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

The default value is false; when true, the consistency of PDESYS is checked before proceeding. Note that one could construct the determining system for the symmetries of an inconsistent PDESYS, in which case the solution of the determining system would be meaningless.

By default DeterminingPDE processes PDESYS as given. If, in PDESYS, all the integrability conditions that can be derived from its equations are explicitly present, then the resulting determining system for the symmetries is assured to have for solution all the symmetries of PDESYS. Otherwise, the returned determining system may in rare situations also (correctly) include equations that however may restrict the possible values of the symmetry solutions, and hence some of the symmetries be missed. Note also that, for a system to be closed, it is not sufficient to have it in reduced involutive or equivalent form produced by the casesplit and rifsimp commands or the DifferentialAlgebra package. To assure that all the integrability conditions are explicitly present use the option closesystem - see related example in the Examples section. See also the Library routine CloseSystem. This option is ignored when computing integrating factors or conserved currents

Use this option, giving just the keyword conservedcurrents, to request the determining system satisfied by the conserved currents. Each component of the conserved current will be represented by a different function depending on all the jet space variables including derivatives up to the maximum differential order found in PDESYS minus one. This option can also be used as an equation, in which case the right-hand side can be a single function, and the components of the conserved currents will be taken to be different functions all of them having the same functionality as the one indicated in the right-hand side. Alternatively you can specify the right-hand side as a list directly indicating the functionality of each of the components of the conserved currents.

Indicates the dependency of the infinitesimals, integrating factors or conserved currents. Default value is all the jet space variables associated to point transformations. To change this default use this option with a name variable or a set or list of name variables in the right-hand side.

Default value is true, and so the output system is simplified with respect to its integrability conditions before being returned. This default frequently simplifies the determining PDE system significatively, but in same cases the mathematical computations to achieve the result are lengthy. To skip that simplification and have DeterminingPDE returning faster pass this option with false in the right-hand side.

Use this option, giving just the keyword integratingfactors, to request the determining system satisfied by the integrating factors. Each component of the conserved current will be represented by a different function depending on all the jet space variables including derivatives up to the maximum differential order found in PDESYS minus one. This option can also be used as an equation, in which case the right-hand side can be a single function, and the components of the integrating factors will be taken to be different functions all of them having the same functionality as the one indicated in the right-hand side. Alternatively you can specify the right-hand side as a list directly indicating the functionality of each of the components of the integrating factors.

The default value is jetvariables so that the output has all the dependent variables and its derivatives expressed in jet notation. The value jetODE can be used provided that the input is a single ODE; the output is then expressed using the jetODE notation of DEtools,Lie, the package for ODE symmetries developed before the introduction of symmetry commands in PDEtools.

By default the output system is computed modulo the given PDESYS; when this option is passed with false in the right-hand side, this computational step is skipped.

The default value is simplify/size. Use this option to specify the simplifier of your choice. To indicate that no simplification is preferred pass the keyword none in the right-hand side.

By default the output system is split, that is, conformed by the coefficients of different powers of independent objects (e.g.: when computing point symmetries, the system's equations are the coefficients of different powers of the derivatives of the unknowns). That frequently makes the output to be an overdetermined system. When this option is passed with false in the right-hand side, the output system consists of a single partial differential equation.

By default DeterminingPDE returns a system whose solutions are the point symmetries of the given PDESYS. You can also request the determining system for symmetries of evolutionary, contact or general types, where in this context general means infinitesimals depending on derivatives of each unknown of the system up to order , where  is the differential order of the unknown as found in the system, and evolutionary is like general but with the  infinitesimals (related to transformations of the independent variables) all equal to zero. By differential order of the unknown as found in the system it is meant: take the maximum differential order of an unknown within each equation of the PDE system, then take the minimum of those numbers, so that it is possible to return a determining system moduloPDESYS.

This option is used to indicate the root names to be used to represent the infinitesimals instead of the default  and .

Consider first a simple case; use diff_table and declare to avoid redundancies in the input (you can enter things using jetvariableswithbrackets jet notation) and in the display of the output (expressed using jetvariables jet notation)

The determining PDE system whose solutions are the infinitesimals

The system returned has the leading derivatives isolated on the left-hand-sides, and the jet notation used by default is jetvariables, so  is represented by ,  by , and so on. The derivatives of the  and  infinitesimals are displayed indexed, in jetvariableswithbrackets notation, because of the use of declare.

Note the labels for the infinitesimals: they are of the form  and , where  and  are respectively the names of independent and dependent variables; these are the infinitesimals associated to symmetry transformations of the corresponding (independent or dependent) variable.

You can also specify that you want a different jet notation, for instance jetnumbers (a less mathematical notation, sometimes preferred for programming purposes), in which case the labels identifying each infinitesimal will also appear in jetnumbers notation, as in  and

The system returned by DeterminingPDE is automatically simplified with respect to its integrability conditions. This simplification is normally desirable, but in some cases it can be expensive in time or depending on the problem even a bottleneck in the computation of symmetries. To skip this simplification use the option integrabilityconditions = false, in which case the leading derivatives are not isolated; compare for instance  with this other equivalent determining system

The fastest (and less simplified) way to compute the determining PDE for the symmetries includes skipping the automatic splitting into coefficients of independent objects, in which case the output consists of a set with a single determining PDE, not a system

Note in the output above the presence of  and  representing derivatives of  written in jetnotation. To see the last computed output with all its functionality and structure explicit, not using the enhanced compact mathematical display due to declare, use show

Note in the above that although the infinitesimals and its derivatives are not displayed in jet notation anymore, the unknown  and its derivatives ,  are still in jet notation - they enter the expression actually as names, not functions, but representing the functions. As usual, to convert into function notation use FromJet.

The infinitesimals are obtained solving this determining system - we take the most simplified version of it, , computed lines above

You can compute these infinitesimals directly from PDE1 as well. By default, whenever, it is possible Infinitesimals will present the symmetry groups with all its independent members explicitly, that is, with the integration constants specialized, as in

The determining PDE system for the integrating factors of PDE1

The determining PDE system for the conserved currents of PDE1 that depend only on  and

To compute the actual integrating factors or conserved currents - so not just their determining equations- see IntegratingFactors and ConservedCurrents.

The Laplace equation  in spherical coordinates with a source term  symmetries

The infinitesimals are computed either by Infinitesimals by calling DeterminingPDE to obtain the determining system then solving it calling pdsolve, finally splitting the result according to the integration constants found in pdsolve's solution

An ODE example (see more commands for handling ODE symmetry problems also in DEtools,Lie) and a dynamical symmetry represented by infinitesimals depending on the first derivative y'

The corresponding determining system directly in jetODE notation, that is the notation used in DEtools,Lie, the package for symmetries and integrating factors for ODEs developed before the symmetry commands of PDEtools

An ODE in disguise, appearing as a PDE, and the corresponding determining PDE system for a very particular form  of its infinitesimals, depending on two arbitrary functions  and

The output above is zero because S2 is already a symmetry of ODE2, thus canceling completely the determining system. To test symmetries given as lists of infinitesimals use the SymmetryTest command

The determining system for symmetries in evolutionary form for an ODE, directly in jetODE notation

By default DeterminingPDE 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 always true otherwise. Consider for instance:

The determining system for the point symmetries of DE and its corresponding solution are

The system DE, however, does not include the integrability condition  that can be derived from its equations. To compute a version of DE that is closed, i.e., that it includes all its integrability conditions explicitly you can use the Library routine CloseSystem or directly pass the closesystem option to DeterminingPDE

The resulting determining system for the closed version of DE admits solutions more general than (6.28)

Comparing, in (6.32)  is entirely arbitrary while in the infinitesimals (6.28) it is not. The closesystem option is also available when calling Infinitesimals, InvariantSolutions and SimilaritySolutions - all of them will forward the option to DeterminingPDE. For example: the same most general form of the infinitesimals (6.32) can be computed directly via

The DeterminingPDE 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

The DeterminingPDE for this system

To compute now the exact form of the symmetry infinitesimals you can either solve this PDE system for the conmutatice and anticommutative functions using pdsolve, or directly pass the system to Infinitesimals that will perform all the steps automatically.

declare, diff_table, Infinitesimals, PDEtools, Physics, Setup, SymmetryTest