Tue Feb 10 12:28:25 2004 run on Linux
%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 differential equation
DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
The symmetry generators are
GEN(1):=dx(1)
GEN(2):=dx(2)
GEN(3):=dx(2)*x(1) + du(1)
2
GEN(4):=dx(1)*x(1) + dx(2)*x(2)*x(1) + du(1)*( - u(1)*x(1) + x(2))
GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
The non-vanishing commutators of the finite subgroup
COMM(1,3):= dx(2)
COMM(1,4):= 2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
COMM(1,5):= 2*dx(1)
COMM(2,4):= dx(2)*x(1) + du(1)
COMM(2,5):= dx(2)
COMM(3,5):= - dx(2)*x(1) - du(1)
2
COMM(4,5):= - 2*dx(1)*x(1)
- 2*dx(2)*x(2)*x(1)
+ 2*du(1)*(u(1)*x(1) - x(2))
%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 differential equation
DEQ(1):=3*u(1,3,3)
+u(1,2,2,2,2)
+6*u(1,2,2)*u(1)
2
+6*u(1,2)
+4*u(1,1,2)
The symmetry generators are
GEN(1):=3*dx(2)*c(12) + 2*du(1)*df(c(12),x(1))
GEN(2):= 6*dx(2)*df(c(9),x(1))*x(3)
- 9*dx(3)*c(9)
+ 4*du(1)*df(c(9),x(1),2)*x(3)
GEN(3):= 27*dx(1)*xi(1)
2
3*dx(2)*( - 2*df(xi(1),x(1),2)*x(3) + 3*df(xi(1),x(1))*x(2))
+ 18*dx(3)*df(xi(1),x(1))*x(3)
2*du(1)*(
2
-2*df(xi(1),x(1),3)*x(3)
+3*df(xi(1),x(1),2)*x(2)
-9*df(xi(1),x(1))*u(1))
The remaining dependencies
xi(1) depends on x(1)
c(12) depends on x(1)
c(9) depends on x(1)
%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 differential equation
DEQ(1):=
-3*u(1,3,3)
+6*u(1,3)*u(1,2,2)
-u(1,2,2,2,2)
2
+6*u(1,2,2)*u(1,2)
+u(1,1,2)
The symmetry generators are
GEN(1):=du(1)*c(16)
GEN(2):=6*dx(2)*c(14) + du(1)*df(c(14),x(1))*x(3)
GEN(3):= 12*dx(2)*df(c(11),x(1))*x(3)
+ 72*dx(3)*c(11)
2
+ du(1)*(df(c(11),x(1),2)*x(3) + 6*df(c(11),x(1))*x(2))
GEN(4):= 324*dx(1)*xi(1)
2
+ 18*dx(2)*(df(xi(1),x(1),2)*x(3) + 6*df(xi(1),x(1))*x(2))
+ 216*dx(3)*df(xi(1),x(1))*x(3)
2
+ du(1)*x(3)*(df(xi(1),x(1),3)*x(3) + 18*df(xi(1),x(1),2)*x(2))
The remaining dependencies
xi(1) depends on x(1)
c(16) depends on x(1)
c(14) depends on x(1)
c(11) depends on x(1)
%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 differential equations
DEQ(1):=u(2,2,2)
3
+2*u(2)
2
+2*u(2)*u(1)
+u(1,1)
DEQ(2):=
-u(2,1)
2
+2*u(2) *u(1)
+u(1,2,2)
3
+2*u(1)
The symmetry generators are
GEN(1):=dx(1)
GEN(2):=dx(2)
GEN(3):=du(2)*u(1) + du(1)*u(2)
GEN(4):=2*dx(2)*x(1) - du(2)*u(1)*x(2) - du(1)*u(2)*x(2)
GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) + du(2)*u(2) - du(1)*u(1)
The non-vanishing commutators of the finite subgroup
COMM(1,4):= 2*dx(2)
COMM(1,5):= 2*dx(1)
COMM(2,4):= - du(2)*u(1) - du(1)*u(2)
COMM(2,5):= dx(2)
COMM(3,5):= 2*du(2)*u(1) - 2*du(1)*u(2)
COMM(4,5):= - 2*dx(2)*x(1) - du(2)*u(1)*x(2) + 3*du(1)*u(2)*x(2)
%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()$
Determining system is not completely solved
The remaining equations are
GL(1):=df(c(5),x(2),2) + df(c(5),x(1))
GL(2):=df(c(5),x(2),x(1)) + df(c(5),x(2),3)
The remaining dependencies
c(5) depends on x(1),x(2)
Number of functions is 21
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 differential equations
DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
DEQ(2):=u(2,2,2) + u(2,1)
DEQ(3):= - 2*u(2,2) + u(2)*u(1)
2
DEQ(4):=4*u(2,1) + 2*u(2)*u(1,2) + u(2)*u(1)
The symmetry generators are
GEN(1):=dx(1)
GEN(2):=dx(2)
GEN(3):=du(2)*u(2)
GEN(4):=2*dx(2)*x(1) + du(2)*u(2)*x(2) + 2*du(1)
2
GEN(5):= 4*dx(1)*x(1)
+ 4*dx(2)*x(2)*x(1)
4*du(1)*( - u(1)*x(1) + x(2))
2
+ du(2)*u(2)*(x(2) - 2*x(1))
GEN(6):=4*dx(1)*x(1) + 2*dx(2)*x(2) - du(2)*u(2) - 2*du(1)*u(1)
GEN(7):=du(2)*c(5)*u(2) + du(1)*(2*df(c(5),x(2)) - c(5)*u(1))
The remaining dependencies
c(5) depends on x(1),x(2)
Constraints
df(c(5),x(1)):= - df(c(5),x(2),2)
df(c(5),x(2),x(1)):= - df(c(5),x(2),3)
The non-vanishing commutators of the finite subgroup
COMM(1,4):= 2*dx(2)
COMM(1,5):= 8*dx(1)*x(1) + 4*dx(2)*x(2) - 2*du(2)*u(2) - 4*du(1)*u(1)
COMM(1,6):= 4*dx(1)
COMM(2,4):= du(2)*u(2)
COMM(2,5):= 4*dx(2)*x(1) + 2*du(2)*u(2)*x(2) + 4*du(1)
COMM(2,6):= 2*dx(2)
COMM(4,6):= - 4*dx(2)*x(1) - 2*du(2)*u(2)*x(2) - 4*du(1)
2
COMM(5,6):= - 16*dx(1)*x(1)
- 16*dx(2)*x(2)*x(1)
+ 16*du(1)*(u(1)*x(1) - x(2))
2
4*du(2)*u(2)*( - x(2) + 2*x(1))
%The symmetries of the subsystem comprising equation 1 and 3 are
%obtained by
cresys(deq 1,deq 3)$
simpsys()$
result()$
The differential equations
DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
DEQ(3):= - 2*u(2,2) + u(2)*u(1)
The symmetry generators are
GEN(1):=dx(1)
GEN(2):=dx(2)
GEN(3):=du(2)
GEN(4):=2*dx(2)*x(1) + du(2)*x(2) + 2*du(1)
GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
2
GEN(6):= 4*dx(1)*x(1)
+ 4*dx(2)*x(2)*x(1)
4*du(1)*( - u(1)*x(1) + x(2))
2
+ du(2)*x(2)
GEN(7):=du(2)*c(11)
The remaining dependencies
c(11) depends on x(1)
The non-vanishing commutators of the finite subgroup
COMM(1,4):= 2*dx(2)
COMM(1,5):= 2*dx(1)
COMM(1,6):= 8*dx(1)*x(1) + 4*dx(2)*x(2) - 4*du(1)*u(1)
COMM(2,4):= du(2)
COMM(2,5):= dx(2)
COMM(2,6):= 4*dx(2)*x(1) + 2*du(2)*x(2) + 4*du(1)
COMM(4,5):= - 2*dx(2)*x(1) - du(2)*x(2) - 2*du(1)
2
COMM(5,6):= 8*dx(1)*x(1)
+ 8*dx(2)*x(2)*x(1)
8*du(1)*( - u(1)*x(1) + x(2))
2
+ 2*du(2)*x(2)
%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.
end;
Time for test: 940 ms, plus GC time: 20 ms