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Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : PDEtools/SymmetryCommutator

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SymmetryCommutator

  

return the commutator of two infinitesimals of a symmetry transformation

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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, S1,S2=S1@S2S2@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 S1,S2 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

withPDEtools,SymmetryCommutator,InfinitesimalGenerator

SymmetryCommutator,InfinitesimalGenerator

(1)

Consider two lists of infinitesimals corresponding to a symmetry transformation where there are two independent variables and one dependent variable, ux,t.

S1,S2_ξx=x,_ξt=1,_ηu=t,_ξx=1,_ξt=1t,_ηu=x2

S1,S2:=_ξx=x,_ξt=1,_ηu=t,_ξx=1,_ξt=1t,_ηu=x2

(2)

The corresponding infinitesimal generators in operator format are

G1InfinitesimalGeneratorS1,ux,t

G1:=f→xxf+tf+tuf

(3)

G2InfinitesimalGeneratorS2,ux,t

G2:=f→xf+tft+x2uf

(4)

The symmetry commutator is S1,S2=S1@S2S2@S1; when S1 is a operator, the output is then a differential operator

SymmetryCommutatorG1,G2,ux,t

f→xftft2+2tx21uft

(5)

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

SymmetryCommutatorG1,G2,ux,t,output=list

_ξx=1,_ξt=1t2,_ηu=2tx21t

(6)

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

SymmetryCommutatorS1,G2,ux,t

_ξx=1,_ξt=1t2,_ηu=2tx21t

(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.

SymmetryCommutatorS1,S2,ux,t,prolongation=1

_ξx=1,_ξt=1t2,_ηu=2tx21t,_ηu,x=4x,_ηu,t=t2utt3

(8)

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

SymmetryCommutatorS1,S2,ux,t,prolongation=1,notation=jetnumbers

_ξ1=1,_ξ2=1t2,_η1=2tx21t,_η1,1=4x,_η1,2=t2u2t3

(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

SymmetryCommutatorS1,S2,ux,t,prolongation=1,output=operator

f→xftft2+2tx21uft+4xuxf+t2ututft3

(10)

Compatibility

• 

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

• 

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

See Also

InfinitesimalGenerator

PDEtools

ToJet

 


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