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PDEtools[CharacteristicQ] - compute the characteristic of a point symmetry represented by its infinitesimals
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Calling Sequence
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CharacteristicQ(S, DepVars, 'options'='value')
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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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DepVars
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a function or a list of them indicating the dependent variables of the problem
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checktype = ...
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optional - can be true (default) or false, to have or have not inserted a check-of-type for the arguments of the output procedure
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expanded = ...
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optional - can be true or false (default), to have or have not expanded the sums entering the body of the output procedure
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jetnotation = ...
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(optional) can be true (default, the notation found in S), false, jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available
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Description
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The CharacteristicQ command computes the characteristic of a point symmetry represented by its infinitesimals or the corresponding infinitesimal generator operator. That is, for a PDE problem with n independent and m dependent variables, given a related list of infinitesimals ![[xi[1], `...`, xi[n], eta[1], `...`, eta[m]]](/support/helpjp/helpview.aspx?si=7412/file02490/math108.png) , CharacteristicQ computes the procedure
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![m->eta[m]-(∑)x i[j] (d u[m])/(d x[j])](/support/helpjp/helpview.aspx?si=7412/file02490/math116.png)
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where  identifies a dependent variable.
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The sum in the body of this operator returned by CharacteristicQ is not expanded unless explicitly requested using the optional argument expanded. Also, jetnotation is used in this operator and a check-of-type for the value of  is automatically inserted unless explicitly requested otherwise with the optional arguments jetnotation = false and/or checktype = false - see the examples below.
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To avoid having to remember the optional keywords, if you misspell a keyword, or a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.
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Examples
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![[CharacteristicQ, InfinitesimalGenerator]](/support/helpjp/helpview.aspx?si=7412/file02490/math148.png)
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Consider a problem in two independent and two dependent variables u(x, t), v(x, t), and the generic form of infinitesimals for this type of problem
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![F := ([u, v])(x, t)](/support/helpjp/helpview.aspx?si=7412/file02490/math165.png)
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![F := [u(x, t), v(x, t)]](/support/helpjp/helpview.aspx?si=7412/file02490/math168.png)
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, j = [x, t]), seq(eta[j](x, t, u, v), j = [u, v])]](/support/helpjp/helpview.aspx?si=7412/file02490/math172.png)
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, xi[t](x, t, u, v), eta[u](x, t, u, v), eta[v](x, t, u, v)]](/support/helpjp/helpview.aspx?si=7412/file02490/math175.png)
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By default CharacteristicQ returns, fast, an operator in its most abstract form, with a test-type for the value of  and not expanded; essentially, nothing is actually computed until you need it
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![Q := proc (m::(satisfies(proc (m) options operator, arrow; m::posint and m <= 2 end proc))) options operator, arrow; eta[m]-add(xi[j]*(diff(y[m], X[j])), j = 1 .. 2) end proc](/support/helpjp/helpview.aspx?si=7412/file02490/math188.png)
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This resulting characteristic is a function that can then be applied to an integer as large as the number of dependent variables of the problem, in this case two
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-xi[x](x, t, u, v)*u[x]-xi[t](x, t, u, v)*u[t]](/support/helpjp/helpview.aspx?si=7412/file02490/math197.png)
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-xi[x](x, t, u, v)*v[x]-xi[t](x, t, u, v)*v[t]](/support/helpjp/helpview.aspx?si=7412/file02490/math204.png)
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You can instead request to CharacteristicQ for the sum in the mapping to be expanded before returning, or to avoid the check of type of the value of 
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![proc (m) options operator, arrow; eta[m]-xi[x](x, t, u, v)*(diff(y[m], x))-xi[t](x, t, u, v)*(diff(y[m], t)) end proc](/support/helpjp/helpview.aspx?si=7412/file02490/math216.png)
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Instead of passing the symmetry as a list of infinitesimals you can also pass the corresponding infinitesimal generator operator. You construct this operator with InfinitesimalGenerator
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*(diff(f, x))+xi[t](x, t, u, v)*(diff(f, t))+eta[u](x, t, u, v)*(diff(f, u))+eta[v](x, t, u, v)*(diff(f, v)) end proc](/support/helpjp/helpview.aspx?si=7412/file02490/math228.png)
| (8) |
This is the same output as (4.4)
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![proc (m::(satisfies(proc (m) options operator, arrow; m::posint and m <= 2 end proc))) options operator, arrow; eta[m]-add(xi[j]*(diff(y[m], X[j])), j = 1 .. 2) end proc](/support/helpjp/helpview.aspx?si=7412/file02490/math238.png)
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To request the output in function instead of jet notation use
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![Qf := proc (m) options operator, arrow; eta[m]-xi[x](x, t, u(x, t), v(x, t))*(diff(y[m], x))-xi[t](x, t, u(x, t), v(x, t))*(diff(y[m], t)) end proc](/support/helpjp/helpview.aspx?si=7412/file02490/math247.png)
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Compare for instance this output with the output of 
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, v(x, t))-xi[x](x, t, u(x, t), v(x, t))*(diff(u(x, t), x))-xi[t](x, t, u(x, t), v(x, t))*(diff(u(x, t), t))](/support/helpjp/helpview.aspx?si=7412/file02490/math257.png)
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