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Sat May 30 16:25:35 PDT 1992 REDUCE 3.4.1, 15-Jul-92 ... 1: 1: 2: 2: (SPDE) 3: 3: 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; 5: 5: Time: 50473 ms plus GC time: 1513 ms 6: 6: Quitting Sat May 30 16:26:28 PDT 1992