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The package SPDE provides a set of functions which may be used to determine the symmetry group of Lie- or point-symmetries of a given system of partial differential equations. In many cases the determining system is solved completely automatically. In other cases the user has to provide additional input information for the solution algorithm to terminate.
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: [Sch85b, Sch88, Sch87].
The symmetry analysis of partial differential equations logically falls into three parts. Accordingly the most important functions provided by the package are:
Some other useful functions for obtaining various kinds of output are:
There are several global variables defined by the system which should not be used for any other purpose than that given in Table 20.17 and 20.18. The three globals of the type integer are:
In addition there are the following global variables of type operator:
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 20.18. 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 20.16) 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 controlled 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 section 7.27). 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.
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 intermediate 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)
Constraint
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 “Constraint” 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 final section.
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 assignments 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 correctness 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 ZETAs 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))
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.
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