQInvariants can also be used as optional arguments for the group InvariantSolutions command when computing solutions to PDE systems.

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.

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

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

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

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

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

$\mathrm{\Phi }≔\mathrm{CharacteristicQInvariants}\left(S\,u\left(x\,t\right)\right)$

$\mathrm{\Phi }≔-\ln \left(x\right)+t,\frac{u}{x}$

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

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

$\mathrm{normal}\left(G\left(F\left(\mathrm{\Phi }\right)\right)\right)$

$0$

$\mathrm{CharacteristicQInvariants}\left(S\,u\left(x\,t\right)\,\mathrm{jetnotation}=\mathrm{jetvariableswithbrackets}\right)$

$-\ln \left(x\right)+t,\frac{u\left[\right]}{x}$

$\mathrm{CharacteristicQInvariants}\left(S\,u\left(x\,t\right)\,\mathrm{jetnotation}=\mathrm{false}\right)$

$-\ln \left(x\right)+t,\frac{u\left(x\,t\right)}{x}$