The CharacteristicQ command computes the characteristic of a point symmetry represented by its infinitesimals or the corresponding infinitesimal generator operator. That is, for a PDE problem with n independent and m dependent variables, given a related list of infinitesimals $\left[{\mathrm{\xi }}_1\,\mathrm{...}\,{\mathrm{\xi }}_n\,{\mathrm{\eta }}_1\,\mathrm{...}\,{\mathrm{\eta }}_m\right]$, CharacteristicQ computes the procedure

where $m$ identifies a dependent variable.

The sum in the body of this operator returned by CharacteristicQ is not expanded unless explicitly requested using the optional argument expanded. Also, jetnotation is used in this operator and a check-of-type for the value of $m$ is automatically inserted unless explicitly requested otherwise with the optional arguments jetnotation = false and/or checktype = false - see the examples below.

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

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

$\left[\mathrm{CharacteristicQ}\,\mathrm{InfinitesimalGenerator}\right]$

$F≔\left[u\,v\right]\left(x\,t\right)$

$F≔\left[u\left(x\,t\right)\,v\left(x\,t\right)\right]$

$S≔\left[\mathrm{seq}\left(\mathrm{\xi }\left[j\right]\left(x\,t\,u\,v\right)\,j=\left[x\,t\right]\right)\,\mathrm{seq}\left(\mathrm{\eta }\left[j\right]\left(x\,t\,u\,v\right)\,j=\left[u\,v\right]\right)\right]$

$S≔\left[{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\,{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\,{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\,{\mathrm{\eta }}_v\left(x\,t\,u\,v\right)\right]$

$Q≔\mathrm{CharacteristicQ}\left(S\,F\right)$

$Q≔m::\mathrm{satisfies}\left(m\mapsto m::{\mathbb{Z}}^{+}\mathbf{and}m\le 2\right)\mapsto {\mathrm{\eta }}_m-\mathrm{add}\left({\mathrm{\xi }}_j\cdot \mathrm{diff}\left(y_m\,X_j\right)\,j=1..2\right)$

$Q\left(1\right)$

${\mathrm{\eta }}_u\left(x\,t\,u\,v\right)-{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)u_x-{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)u_t$

$Q\left(2\right)$

${\mathrm{\eta }}_v\left(x\,t\,u\,v\right)-{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)v_x-{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)v_t$

$\mathrm{CharacteristicQ}\left(S\,F\,\mathrm{expanded}\,\mathrm{checktype}=\mathrm{false}\right)$

$m\mapsto {\mathrm{\eta }}_m-{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\cdot \mathrm{diff}\left(y_m\,x\right)-{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\cdot \mathrm{diff}\left(y_m\,t\right)$

$G≔\mathrm{InfinitesimalGenerator}\left(S\,F\right)$

$G≔f\→{\mathrm{\ξ}}_x\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial x}f\right)\+{\mathrm{\ξ}}_t\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial t}f\right)\+{\mathrm{\η}}_u\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial u}f\right)\+{\mathrm{\η}}_v\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial v}f\right)$

$\mathrm{CharacteristicQ}\left(G\,F\right)$

$m::\mathrm{satisfies}\left(m\mapsto m::{\mathbb{Z}}^{+}\mathbf{and}m\le 2\right)\mapsto {\mathrm{\eta }}_m-\mathrm{add}\left({\mathrm{\xi }}_j\cdot \mathrm{diff}\left(y_m\,X_j\right)\,j=1..2\right)$

$\mathrm{Qf}≔\mathrm{CharacteristicQ}\left(S\,F\,\mathrm{expanded}\,\mathrm{checktype}=\mathrm{false}\,\mathrm{jetnotation}=\mathrm{false}\right)$

$\mathrm{Qf}≔m\mapsto {\mathrm{\eta }}_m-{\mathrm{\xi }}_x\left(x\,t\,u\left(x\,t\right)\,v\left(x\,t\right)\right)\cdot \mathrm{diff}\left(y_m\,x\right)-{\mathrm{\xi }}_t\left(x\,t\,u\left(x\,t\right)\,v\left(x\,t\right)\right)\cdot \mathrm{diff}\left(y_m\,t\right)$

$\mathrm{Qf}\left(1\right)$

${\mathrm{\eta }}_u\left(x\,t\,u\left(x\,t\right)\,v\left(x\,t\right)\right)-{\mathrm{\xi }}_x\left(x\,t\,u\left(x\,t\right)\,v\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)-{\mathrm{\xi }}_t\left(x\,t\,u\left(x\,t\right)\,v\left(x\,t\right)\right)\left(\frac{\partial }{\partial t}u\left(x\,t\right)\right)$