Infinitesimals(PDESYS, DepVars, S, 'options'='value')

Consider first a simple case; use diff_table and declare to avoid redundancies in the input and in the display of the output

To see the last computed output (in this case, ) with all its functionality and structure explicit, not using the enhanced compact mathematical display above, use show

The infinitesimals of (point) symmetry generators for  are

This result is a sequence of lists of infinitesimals components, with equations where the left-hand-side is an infinitesimal label including the functionality with which that infinitesimal was searched, and the right-hand-sides show the values found for them. To avoid displaying the functionality in these labels, use displayfunctionality = false.

To rewrite these list of infinitesimals returned by Infinitesimals as the corresponding infinitesimal generator differential operators, for example the last one of the result above

use InfinitesimalGenerator, as in

The infinitesimals returned by Infinitesimals are computed by solving the determining PDE system returned by DeterminingPDE

Note in the output above that  is  displayed in jet notation (because of using declare),  is  and so on. So, another way to compute the infinitesimals is to first compute this determining system  and then solve it using pdsolve, as in .

To see the lists of infinitesimals before specializing into cases the integration constants of the solution of DetSys use the optional argument split = false

To see this output in the jetnumbers jet notation use ToJet or call Infinitesimals with the optional argument jetnotation = jetnumbers

This notation is less readable than the default jetvariables but may be easier to manipulate within programs. Note the correspondence between the numbers indexing the infinitesimal label  and the positions of independent variables in the single function of this problem . The first number indexing the infinitesimal label  identifies the dependent variable to which the infinitesimal corresponds (in this example there is only one, ). You can tell that these infinitesimals are related to point symmetry transformations because they where searched as depending on () - not on the partial derivatives of . To generate these infinitesimal labels, in jetnumbers or jetvariables notation, prolonged or not, see the GenerateInfinitesimalLabels command of the PDEtools programming Library.

You can test for the correctness of the output of Infinitesimals using the SymmetryTest command in this way

Alternatively you can search for restricted forms of the infinitesimals. The following generates a search for infinitesimals of polynomial type with degree 0 (all the infinitesimals are constants):

Infinitesimals of polynomial type such that the degree of  and of  are respectively at most 1 and 2, also avoid displaying the functionality on the left-hand-sides:

Infinitesimals depending on either of ,  or

Infinitesimals depending on either of  or

Infinitesimals involving the product or sum of only two functions  and  in a particular form

The option HINT = [f(x)*g(t), f(x) + g(t), f(x)*g(t)] produces essentially the same output.

An example with two independent and two dependent variables

To input the system saving redundant input use diff_table

The infinitesimals

You can also specialize the search further, indicating both a type of symmetry, for example: evolutionary, and a type of functionality, for example: polynomial. In their general form, evolutionary symmetries are dynamical in that they depend on derivatives of the functions defining the jet space, here  and

As expected, in the result above there are dynamical symmetries (the last three), involving the derivatives  and . Dynamical symmetrics can also be tested for correctness with SymmetryTest

The infinitesimals of a PDE system are the key building blocks with which you can construct invariant solutions, invariant functions, transformations leaving the PDE system invariant or transformations reducing the number of its independent variables, etc.

For example, this is the infinitesimal generator associated to the fourth list of infinitesimals in  prolonged to order 1 (that is: appliable to functions depending on up to first order partial derivatives of  and )

These are the invariants up to order 1 associated to

Therefore, if you apply  to each of these invariants in the sequence above, you are expected to obtain a sequence of zeros

This is the transformation associated to  leaving  invariant

In the above  is the parameter of the (Lie) group of transformations. These other are the transformation equations and their inverses, associated to , reducing the number of independent variables of

An example where the infinitesimals depend on arbitrary functions

To specialize the arbitrary functions  in the output above use the option specialize_Fn (see the Options section)

To specialize these functions taking, for instance, the product or sum of their arguments, use

An example that is not linear in the highest derivative

By default Infinitesimals processes DE systems as given. If all the integrability conditions that can be derived from the system's equations are explicitly present, then the resulting determining system for the symmetry infinitesimals has for solution all the symmetries. This is not true otherwise. Consider for instance:

The infinitesimals of point symmetries of  are

However, the system  does not include the integrability condition  that can be derived from its equations. You can see that  is compatible with DE by changing the ordering for the independent variables (see casesplit)

To see a version of  that is closed, i.e., that it includes all its integrability conditions explicitly use the Library routine CloseSystem

When the system is closed, the resulting determining system and hence infinitesimals that solve it include some not present in  above

Compare for instance the arbitrary functions found in the output above for  and  with the ones found in . Another way to see how  are more restricted is to compute the arbitrary functions present in each result

Instead of closing the system interactively, you can request directly to Infinitesimals to close the given system  before computing the determining system for its symmetries by using the option closesystem

The Infinitesimals command also works with anticommutative variables, natively, without using the approach explained in PerformOnAnticommutativeSystem.

Set first  and  as suffixes for variables of type/anticommutative (see Setup)

A PDE system example with two unknown anticommutative functions of four variables, two commutative and two anticommutative; to avoid redundant typing in the input that follows and redundant display of information on the screen let's use PDEtools:-diff_table PDEtools:-declare

Now we can enter derivatives directly as the function's name indexed by the differentiation variables and see the display the same way; two PDEs

Set for instance the generic form of the infinitesimals for a PDE system like this one formed by pde[1] and pde[2]. For this purpose, we need anticommutative infinitesimals for the dependent variable  and two of the independent variables,  and ; we use here the capital greek letters  and  for the anticommutative infinitesimal symmetry generators and the corresponding lower case greek letters for commutative ones

The corresponding InfinitesimalGenerator

To compute the symmetry infinitesimals you can now compute the DeterminingPDE for this system, then call pdsolve to solve it, or directly call Infinitesimals that performs all that and some other convenient algebraic manipulations automatically; the related result comes specialized

To verify this result you can use SymmetryTest - it also handles anticommutative variables natively.

To see these three list of symmetry infinitesimals in a more readable form with a label on the left-hand-side, you can use

declare, DeterminingPDE, diff_table, InfinitesimalGenerator, PDEtools, PDEtools/Library, Physics, Setup, SimilarityTransformation, SymmetryTest, SymmetryTransformation