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PDEtools

 SymmetryTest
 tests whether a given list of infinitesimals represents a symmetry of a given PDE system.

 Calling Sequence SymmetryTest(S, PDESYS, DepVars)

Parameters

 S - a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator PDESYS - a PDE or a set or list of them; it can include ODEs and not differential equations DepVars - optional - may be required; a function or a list of them indicating the dependent variables of the problem

Description

 • SymmetryTest tests whether a symmetry, given as a list of infinitesimals S or as the corresponding infinitesimal generator differential operator, is a symmetry of a given PDE system PDESYS; if so, S satisfies the determining PDE for PDESYS.
 • If DepVars is not specified, SymmetryTest will consider all the differentiated unknown functions in PDESYS as unknown of the problems.

Examples

Consider the wave equation in four dimensions to avoid redundant typing on input and on the display use diff_table and declare

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$
 > $U≔\mathrm{diff_table}\left(u\left(x,y,z,t\right)\right):$
 > $\mathrm{declare}\left(U\left[\right]\right)$
 ${u}{}\left({x}{,}{y}{,}{z}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{u}$ (1)
 > $\mathrm{pde}\left[1\right]≔U\left[x,x\right]+U\left[y,y\right]+U\left[z,z\right]-U\left[t,t\right]=0$
 ${{\mathrm{pde}}}_{{1}}{≔}{{u}}_{{x}{,}{x}}{+}{{u}}_{{y}{,}{y}}{+}{{u}}_{{z}{,}{z}}{-}{{u}}_{{t}{,}{t}}{=}{0}$ (2)

Compute the infinitesimals of point symmetry transformations leaving invariant pde[1] and test for correctness the first list

 > $\mathrm{declare}\left(\left(\mathrm{_ξ},\mathrm{_η}\right)\left(x,y,z,t,u\right)\right)$
 ${\mathrm{_xi}}{}\left({x}{,}{y}{,}{z}{,}{t}{,}{u}\right){}{\mathrm{will now be displayed as}}{}{\mathrm{_ξ}}$
 ${\mathrm{_eta}}{}\left({x}{,}{y}{,}{z}{,}{t}{,}{u}\right){}{\mathrm{will now be displayed as}}{}{\mathrm{_η}}$ (3)
 > $S≔\mathrm{Infinitesimals}\left(\mathrm{pde}\left[1\right]\right)$
 ${S}{≔}\left[{{\mathrm{_ξ}}}_{{x}}{=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{=}{1}{,}{{\mathrm{_ξ}}}_{{z}}{=}{0}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{=}{0}{,}{{\mathrm{_ξ}}}_{{z}}{=}{1}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{=}{0}{,}{{\mathrm{_ξ}}}_{{z}}{=}{0}{,}{{\mathrm{_ξ}}}_{{t}}{=}{1}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{1}{,}{{\mathrm{_ξ}}}_{{y}}{=}{0}{,}{{\mathrm{_ξ}}}_{{z}}{=}{0}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{=}{t}{,}{{\mathrm{_ξ}}}_{{z}}{=}{0}{,}{{\mathrm{_ξ}}}_{{t}}{=}{y}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{=}{0}{,}{{\mathrm{_ξ}}}_{{z}}{=}{t}{,}{{\mathrm{_ξ}}}_{{t}}{=}{z}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{t}{,}{{\mathrm{_ξ}}}_{{y}}{=}{0}{,}{{\mathrm{_ξ}}}_{{z}}{=}{0}{,}{{\mathrm{_ξ}}}_{{t}}{=}{x}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{x}{,}{{\mathrm{_ξ}}}_{{y}}{=}{y}{,}{{\mathrm{_ξ}}}_{{z}}{=}{z}{,}{{\mathrm{_ξ}}}_{{t}}{=}{t}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{=}{0}{,}{{\mathrm{_ξ}}}_{{z}}{=}{0}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{u}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{=}{z}{,}{{\mathrm{_ξ}}}_{{z}}{=}{-}{y}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{z}{,}{{\mathrm{_ξ}}}_{{y}}{=}{0}{,}{{\mathrm{_ξ}}}_{{z}}{=}{-}{x}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{y}{,}{{\mathrm{_ξ}}}_{{y}}{=}{-}{x}{,}{{\mathrm{_ξ}}}_{{z}}{=}{0}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{x}{}{z}{,}{{\mathrm{_ξ}}}_{{y}}{=}{z}{}{y}{,}{{\mathrm{_ξ}}}_{{z}}{=}\frac{{{z}}^{{2}}}{{2}}{+}\frac{{{t}}^{{2}}}{{2}}{-}\frac{{{x}}^{{2}}}{{2}}{-}\frac{{{y}}^{{2}}}{{2}}{,}{{\mathrm{_ξ}}}_{{t}}{=}{t}{}{z}{,}{{\mathrm{_η}}}_{{u}}{=}{-}{u}{}{z}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{x}{}{y}{,}{{\mathrm{_ξ}}}_{{y}}{=}\frac{{{y}}^{{2}}}{{2}}{+}\frac{{{t}}^{{2}}}{{2}}{-}\frac{{{x}}^{{2}}}{{2}}{-}\frac{{{z}}^{{2}}}{{2}}{,}{{\mathrm{_ξ}}}_{{z}}{=}{z}{}{y}{,}{{\mathrm{_ξ}}}_{{t}}{=}{t}{}{y}{,}{{\mathrm{_η}}}_{{u}}{=}{-}{u}{}{y}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{t}{}{x}{,}{{\mathrm{_ξ}}}_{{y}}{=}{t}{}{y}{,}{{\mathrm{_ξ}}}_{{z}}{=}{t}{}{z}{,}{{\mathrm{_ξ}}}_{{t}}{=}\frac{{{t}}^{{2}}}{{2}}{+}\frac{{{x}}^{{2}}}{{2}}{+}\frac{{{y}}^{{2}}}{{2}}{+}\frac{{{z}}^{{2}}}{{2}}{,}{{\mathrm{_η}}}_{{u}}{=}{-}{u}{}{t}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}\frac{{{z}}^{{2}}}{{2}}{-}\frac{{{x}}^{{2}}}{{2}}{-}\frac{{{t}}^{{2}}}{{2}}{+}\frac{{{y}}^{{2}}}{{2}}{,}{{\mathrm{_ξ}}}_{{y}}{=}{-}{x}{}{y}{,}{{\mathrm{_ξ}}}_{{z}}{=}{-}{x}{}{z}{,}{{\mathrm{_ξ}}}_{{t}}{=}{-}{t}{}{x}{,}{{\mathrm{_η}}}_{{u}}{=}{u}{}{x}\right]$ (4)
 > $\mathrm{SymmetryTest}\left(S\left[1\right],\mathrm{pde}\left[1\right]\right)$
 $\left\{{0}\right\}$ (5)

Test all the lists in one step

 > $\mathrm{map}\left(\mathrm{SymmetryTest},\left[S\right],\mathrm{pde}\left[1\right]\right)$
 $\left[\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}{,}\left\{{0}\right\}\right]$ (6)

SymmetryTest can also test dynamical symmetries, that is, symmetries where the infinitesimals depend on derivatives of the unknown functions of the problem. Consider for instance the sine-Gordon equation

 > $\mathrm{declare}\left(u\left(x,y\right)\right)$
 ${u}{}\left({x}{,}{y}\right){}{\mathrm{will now be displayed as}}{}{u}$ (7)
 > $\mathrm{SGE}≔\mathrm{diff}\left(u\left(x,y\right),x,y\right)=\mathrm{sin}\left(u\left(x,y\right)\right)$
 ${\mathrm{SGE}}{≔}{{u}}_{{x}{,}{y}}{=}{\mathrm{sin}}{}\left({u}\right)$ (8)

The following list of infinitesimals represent a symmetry of SGE

 > $S≔\left[\mathrm{_ξ}\left[1\right]=0,\mathrm{_ξ}\left[2\right]=0,\mathrm{_η}\left[1\right]=u\left[1,1,1\right]+\frac{1}{2}{u\left[1\right]}^{3}\right]$
 ${S}{≔}\left[{{\mathrm{_ξ}}}_{{1}}{=}{0}{,}{{\mathrm{_ξ}}}_{{2}}{=}{0}{,}{{\mathrm{_η}}}_{{1}}{=}{{u}}_{{1}{,}{1}{,}{1}}{+}\frac{{{u}}_{{1}}^{{3}}}{{2}}\right]$ (9)
 > $\mathrm{FromJet}\left(S,u\left(x,y\right)\right)$
 $\left[{{\mathrm{_ξ}}}_{{1}}{=}{0}{,}{{\mathrm{_ξ}}}_{{2}}{=}{0}{,}{{\mathrm{_η}}}_{{1}}{=}{{u}}_{{x}{,}{x}{,}{x}}{+}\frac{{{u}}_{{x}}^{{3}}}{{2}}\right]$ (10)
 > $\mathrm{SymmetryTest}\left(S,\mathrm{SGE}\right)$
 $\left\{{0}\right\}$ (11)