Sat May 30 16:25:35 PDT 1992
REDUCE 3.4.1, 15-Jul-92 ...
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(SPDE)
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Time: 85 ms
4: 4: %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)
+ DU(1)*X(3)
2
*(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(1)*U(2) + DU(2)*U(1)
GEN(4):=2*DX(2)*X(1) - DU(1)*U(2)*X(2) - DU(2)*U(1)*X(2)
GEN(5):=2*DX(1)*X(1) + DX(2)*X(2) - DU(1)*U(1) + DU(2)*U(2)
The non-vanishing commutators of the finite subgroup
COMM(1,4):= 2*DX(2)
COMM(1,5):= 2*DX(1)
COMM(2,4):= - DU(1)*U(2) - DU(2)*U(1)
COMM(2,5):= DX(2)
COMM(3,5):= - 2*DU(1)*U(2) + 2*DU(2)*U(1)
COMM(4,5):= - 2*DX(2)*X(1) + 3*DU(1)*U(2)*X(2) - DU(2)*U(1)*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) + 2*DU(1) + DU(2)*U(2)*X(2)
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) - 2*DU(1)*U(1) - DU(2)*U(2)
GEN(7):=DU(1)*(2*DF(C(5),X(2)) - C(5)*U(1)) + DU(2)*C(5)*U(2)
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) - 4*DU(1)*U(1) - 2*DU(2)*U(2)
COMM(1,6):= 4*DX(1)
COMM(2,4):= DU(2)*U(2)
COMM(2,5):= 4*DX(2)*X(1) + 4*DU(1) + 2*DU(2)*U(2)*X(2)
COMM(2,6):= 2*DX(2)
COMM(4,6):= - 4*DX(2)*X(1) - 4*DU(1) - 2*DU(2)*U(2)*X(2)
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) + 2*DU(1) + DU(2)*X(2)
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) + 4*DU(1) + 2*DU(2)*X(2)
COMM(4,5):= - 2*DX(2)*X(1) - 2*DU(1) - DU(2)*X(2)
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;
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Time: 50473 ms plus GC time: 1513 ms
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Quitting
Sat May 30 16:26:28 PDT 1992