SymmetryCommutator - Maple Help

PDEtools

 SymmetryCommutator
 return the commutator of two infinitesimals of a symmetry transformation

 Calling Sequence SymmetryCommutator(S1, S2, DepVars, options=value)

Parameters

 S1, S2 - two lists with the infinitesimals of a symmetry transformation, either as lists or procedures (infinitesimal generators) DepVars - function or a list of them indicating the dependent variables of the problem jetnotation = ... - (optional) true (default, the notation found in S1), false, jetnumbers, jetvariables, jetvariableswithbrackets or jetODE; to respectively return or not using the different jet notations available output = ... - (optional) list or operator; specifies whether the output should be a list of infinitesimal components or its corresponding infinitesimal generator differential operator prolongation = ... - (optional) positive integer indicating the desired prolongation order of the commutator; default is the prolongation order found in the given S1.

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{SymmetryCommutator},\mathrm{InfinitesimalGenerator}\right)$
 $\left[{\mathrm{SymmetryCommutator}}{,}{\mathrm{InfinitesimalGenerator}}\right]$ (1)

Consider two lists of infinitesimals corresponding to a symmetry transformation where there are two independent variables and one dependent variable, $u\left(x,t\right)$.

 > $\mathrm{S1},\mathrm{S2}≔\left[\mathrm{_ξ}\left[x\right]=x,\mathrm{_ξ}\left[t\right]=1,\mathrm{_η}\left[u\right]=t\right],\left[\mathrm{_ξ}\left[x\right]=1,\mathrm{_ξ}\left[t\right]=\frac{1}{t},\mathrm{_η}\left[u\right]={x}^{2}\right]$
 ${\mathrm{S1}}{,}{\mathrm{S2}}{≔}\left[{{\mathrm{_ξ}}}_{{x}}{=}{x}{,}{{\mathrm{_ξ}}}_{{t}}{=}{1}{,}{{\mathrm{_η}}}_{{u}}{=}{t}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}{1}{,}{{\mathrm{_ξ}}}_{{t}}{=}\frac{{1}}{{t}}{,}{{\mathrm{_η}}}_{{u}}{=}{{x}}^{{2}}\right]$ (2)

The corresponding infinitesimal generators in operator format are

 > $\mathrm{G1}≔\mathrm{InfinitesimalGenerator}\left(\mathrm{S1},u\left(x,t\right)\right)$
 ${\mathrm{G1}}{≔}{f}{→}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){+}\frac{{\partial }}{{\partial }{t}}{}{f}{+}{t}{}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right)$ (3)
 > $\mathrm{G2}≔\mathrm{InfinitesimalGenerator}\left(\mathrm{S2},u\left(x,t\right)\right)$
 ${\mathrm{G2}}{≔}{f}{→}\frac{{\partial }}{{\partial }{x}}{}{f}{+}\frac{\frac{{\partial }}{{\partial }{t}}{}{f}}{{t}}{+}{{x}}^{{2}}{}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right)$ (4)

The symmetry commutator is $\left[\mathrm{S1},\mathrm{S2}\right]=\mathrm{S1}@\mathrm{S2}-\mathrm{S2}@\mathrm{S1}$; when S1 is a operator, the output is then a differential operator

 > $\mathrm{SymmetryCommutator}\left(\mathrm{G1},\mathrm{G2},u\left(x,t\right)\right)$
 ${f}{→}{-}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){-}\frac{\frac{{\partial }}{{\partial }{t}}{}{f}}{{{t}}^{{2}}}{+}\frac{\left({2}{}{t}{}{{x}}^{{2}}{-}{1}\right){}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right)}{{t}}$ (5)

The output can be requested as an ordered list of infinitesimal components

 > $\mathrm{SymmetryCommutator}\left(\mathrm{G1},\mathrm{G2},u\left(x,t\right),\mathrm{output}=\mathrm{list}\right)$
 $\left[{{\mathrm{_ξ}}}_{{x}}{=}{-1}{,}{{\mathrm{_ξ}}}_{{t}}{=}{-}\frac{{1}}{{{t}}^{{2}}}{,}{{\mathrm{_η}}}_{{u}}{=}\frac{{2}{}{t}{}{{x}}^{{2}}{-}{1}}{{t}}\right]$ (6)

The input can also be given in mixed formats, in which case the output is returned in the format of the first infinitesimal

 > $\mathrm{SymmetryCommutator}\left(\mathrm{S1},\mathrm{G2},u\left(x,t\right)\right)$
 $\left[{{\mathrm{_ξ}}}_{{x}}{=}{-1}{,}{{\mathrm{_ξ}}}_{{t}}{=}{-}\frac{{1}}{{{t}}^{{2}}}{,}{{\mathrm{_η}}}_{{u}}{=}\frac{{2}{}{t}{}{{x}}^{{2}}{-}{1}}{{t}}\right]$ (7)

The prolongation order of the commutator is by default the one of the given infinitesimals, but can also be specified using the optional argument prolongation = n, where n is a positive integer.

 > $\mathrm{SymmetryCommutator}\left(\mathrm{S1},\mathrm{S2},u\left(x,t\right),\mathrm{prolongation}=1\right)$
 $\left[{{\mathrm{_ξ}}}_{{x}}{=}{-1}{,}{{\mathrm{_ξ}}}_{{t}}{=}{-}\frac{{1}}{{{t}}^{{2}}}{,}{{\mathrm{_η}}}_{{u}}{=}\frac{{2}{}{t}{}{{x}}^{{2}}{-}{1}}{{t}}{,}{{\mathrm{_η}}}_{{u}{,}\left[{x}\right]}{=}{4}{}{x}{,}{{\mathrm{_η}}}_{{u}{,}\left[{t}\right]}{=}\frac{{t}{-}{2}{}{{u}}_{{t}}}{{{t}}^{{3}}}\right]$ (8)

To request the output in a different notation, for instance jetnumbers (see ToJet), use the optional argument jetnotation = ....

 > $\mathrm{SymmetryCommutator}\left(\mathrm{S1},\mathrm{S2},u\left(x,t\right),\mathrm{prolongation}=1,\mathrm{notation}=\mathrm{jetnumbers}\right)$
 $\left[{{\mathrm{_ξ}}}_{{1}}{=}{-1}{,}{{\mathrm{_ξ}}}_{{2}}{=}{-}\frac{{1}}{{{t}}^{{2}}}{,}{{\mathrm{_η}}}_{{1}}{=}\frac{{2}{}{t}{}{{x}}^{{2}}{-}{1}}{{t}}{,}{{\mathrm{_η}}}_{{1}{,}\left[{1}\right]}{=}{4}{}{x}{,}{{\mathrm{_η}}}_{{1}{,}\left[{2}\right]}{=}\frac{{t}{-}{2}{}{{u}}_{{2}}}{{{t}}^{{3}}}\right]$ (9)

Note that in the output above the infinitesimals (right-hand-sides) and also their labels (left-hand-sides) are written in jetnumbers notation. You can also specify the output format to be an operator

 > $\mathrm{SymmetryCommutator}\left(\mathrm{S1},\mathrm{S2},u\left(x,t\right),\mathrm{prolongation}=1,\mathrm{output}=\mathrm{operator}\right)$
 ${f}{→}{-}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){-}\frac{\frac{{\partial }}{{\partial }{t}}{}{f}}{{{t}}^{{2}}}{+}\frac{\left({2}{}{t}{}{{x}}^{{2}}{-}{1}\right){}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right)}{{t}}{+}{4}{}{x}{}\left(\frac{{\partial }}{{\partial }{{u}}_{{x}}}{}{f}\right){+}\frac{\left({t}{-}{2}{}{{u}}_{{t}}\right){}\left(\frac{{\partial }}{{\partial }{{u}}_{{t}}}{}{f}\right)}{{{t}}^{{3}}}$ (10)

Compatibility

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