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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... %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(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; (TIME: spde 19070 19790)