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PDEtools[SymmetryTest] - tests whether a given list of infinitesimals represents a symmetry of a given PDE system.
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Calling Sequence
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SymmetryTest(S, PDESYS, DepVars)
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Parameters
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S
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a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator
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PDESYS
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a PDE or a set or list of them; it can include ODEs and not differential equations
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DepVars
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optional - may be required; a function or a list of them indicating the dependent variables of the problem
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Description
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If DepVars is not specified, SymmetryTest will consider all the differentiated unknown functions in PDESYS as unknown of the problems.
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Examples
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Consider the wave equation in four dimensions to avoid redundant typing on input and on the display use diff_table and declare
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![declare(U[])](/support/helpjp/helpview.aspx?si=7201/file02517/math106.png)
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| (1) |
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![pde[1] := U[x, x]+U[y, y]+U[z, z]-U[t, t] = 0](/support/helpjp/helpview.aspx?si=7201/file02517/math113.png)
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![pde[1] := u[x, x]+u[y, y]+u[z, z]-u[t, t] = 0](/support/helpjp/helpview.aspx?si=7201/file02517/math116.png)
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Compute the infinitesimals of point symmetry transformations leaving invariant pde[1] and test for correctness the first list
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![S := Infinitesimals(pde[1])](/support/helpjp/helpview.aspx?si=7201/file02517/math138.png)
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![SymmetryTest(S[1], pde[1])](/support/helpjp/helpview.aspx?si=7201/file02517/math145.png)
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Test all the lists in one step
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![map(SymmetryTest, [S], pde[1])](/support/helpjp/helpview.aspx?si=7201/file02517/math154.png)
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![[{0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}, {0}]](/support/helpjp/helpview.aspx?si=7201/file02517/math157.png)
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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
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![SGE := u[x, y] = sin(u)](/support/helpjp/helpview.aspx?si=7201/file02517/math179.png)
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The following list of infinitesimals represent a symmetry of SGE
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![S := [_xi[1] = 0, _xi[2] = 0, _eta[1] = u[1, 1, 1]+(1/2)*u[1]^3]](/support/helpjp/helpview.aspx?si=7201/file02517/math189.png)
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![S := [_xi[1] = 0, _xi[2] = 0, _eta[1] = u[1, 1, 1]+(1/2)*u[1]^3]](/support/helpjp/helpview.aspx?si=7201/file02517/math192.png)
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![[_xi[1] = 0, _xi[2] = 0, _eta[1] = u[x, x, x]+(1/2)*u[x]^3]](/support/helpjp/helpview.aspx?si=7201/file02517/math199.png)
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| (11) |
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