Given a pair of infinitesimals of a symmetry transformation, S1 and S2, either as lists or differential operators (see infinitesimal generator), the SymmetryCommutator command returns the commutator of these symmetries, $\left[\mathrm{S1}\,\mathrm{S2}\right]=\mathrm{S1}@\mathrm{S2}-\mathrm{S2}@\mathrm{S1}$.

If S1 and S2 are infinitesimal generator differential operators, the result is also a differential operator. If S1 and S2 are given as lists, then the coefficients in that differential operator, so conforming the infinitesimal, are returned within a list. When S1 and S2 are not of the same kind (operator or list), the output is in the format of S1 unless indicated otherwise using the option output = ...

The prolongation order of the commutator $\left[\mathrm{S1}\,\mathrm{S2}\right]$ returned is the one found in S1 unless indicated otherwise using the option prolongation = n where n is a non-negative integer.

The jet notation used in the output is the one of S1 unless indicated otherwise using the option jetnotation = ... where the right-hand side is any of jetnumbers' (default), 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.

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

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

$\mathrm{S1},\mathrm{S2}≔\left[\mathrm{\_\ξ}\left[x\right]=x\,\mathrm{\_\ξ}\left[t\right]=1\,\mathrm{\_\η}\left[u\right]=t\right],\left[\mathrm{\_\ξ}\left[x\right]=1\,\mathrm{\_\ξ}\left[t\right]=\frac{1}{t}\,\mathrm{\_\η}\left[u\right]=x^2\right]$

$\mathrm{S1},\mathrm{S2}≔\left[{\mathrm{\_\ξ}}_x=x\,{\mathrm{\_\ξ}}_t=1\,{\mathrm{\_\η}}_u=t\right],\left[{\mathrm{\_\ξ}}_x=1\,{\mathrm{\_\ξ}}_t=\frac{1}{t}\,{\mathrm{\_\η}}_u=x^2\right]$

$\mathrm{G1}≔\mathrm{InfinitesimalGenerator}\left(\mathrm{S1}\,u\left(x\,t\right)\right)$

$\mathrm{G1}≔f\→x\left(\frac{\partial }{\partial x}f\right)\+\frac{\partial }{\partial t}f\+t\left(\frac{\partial }{\partial u}f\right)$

$\mathrm{G2}≔\mathrm{InfinitesimalGenerator}\left(\mathrm{S2}\,u\left(x\,t\right)\right)$

$\mathrm{G2}≔f\→\frac{\partial }{\partial x}f\+\frac{\frac{\partial }{\partial t}f}{t}\+x^2\left(\frac{\partial }{\partial u}f\right)$

$\mathrm{SymmetryCommutator}\left(\mathrm{G1}\,\mathrm{G2}\,u\left(x\,t\right)\right)$

$f\→-\left(\frac{\partial }{\partial x}f\right)-\frac{\frac{\partial }{\partial t}f}{t^2}\+\frac{\left(2t{x}^2-1\right)\left(\frac{\partial }{\partial u}f\right)}{t}$

$\mathrm{SymmetryCommutator}\left(\mathrm{G1}\,\mathrm{G2}\,u\left(x\,t\right)\,\mathrm{output}=\mathrm{list}\right)$

$\left[{\mathrm{\_\ξ}}_x=\mathrm{-1}\,{\mathrm{\_\ξ}}_t=-\frac{1}{t^2}\,{\mathrm{\_\η}}_u=\frac{2t{x}^2-1}{t}\right]$

$\mathrm{SymmetryCommutator}\left(\mathrm{S1}\,\mathrm{G2}\,u\left(x\,t\right)\right)$

$\left[{\mathrm{\_\ξ}}_x=\mathrm{-1}\,{\mathrm{\_\ξ}}_t=-\frac{1}{t^2}\,{\mathrm{\_\η}}_u=\frac{2t{x}^2-1}{t}\right]$

$\mathrm{SymmetryCommutator}\left(\mathrm{S1}\,\mathrm{S2}\,u\left(x\,t\right)\,\mathrm{prolongation}=1\right)$

$\left[{\mathrm{\_\ξ}}_x=\mathrm{-1}\,{\mathrm{\_\ξ}}_t=-\frac{1}{t^2}\,{\mathrm{\_\η}}_u=\frac{2t{x}^2-1}{t}\,{\mathrm{\_\η}}_{u,\left[x\right]}=4x\,{\mathrm{\_\η}}_{u,\left[t\right]}=\frac{t-2u_t}{t^3}\right]$

$\mathrm{SymmetryCommutator}\left(\mathrm{S1}\,\mathrm{S2}\,u\left(x\,t\right)\,\mathrm{prolongation}=1\,\mathrm{notation}=\mathrm{jetnumbers}\right)$

$\left[{\mathrm{\_\ξ}}_1=\mathrm{-1}\,{\mathrm{\_\ξ}}_2=-\frac{1}{t^2}\,{\mathrm{\_\η}}_1=\frac{2t{x}^2-1}{t}\,{\mathrm{\_\η}}_{1,\left[1\right]}=4x\,{\mathrm{\_\η}}_{1,\left[2\right]}=\frac{t-2u_2}{t^3}\right]$

$\mathrm{SymmetryCommutator}\left(\mathrm{S1}\,\mathrm{S2}\,u\left(x\,t\right)\,\mathrm{prolongation}=1\,\mathrm{output}=\mathrm{operator}\right)$

$f\→-\left(\frac{\partial }{\partial x}f\right)-\frac{\frac{\partial }{\partial t}f}{t^2}\+\frac{\left(2t{x}^2-1\right)\left(\frac{\partial }{\partial u}f\right)}{t}\+4x\left(\frac{\partial }{\partial u_x}f\right)\+\frac{\left(t-2u_t\right)\left(\frac{\partial }{\partial u_t}f\right)}{t^3}$

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

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