The Package SPDE for Determining Symmetries
of Partial Differential Equations
User's Manual
Fritz Schwarz
GMD, Institut F1
Postfach 1240
5205 St. Augustin
West Germany
Tel. 02241-142782
EARN ID: GF1002@DBNGMD21
1. General Information.
The package SPDE provides a set of functions which may be applied
to determine the symmetry group of Lie-or point-symmetries of a
given system of partial differential equations. Preferably it is
used interactively on a computer terminal. In many cases the
determining system is solved completely automatically. In some
other cases the user has to provide some additional input
information for the solution algorithm to terminate. The package
should only be used in compiled form.
For all theoretical questions, a description of the algorithm and
numerous examples the following articles should be consulted:
"Automatically Determining Symmetries of Partial Differential
Equations", Computing vol. 34, page 91-106(1985) and vol. 36, page
279-280(1986), "Symmetries of Differential Equations: From Sophus
Lie to Computer Algebra", SIAM Review, to appear, and Chapter 2
of the Lecture Notes "Computer Algebra and Differential Equations
of Mathematical Physics'', to appear.
2. Description of the System Functions and Variables.
The symmetry analysis of partial differential equations logically
falls into three parts. Accordingly the most important functions
provided by the package are:
----------------------------------------------------------------
| Function name | Operation |
|--------------------------------------------------------------|
|CRESYS(<arguments>)| Constructs determining system |
|--------------------------------------------------------------|
| SIMPSYS() | Solves determining system |
|--------------------------------------------------------------|
| RESULT() | Prints infinitesimal generators |
| | and commutator table |
----------------------------------------------------------------
Table 1.
Some other useful functions for obtaining various kinds of output
are:
----------------------------------------------------------------
| PRSYS() | Prints determining system |
|--------------------------------------------------------------|
| PRGEN() | Prints infinitesimal generators |
|--------------------------------------------------------------|
| COMM(U,V) | Prints commutator of generators U and V |
----------------------------------------------------------------
Table 2.
There are several global variables defined by the system which
should not be used for any other purpose than that given
in Tables 3 and 4. The three globals of the type integer are:
---------------------------------------------------------------
| Variable name | Meaning |
|----------------------|--------------------------------------|
| NN | Number of independent variables |
|----------------------|--------------------------------------|
| MM | Number of dependent variables |
|----------------------|--------------------------------------|
| PCLASS=0, 1 or 2 | Controls amount of output |
---------------------------------------------------------------
Table 3.
In addition there are the following global variables of type
operator:
---------------------------------------------------------------
| Variable name | Meaning |
|----------------------|--------------------------------------|
| X(I) | Independent variable x |
| | i |
|----------------------|--------------------------------------|
| | alfa |
| U(ALFA) | Dependent variable u |
|----------------------|--------------------------------------|
| | alfa |
| U(ALFA,I) | Derivative of u w.r.t. x |
| | i |
|----------------------|--------------------------------------|
| DEQ(I) | i-th differential equation |
|----------------------|--------------------------------------|
| SDER(I) | Derivative w.r.t. which DEQ(I) |
| | is resolved |
|----------------------|--------------------------------------|
| GL(I) | i-th equation of determining system |
|----------------------|--------------------------------------|
| GEN(I) | i-th infinitesimal generator |
|----------------------|--------------------------------------|
| XI(I), ETA(ALFA) | See definition given in the |
| | |
| ZETA(ALFA,I) | references quoted in Section 1. |
|----------------------|--------------------------------------|
| C(I) | i-th function used for substitution |
---------------------------------------------------------------
Table 4.
The differential equations of the system at issue have to be assigned
as values to the operator deq i applying the notation which is
defined in Table 4. The entries in the third and the last line of
that Table have obvious extensions to higher derivatives.
The derivative w.r.t. which the i-th differential equation deq i is
resolved has to be assigned to sder i. Exception: If there is a
single differential equation and no assignment has been made by the
user, the highest derivative is taken by default.
When the appropriate assignments are made to the variable deq,
the values of NN and MM ( Table 2 ) are determined automatically,
i.e. they have not to be assigned by the user.
The function CRESYS may be called with any number of arguments,
i.e.
CRESYS(); or CRESYS(deq 1,deq 2,... );
are legal calls. If it is called without any argument, all current
assignments to deq are taken into account. Example: If deq 1, deq 2
and deq 3 have been assigned a differential equation and the
symmetry group of the full system comprising all three equations
is desired, equivalent calls are
CRESYS(); or CRESYS(deq 1,deq 2,deq 3);
The first alternative saves some typing. If later in the session
the symmetry group of deq 1 alone has to be determined, the correct
call is
CRESYS deq 1;
After the determining system has bee created, SIMPSYS which has
no arguments may be called for solving it. The amount of
intermediate output produced by SIMPSYS is controled by the
global variable PCLASS with the default value 0. With PCLASS equal
to 0, no intermediate steps are shown. With PCLASS equal to 1, all
intermediate steps are displayed so that the solution algorithm
may be followed through in detail. Each time the algorithm passes
through the top of the main solution loop the message
Entering main loop
is written. PCLASS equal 2 produces a lot of LISP output and is of
no interest for the normal user.
If with PCLASS=0 the procedure SIMPSYS terminates without any
response, the determining system is completely solved. In some
cases SIMPSYS does not solve the determining system completely in a
single run. In general this is true if there are only genuine
differential equations left which the algorithm cannot handle at
present. If a case like this occurs, SIMPSYS returns the remaining
equations of the determining system. To proceed with the solution
algorithm, appropriate assignments have to be transmitted by the
user, e.g. the explicit solution for one of the returned
differential equations. Any new functions which are introduced
thereby must be operators of the form c(k) with the correct
dependencies generated by a depend statement ( see REDUCE User's
Guide ). Its enumeration has to be chosen in agreement with the
current number of functions which have alreday been introduced.
This value is returned by SIMPSYS too.
After the determining system has been solved, the procedure RESULT
which has no arguments may be called. It displays the infinitesimal
generators and its non-vanishing commutators.
2. How to Use the Package. Examples.
In this Section it is explained by way of several examples how the
package SPDE is used interactively to determine the symmetry group
of partial differential equations. Consider first the diffusion
equation which in the notation given above may be written as
deq 1:=u(1,1)+u(1,2,2);
It has been assigned as the value of deq 1 by this statement.
There is no need to assign a value to sder 1 here because the
system comprises only a single equation.
The determining system is constructed by calling
CRESYS(); or CRESYS deq 1;
The latter call is compulsory if there are other assignments to the
operator deq i than for i=1.
The error message
***** Differential equations not defined
appears if there are no differential equations assigned to any deq.
If the user wants the determining system displayed for inspection
before starting the solution algorithm he may call
PRSYS();
and gets the answer
GL(1):=2*DF(ETA(1),U(1),X(2)) - DF(XI(2),X(2),2) - DF(XI(2),X(1))
GL(2):=DF(ETA(1),U(1),2) - 2*DF(XI(2),U(1),X(2))
GL(3):=DF(ETA(1),X(2),2) + DF(ETA(1),X(1))
GL(4):=DF(XI(2),U(1),2)
GL(5):=DF(XI(2),U(1)) - DF(XI(1),U(1),X(2))
GL(6):=2*DF(XI(2),X(2)) - DF(XI(1),X(2),2) - DF(XI(1),X(1))
GL(7):=DF(XI(1),U(1),2)
GL(8):=DF(XI(1),U(1))
GL(9):=DF(XI(1),X(2))
The remaining dependencies
XI(2) depends on U(1),X(2),X(1)
XI(1) depends on U(1),X(2),X(1)
ETA(1) depends on U(1),X(2),X(1)
The last message means that all three functions XI(1), XI(2) and
ETA(1) depend on X(1), X(2) and U(1). Without this information the
nine equations GL(1) to GL(9) forming the determining system are
meaningless. Now the solution algorithm may be activated by calling
SIMPSYS();
If the print flag PCLASS has its default value which is 0 no inter-
mediate output is produced and the answer is
Determining system is not completely solved
The remaining equations are
GL(1):=DF(C(1),X(2),2) + DF(C(1),X(1))
Number of functions is 16
The remaining dependencies
C(1) depends on X(2),X(1)
With PCLASS equal to 1 about 6 pages of intermediate output are
obtained. It allows the user to follow through each step of the
solution algorithm.
In this example the algorithm did not solve the determining system
completely as it is shown by the last message. This was to be
expected because the diffusion equation is linear and therefore the
symmetry group contains a generator depending on a function which
solves the original differential equation. In cases like this the
user has to provide some additional information to the system so
that the solution algorithm may continue. In the example under
consideration the appropriate input is
DF(C(1),X(1)) := - DF(C(1),X(2),2);
If now the solution algorithm is activated again by
SIMPSYS();
the solution algorithm terminates without any further message, i.e.
there are no equations of the determining system left unsolved. To
obtain the symmetry generators one has to say finally
RESULT();
and obtains the answer
The differential equation
DEQ(1):=U(1,2,2) + U(1,1)
The symmetry generators are
GEN(1):= DX(1)
GEN(2):= DX(2)
GEN(3):= 2*DX(2)*X(1) + DU(1)*U(1)*X(2)
GEN(4):= DU(1)*U(1)
GEN(5):= 2*DX(1)*X(1) + DX(2)*X(2)
2
GEN(6):= 4*DX(1)*X(1)
+ 4*DX(2)*X(2)*X(1)
2
+ DU(1)*U(1)*(X(2) - 2*X(1))
GEN(7):= DU(1)*C(1)
The remaining dependencies
C(1) depends on X(2),X(1)
Constraints
DF(C(1),X(1)):= - DF(C(1),X(2),2)
The non-vanishing commutators of the finite subgroup
COMM(1,3):= 2*DX(2)
COMM(1,5):= 2*DX(1)
COMM(1,6):= 8*DX(1)*X(1) + 4*DX(2)*X(2) - 2*DU(1)*U(1)
COMM(2,3):= DU(1)*U(1)
COMM(2,5):= DX(2)
COMM(2,6):= 4*DX(2)*X(1) + 2*DU(1)*U(1)*X(2)
COMM(3,5):= - (2*DX(2)*X(1) + DU(1)*U(1)*X(2))
2
COMM(5,6):= 8*DX(1)*X(1)
+ 8*DX(2)*X(2)*X(1)
2
+ 2*DU(1)*U(1)*(X(2) - 2*X(1))
The message "Constraints" which appears after the symmetry
generators are displayed means that the function c(1) depends on
x(1) and x(2) and satisfies the diffusion equation.
More examples which may used for test runs are given in the
Appendix.
If the user wants to test a certain ansatz of a symmetry generator
for given differential equations, the correct proceeding is as
follows. Create the determining system as described above. Make
the appropriate assignements for the generator and call PRSYS()
after that. The determining system with this ansatz substituted
is returned. Example: Assume again that the determining system
for the diffusion equation has been created. To check the
correctnes for example of generator GEN 3 which has been obtained
above, the assignments
XI(1):=0; XI(2):=2*X(1); ETA(1):=X(2)*U(1);
have to be made. If now PRSYS() is called all GL(K) are zero
proving the correctness of this generator.
Sometimes a user only wants to know some of the functions ZETA for
for various values of its possible arguments and given values of MM
and NN. In these cases the user has to assign the desired values of
MM and NN and may call the ZETA's after that. Example:
MM:=1; NN:=2;
FACTOR U(1,2),U(1,1),U(1,1,2),U(1,1,1);
ON LIST;
ZETA(1,1);
-U(1,2)*U(1,1)*DF(XI(2),U(1))
-U(1,2)*DF(XI(2),X(1))
2
-U(1,1) *DF(XI(1),U(1))
+U(1,1)*(DF(ETA(1),U(1)) -DF(XI(1),X(1)))
+DF(ETA(1),X(1))
ZETA(1,1,1);
-2*U(1,1,2)*U(1,1)*DF(XI(2),U(1))
-2*U(1,1,2)*DF(XI(2),X(1))
-U(1,1,1)*U(1,2)*DF(XI(2),U(1))
-3*U(1,1,1)*U(1,1)*DF(XI(1),U(1))
+U(1,1,1)*(DF(ETA(1),U(1)) -2*DF(XI(1),X(1)))
2
-U(1,2)*U(1,1) *DF(XI(2),U(1),2)
-2*U(1,2)*U(1,1)*DF(XI(2),U(1),X(1))
-U(1,2)*DF(XI(2),X(1),2)
3
-U(1,1) *DF(XI(1),U(1),2)
2
+U(1,1) *(DF(ETA(1),U(1),2) -2*DF(XI(1),U(1),X(1)))
+U(1,1)*(2*DF(ETA(1),U(1),X(1)) -DF(XI(1),X(1),2))
+DF(ETA(1),X(1),2)
If by error no values to MM or NN and have been assigned the message
***** Number of variables not defined
is returned. Often the functions ZETA are desired for special
values of its arguments ETA(ALFA) and XI(K). To this end they have
to be assigned first to some other variable. After that they may be
evaluated for the special arguments. In the previous example
this may be achieved by
Z11:=ZETA(1,1)$ Z111:=ZETA(1,1,1)$
Now assign the following values to XI 1, XI 2 and ETA 1:
XI 1:=4*X(1)**2; XI 2:=4*X(2)*X(1);
ETA 1:=U(1)*(X(2)**2 - 2*X(1));
They correspond to the generator GEN 6 of the diffusion equation
which has been obtained above. Now the desired expressions are
obtained by calling
Z11;
2
- (4*U(1,2)*X(2) - U(1,1)*X(2) + 10*U(1,1)*X(1) + 2*U(1))
Z111;
2
- (8*U(1,1,2)*X(2) - U(1,1,1)*X(2) + 18*U(1,1,1)*X(1) + 12*U(1,1))
%Appendix (Testfile).
%This appendix is a test file. The symmetry groups for various
%equations or systems of equations are determined. The variable
%PCLASS has the default value 0 and may be changed by the user
%before running it. The output may be compared with the results
%which are given in the references.
%The Burgers equations
deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
cresys deq 1$ simpsys()$ result()$
%The Kadomtsev-Petviashvili equation
deq 1:=3*u(1,3,3)+u(1,2,2,2,2)+6*u(1,2,2)*u 1
+6*u(1,2)**2+4*u(1,1,2)$
cresys deq 1$ simpsys()$ result()$
%The modified Kadomtsev-Petviashvili equation
deq 1:=u(1,1,2)-u(1,2,2,2,2)-3*u(1,3,3)
+6*u(1,2)**2*u(1,2,2)+6*u(1,3)*u(1,2,2)$
cresys deq 1$ simpsys()$ result()$
%The real- and the imaginary part of the nonlinear Schroedinger
%equation
deq 1:= u(1,1)+u(2,2,2)+2*u 1**2*u 2+2*u 2**3$
deq 2:=-u(2,1)+u(1,2,2)+2*u 1*u 2**2+2*u 1**3$
%Because this is not a single equation the two assignments
sder 1:=u(2,2,2)$ sder 2:=u(1,2,2)$
%are necessary.
cresys()$ simpsys()$ result()$
%The symmetries of the system comprising the four equations
deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
deq 2:=u(2,1)+u(2,2,2)$
deq 3:=u 1*u 2-2*u(2,2)$
deq 4:=4*u(2,1)+u 2*(u 1**2+2*u(1,2))$
sder 1:=u(1,2,2)$ sder 2:=u(2,2,2)$ sder 3:=u(2,2)$ sder 4:=u(2,1)$
%is obtained by calling
cresys()$ simpsys()$
df(c 5,x 1):=-df(c 5,x 2,2)$
df(c 5,x 2,x 1):=-df(c 5,x 2,3)$
simpsys()$ result()$
%The symmetries of the subsystem comprising equation 1 and 3 are
%obtained by
cresys(deq 1,deq 3)$ simpsys()$ result()$
%The result for all possible subsystems is discussed in detail in
%''Symmetries and Involution Systems: Some Experiments in Computer
%Algebra'', contribution to the Proceedings of the Oberwolfach
%Meeting on Nonlinear Evolution Equations, Summer 1986, to appear.