reduce-historical/r34/xlog/spde.log

Sat Jun 29 14:12:57 PDT 1991
REDUCE 3.4, 15-Jul-91 ...

1: 1: 
2: 2: 
3: 3:  %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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Quitting
Sat Jun 29 14:15:09 PDT 1991