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SymmetryTransformation

  

computes the finite form of the (symmetry) transformation leaving invariant any PDE system admitting a given symmetry

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

SymmetryTransformation(S, DepVars, NewVars, 'options'='value')

Parameters

S

-

a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator

DepVars

-

a function or a list of functions indicating the dependent variables of the problem

NewVars

-

optional - a function or a list of functions representing the new dependent variables

jetnotation = ...

-

(optional) can be true (same as jetvariables), false (default), jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available

simplifier = ...

-

optional - indicates the simplifier to be used instead of the default simplify/size

redefinegroupparameter

-

optional - to simplify a subexpression that involves the Lie group parameter replacing it by another group parameter

Description

• 

Given a list with the infinitesimals of a symmetry symmetry transformation.

• 

When there is only one dependent variable, DepVars and NewVars can be a function. Otherwise they must be a list of functions representing dependent variables. If NewVars are not given, SymmetryTransformation will generate a list of globals to represent them.

• 

You can optionally specify a simplifier, to be used instead of the default which is simplify/size, as well as requesting the output to be in jet notation by respectively using the optional arguments simplifier = ... and jetnotation. Note that the option simplifier = ... can be used not just to "simplify" the output but also to post-process this output in the way you want, for instance using a procedure that you have written to discard, change or do what you find necessary with the transformation.

• 

In some cases, the Lie group parameter introduced by SymmetryTransformation appears embedded into a subexpression, for example as in ⅇ_ϵ, and only appears through functions of that subexpression. To have these cases returned with _ϵ instead of - say - ⅇ_ϵ, use the option redefinegroupparameter.

• 

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,SymmetryTransformation,ChangeSymmetry,InfinitesimalGenerator

SymmetryTransformation,ChangeSymmetry,InfinitesimalGenerator

(1)

Consider a PDE problem with two independent variables and one dependent variable, ux,t, and consider the list of infinitesimals of a symmetry group

S_ξx=x,_ξt=1,_ηu=u

S:=_ξx=x,_ξt=1,_ηu=u

(2)

In the input above you can also enter the symmetry S without infinitesimals' labels, as in x,1,u. The corresponding infinitesimal generator is

GInfinitesimalGeneratorS,ux,t

G:=f→xxf+tf+uuf

(3)

A PDESYS is invariant under the symmetry transformation generated by G in that GPDESYS=0, where, in this formula, G represents the prolongation necessary to act on PDESYS (see InfinitesimalGenerator).

The actual form of this finite, one-parameter, symmetry transformation relating the original variables t,x,ux,t to new variables, r,s,vr,s, that leaves invariant any PDE system admitting the symmetry represented by G above is obtained via

SymmetryTransformationS,ux,t,vr,s

r=xⅇ_ε,s=_ε+t,vr,s=ⅇ_εux,t

(4)

where _ϵ is a (Lie group) transformation parameter. To express this transformation using jetnotation use

SymmetryTransformationS,ux,t,vr,s,jetnotation

r=xⅇ_ε,s=_ε+t,v=ⅇ_εu

(5)

SymmetryTransformationS,ux,t,vr,s,jetnotation=jetnumbers

r=xⅇ_ε,s=_ε+t,v[]=ⅇ_εu[]

(6)

That this transformation leaves invariant any PDE system invariant under G above is visible in the fact that it also leaves invariant the infinitesimals S; to verify this you can use ChangeSymmetry

TR,NewVarssolve,x,t,ux,t,maplhs,

TR,NewVars:=t=s_ε,x=rⅇ_ε,ux,t=vr,sⅇ_ε,r,s,vr,s

(7)

ChangeSymmetryTR,S,ux,t,NewVars

_ξr=r,_ξs=1,_ηv=v

(8)

which is the same as S (but written in terms of vr,s instead of ux,t). So to this list of infinitesimals corresponds, written in terms of vr,s, this infinitesimal generator

InfinitesimalGenerator,vr,s

f→rrf+sf+vvf

(9)

which is also equal to G, only written in terms of vr,s.

If the new variables, vr,s,  are not indicated, variables prefixed by the underscore _ to represent the new variables are introduced

SymmetryTransformationS,ux,t

_t1=xⅇ_ε,_t2=_ε+t,_u1_t1,_t2=ⅇ_εux,t

(10)

An example where the Lie group parameter _ϵ appears only through the subexpression ⅇ_ϵ

SymmetryTransformation0,0,z,0,0,ux,y,z,t

_t1=x,_t2=y,_t3=zⅇ_ε,_t4=t,_u1_t1,_t2,_t3,_t4=ux,y,z,t

(11)

A symmetry transformation with the parameter redefined

SymmetryTransformation0,0,z,0,0,ux,y,z,t,'redefinegroupparameter'

_t1=x,_t2=y,_t3=z_ε,_t4=t,_u1_t1,_t2,_t3,_t4=ux,y,z,t

(12)

See Also

CanonicalCoordinates

ChangeSymmetry

InfinitesimalGenerator

Invariants

InvariantSolutions

InvariantTransformation

PDEtools

SimilarityTransformation

SymmetrySolutions

 


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