Tue Apr 15 00:33:02 2008 run on win32
%*******************************************************************%
%                                                                   %
%                      L I E P D E . T S T                          %
%                      -------------------                          %
%  liepde.tst contains test examples for the program liepde.red.    %
%                                                                   %
%  Author of this file: Thomas Wolf                                 %
%  Date:  21. April 1998, 6. May 2003                               %
%                                                                   %
%  Details about the syntax of liepde.red are given in liepde.tex.  %
%                                                                   %
%  To run this demo you need to load liepde and crack through       %
%     load crack,liepde$                                            %
%  and to read in this file as                                      %
%     in "liepde.tst";                                              %
%  If you got the source code of a newer version of liepde then     %
%  either read it in through                                        %
%     in "crack.red","liepde.red"$                                  %
%  (with the appropriate directory name in front of liepde.red)     %
%  or, to speed up the calculation, you compile before with         %
%     faslout "crack"$                                              %
%     in "crack.red"$                                               %
%     faslend$                                                      %
%     faslout "liepde"$                                             %
%     in "liepde.red"$                                              %
%     faslend$                                                      %
%  and then load both it with                                       %
%     load crack,liepde$                                            %
%                                                                   %
%*******************************************************************%

load crack;



lisp(depl!*:=nil)$

     % clearing of all dependences
setcrackflags()$


lisp(print_:=nil)$


on dfprint$



comment
-------------------------------------------------------
The following runs demonstrate the program LIEPDE for 
the computation of infinitesimal symmetries. Times given
below refer to a 8 MB session under LINUX on a 133 MHz
Pentium PC with the CRACK version of April 1998 running
PSL Reduce.
-------------------------------------------------------;


lisp(prelim_:=nil)$

      % not necessary as this is the default value
lisp(individual_:=nil)$

  % not necessary as this is the default value

comment
-------------------------------------------------------
The first example is a single ODE with a parametric
function f=f(x) for which point symmetries are to be
determined.
(Time ~ 6 sec.);

write"-------------------------------------------------------";


-------------------------------------------------------


lisp(freeint_:=nil)$

 % This enables the solution of differential equ.s in
                     % which unevaluated integrals remain. This becomes
                     % necessary in this example through the parametric
                     % function f=f(x)
depend y,x$


depend f,x$


liepde({df(y,x,2)=-(y+3*f)*df(y,x)+y**3-f*y**2-(2*f**2+df(f,x))*y, 
        {y}, {x}},
       {"point"},{},{})$



The ODE under investigation is :

                                 2        2    3
y  = - f *y - 3*y *f - y *y - 2*f *y - f*y  + y
 2x     x        x      x

for the function(s) : 

y(x)



The symmetries are:

-------- 1. Symmetry:

         int(f,x)
xi_x= - e

       int(f,x)
eta_y=e        *f*y

-------- 2. Symmetry:

         int(f,x)          1
xi_x= - e        *int(-----------,x)
                        int(f,x)
                       e

       int(f,x)          1
eta_y=e        *int(-----------,x)*f*y + y
                      int(f,x)
                     e

-------- 


{{},

           int(f,x)          1                int(f,x)
 {xi_x= - e        *int(-----------,x)*c_5 - e        *c_6,
                          int(f,x)
                         e

         int(f,x)          1                    int(f,x)
  eta_y=e        *int(-----------,x)*c_5*f*y + e        *c_6*f*y + c_5*y},
                        int(f,x)
                       e

 {c_6,c_5}}

nodepnd {y,f}$


lisp(freeint_:=t)$

   % Because the simplification of differential
                     % expressions which involve unevaluated integrals
                     % may provide difficulties such solutions involving
                     % unevaluated integrals are disabled.

comment
-------------------------------------------------------
The following example demonstrates a number of things.
The Burgers equation is investigated concerning third
order symmetries. The equation is used to substitute
df(u,t) and all derivatives of df(u,t). This computation
also shows that any equations that remain unsolved are
returned, like in this case the heat quation.
(Time ~ 15 sec.);

write"-------------------------------------------------------";


-------------------------------------------------------


nodepnd {u}$


depend u,t,x$


liepde({df(u,t)=df(u,x,2)+df(u,x)**2,{u},{t,x}},{"general",3},{},{})$



The PDE under investigation is :

           2
u =u   + u
 t  2x    x

for the function(s) : 

u(t,x)



The symmetries are:

-------- 1. Symmetry:

xi_t=0

xi_x=0

                    2
eta_u=2*u  *t + 2*u  *t + u *x
         2x        x       x

-------- 2. Symmetry:

xi_t=0

xi_x=0

             2       2  2                     2
eta_u=4*u  *t  + 4*u  *t  + 4*u *t*x + 2*t + x
         2x         x          x

-------- 3. Symmetry:

xi_t=0

xi_x=0

              2
eta_u=u   + u
       2x    x

-------- 4. Symmetry:

xi_t=0

xi_x=0

                                         3       2
eta_u=2*u  *t + 6*u  *u *t + u  *x + 2*u  *t + u  *x
         3x        2x  x      2x        x       x

-------- 5. Symmetry:

xi_t=0

xi_x=0

             3              3           2         3  3        2  2            2
eta_u=8*u  *t  + 24*u  *u *t  + 12*u  *t *x + 8*u  *t  + 12*u  *t *x + 12*u *t
         3x          2x  x          2x           x           x             x

           2            3
 + 6*u *t*x  + 6*t*x + x
      x

-------- 6. Symmetry:

xi_t=0

xi_x=0

             2              2                   3  2       2           2
eta_u=4*u  *t  + 12*u  *u *t  + 4*u  *t*x + 4*u  *t  + 4*u  *t*x + u *x  - x
         3x          2x  x         2x          x          x         x

-------- 7. Symmetry:

xi_t=0

xi_x=0

                         3
eta_u=u   + 3*u  *u  + u
       3x      2x  x    x

-------- 8. Symmetry:

xi_t=0

xi_x=0

eta_u=2*u *t + x
         x

-------- 9. Symmetry:

xi_t=0

xi_x=0

eta_u=u
       x

-------- 10. Symmetry:

xi_t=0

xi_x=0

eta_u=1

-------- 

Further symmetries:

xi_t=0

xi_x=0

       c_25
eta_u=------
         u
        e

with c_25(t,x)
which still have to satisfy: 

0=c_25  - c_25
      t       2x


{{c_25  - c_25  },
      t       2x

 {xi_t=0,

  xi_x=0,

          u               u           2      u           3      u
  eta_u=(e *u  *c_31 + 4*e *u  *c_32*t  + 8*e *u  *c_33*t  + 2*e *u  *c_34*t
             3x              3x                 3x                 3x

               u                   u              2       u              3
          + 3*e *u  *u *c_31 + 12*e *u  *u *c_32*t  + 24*e *u  *u *c_33*t
                  2x  x               2x  x                  2x  x

               u                  u               u           2
          + 6*e *u  *u *c_34*t + e *u  *c_28 + 4*e *u  *c_29*t
                  2x  x              2x              2x

               u                 u                    u           2
          + 2*e *u  *c_30*t + 4*e *u  *c_32*t*x + 12*e *u  *c_33*t *x
                  2x                2x                   2x

             u               u   3           u   3       2      u   3       3
          + e *u  *c_34*x + e *u  *c_31 + 4*e *u  *c_32*t  + 8*e *u  *c_33*t
                2x              x               x                  x

               u   3           u   2           u   2       2      u   2
          + 2*e *u  *c_34*t + e *u  *c_28 + 4*e *u  *c_29*t  + 2*e *u  *c_30*t
                  x               x               x                  x

               u   2                u   2       2      u   2
          + 4*e *u  *c_32*t*x + 12*e *u  *c_33*t *x + e *u  *c_34*x
                  x                    x                  x

               u                u              u          2       u          2
          + 4*e *u *c_29*t*x + e *u *c_30*x + e *u *c_32*x  + 12*e *u *c_33*t
                  x                x              x                  x

               u            2    u              u                u
          + 6*e *u *c_33*t*x  + e *u *c_35 + 2*e *u *c_36*t + 2*e *c_29*t
                  x                 x              x

             u       2    u             u             u       3    u
          + e *c_29*x  - e *c_32*x + 6*e *c_33*t*x + e *c_33*x  + e *c_36*x

             u               u
          + e *c_37 + c_25)/e },

 {c_25,

  c_30,

  c_29,

  c_28,

  c_34,

  c_33,

  c_32,

  c_31,

  c_36,

  c_35,

  c_37}}


comment
-------------------------------------------------------
Now the same equation is investigated, this time only
df(u,x,2) and its derivatives are substituted. As a
consequence less jet-variables (u-derivatives of lower
order) are generated in the process of formulating the
symmetry conditions. Less jet-variables in which the
conditions have to be fulfilled identically means less
overdetermined conditions and more solutions which to
compute takes longer than before.
(Time ~ 85 sec.);

write"-------------------------------------------------------";


-------------------------------------------------------


liepde({df(u,x,2)=df(u,t)-df(u,x)**2,{u},{t,x}},{"general",3},{},{})$



The PDE under investigation is :

           2
u  =u  - u
 2x  t    x

for the function(s) : 

u(t,x)



The symmetries are:

-------- 1. Symmetry:

xi_t=0

xi_x=0

                                                                     2
eta_u= - 8*u   *u *t + u    - 4*u    *t - 4*u  *u *t - 2*u  *x - 4*u  *u *t
            t,x  t      t,x      2t,x        2t  x        2t        t   x

       2
 - 2*u  *x + u *u
      t       t  x

-------- 2. Symmetry:

xi_t=0

xi_x=0

                     3            2              2             3              3
eta_u= - 32*u   *u *t  + 12*u   *t  - 12*u   *t*x  - 16*u    *t  - 16*u  *u *t
             t,x  t          t,x          t,x            2t,x          2t  x

           2          2     3        2  2               2               2
 - 24*u  *t *x - 16*u  *u *t  - 24*u  *t *x + 12*u *u *t  - 12*u *u *t*x
       2t            t   x          t             t  x          t  x

                     3         2
 + 24*u *t*x - 2*u *x  + 9*u *x  - 9*x
       t          t         x

-------- 3. Symmetry:

xi_t=0

xi_x=0

                     5             4            3  2             5
eta_u= - 64*u   *u *t  - 160*u   *t  - 80*u   *t *x  - 32*u    *t
             t,x  t           t,x          t,x             2t,x

              5           4          2     5        2  4                4
 - 32*u  *u *t  - 80*u  *t *x - 32*u  *u *t  - 80*u  *t *x - 160*u *u *t
       2t  x          2t            t   x          t              t  x

             3  2           3            2  3           3           2  2
 - 80*u *u *t *x  - 240*u *t *x - 40*u *t *x  - 120*u *t  - 120*u *t *x
       t  x              t            t              x           x

            4       2           3    5
 - 10*u *t*x  - 60*t *x - 20*t*x  - x
       x

-------- 4. Symmetry:

xi_t=0

xi_x=0

                     4            3            2  2             4              4
eta_u= - 32*u   *u *t  + 16*u   *t  - 24*u   *t *x  - 16*u    *t  - 16*u  *u *t
             t,x  t          t,x          t,x             2t,x          2t  x

           3          2     4        2  3               3             2  2
 - 32*u  *t *x - 16*u  *u *t  - 32*u  *t *x + 16*u *u *t  - 24*u *u *t *x
       2t            t   x          t             t  x          t  x

          2             3          2            2       4               3
 + 48*u *t *x - 8*u *t*x  + 84*u *t  + 36*u *t*x  - u *x  + 48*t*x + 8*x
       t           t            x          x         x

-------- 5. Symmetry:

xi_t=0

xi_x=0

                    2                    2            2             2
eta_u= - 8*u   *u *t  + 2*u   *t - u   *x  - 4*u    *t  - 4*u  *u *t
            t,x  t         t,x      t,x         2t,x         2t  x

                   2     2       2                          2
 - 4*u  *t*x - 4*u  *u *t  - 4*u  *t*x + 2*u *u *t - u *u *x  + 2*u *x
      2t          t   x         t           t  x      t  x         t

-------- 6. Symmetry:

xi_t=0

xi_x=0

                                        2
eta_u= - 2*u   *u  - u     - u  *u  - u  *u
            t,x  t    2t,x    2t  x    t   x

-------- 7. Symmetry:

xi_t=0

xi_x=0

eta_u

                                                                  3       2
= - 2*u   *u *x - 2*u  *t - u    *x - 6*u  *u *t - u  *u *x - 2*u  *t - u  *u *x
       t,x  t        3t      2t,x        2t  t      2t  x        t       t   x

-------- 8. Symmetry:

xi_t=0

xi_x=0

                     2                           3           3             2
eta_u= - 48*u   *u *t *x + 12*u   *t*x - 2*u   *x  - 16*u  *t  - 24*u    *t *x
             t,x  t            t,x          t,x          3t          2t,x

              3              2               2        3  3        2     2
 - 48*u  *u *t  - 24*u  *u *t *x - 12*u  *t*x  - 16*u  *t  - 24*u  *u *t *x
       2t  t          2t  x            2t            t           t   x

        2    2                           3         2
 - 12*u  *t*x  + 12*u *u *t*x - 2*u *u *x  + 6*u *x  - 6*u *x + 3
       t             t  x          t  x         t         x

-------- 9. Symmetry:

xi_t=0

xi_x=0

                      4              3              2  3           5
eta_u= - 160*u   *u *t *x + 80*u   *t *x - 40*u   *t *x  - 32*u  *t
              t,x  t            t,x            t,x             3t

             4                5              4             3  2        3  5
 - 80*u    *t *x - 96*u  *u *t  - 80*u  *u *t *x - 80*u  *t *x  - 32*u  *t
       2t,x            2t  t          2t  x            2t             t

        2     4          2  3  2             3               2  3           3
 - 80*u  *u *t *x - 80*u  *t *x  + 80*u *u *t *x - 40*u *u *t *x  + 360*u *t
       t   x            t              t  x            t  x              t

           2  2            4           2              3       5        2
 + 120*u *t *x  - 10*u *t*x  + 420*u *t *x + 60*u *t*x  - u *x  + 120*t
        t             t             x            x         x

          2       4
 + 120*t*x  + 10*x

-------- 10. Symmetry:

xi_t=0

xi_x=0

                      5               4               3  3           6
eta_u= - 384*u   *u *t *x - 960*u   *t *x - 160*u   *t *x  - 64*u  *t
              t,x  t             t,x             t,x             3t

              5                 6               5              5            4  2
 - 192*u    *t *x - 192*u  *u *t  - 192*u  *u *t *x - 480*u  *t  - 240*u  *t *x
        2t,x             2t  t           2t  x             2t           2t

        3  6         2     5           2  5         2  4  2              4
 - 64*u  *t  - 192*u  *u *t *x - 480*u  *t  - 240*u  *t *x  - 960*u *u *t *x
       t            t   x             t            t               t  x

              3  3           4           3  2          2  4           3
 - 160*u *u *t *x  - 720*u *t  - 720*u *t *x  - 60*u *t *x  - 720*u *t *x
        t  x              t           t             t              x

           2  3            5        3        2  2         4    6
 - 240*u *t *x  - 12*u *t*x  - 120*t  - 180*t *x  - 30*t*x  - x
        x             x

-------- 11. Symmetry:

xi_t=0

xi_x=0

                     3              2               3           4
eta_u= - 64*u   *u *t *x + 24*u   *t *x - 8*u   *t*x  - 16*u  *t
             t,x  t            t,x           t,x            3t

             3                4              3             2  2        3  4
 - 32*u    *t *x - 48*u  *u *t  - 32*u  *u *t *x - 24*u  *t *x  - 16*u  *t
       2t,x            2t  t          2t  x            2t             t

        2     3          2  2  2             2                3            2
 - 32*u  *u *t *x - 24*u  *t *x  + 24*u *u *t *x - 8*u *u *t*x  + 24*u *t*x
       t   x            t              t  x           t  x            t

       4                     3              2
 - u *x  - 24*u *t*x + 6*u *x  - 30*t - 15*x
    t          x          x

-------- 12. Symmetry:

xi_t=0

xi_x=0

                                         2                            2
eta_u= - 8*u   *u *t*x + u   *x - 4*u  *t  - 4*u    *t*x - 12*u  *u *t
            t,x  t        t,x        3t         2t,x           2t  t

                       2       3  2       2            2  2
 - 4*u  *u *t*x - u  *x  - 4*u  *t  - 4*u  *u *t*x - u  *x  + u *u *x
      2t  x        2t         t          t   x        t        t  x

-------- 13. Symmetry:

xi_t=0

xi_x=0

                            3
eta_u= - u   - 3*u  *u  - u
          3t      2t  t    t

-------- 14. Symmetry:

xi_t=0

xi_x=0

                                2
eta_u= - u   *x - 2*u  *t - 2*u  *t - u *u *x
          t,x        2t        t       t  x

-------- 15. Symmetry:

xi_t=0

xi_x=0

                  2            3       2  3             2             2
eta_u= - 12*u   *t *x - 8*u  *t  - 8*u  *t  - 12*u *u *t *x - 6*u *t*x
             t,x           2t         t           t  x           t

                  3            2
 + 6*u *t*x - u *x  + 6*t + 3*x
      x        x

-------- 16. Symmetry:

xi_t=0

xi_x=0

                  3             4        2  4             3            3
eta_u= - 32*u   *t *x - 16*u  *t  - 16*u  *t  - 32*u *u *t *x - 48*u *t
             t,x            2t          t           t  x            t

          2  2          2             3       2         2    4
 - 24*u *t *x  - 48*u *t *x - 8*u *t*x  - 12*t  - 12*t*x  - x
       t             x           x

-------- 17. Symmetry:

xi_t=0

xi_x=0

                             2       2  2                       2
eta_u= - 8*u   *t*x - 8*u  *t  - 8*u  *t  - 8*u *u *t*x - 2*u *x  + 2*u *x - 1
            t,x          2t         t          t  x          t         x

-------- 18. Symmetry:

xi_t=0

xi_x=0

              2
eta_u=u   + u
       2t    t

-------- 19. Symmetry:

xi_t=0

xi_x=0

eta_u= - 2*u   *t - 2*u *u *t - u *x
            t,x        t  x      t

-------- 20. Symmetry:

xi_t=0

xi_x=0

                 3            3          2            2           2            3
eta_u= - 8*u   *t  - 8*u *u *t  - 12*u *t *x - 12*u *t  - 6*u *t*x  - 6*t*x - x
            t,x         t  x          t            x         x

-------- 21. Symmetry:

xi_t=0

xi_x=0

                 2            2                  2
eta_u= - 4*u   *t  - 4*u *u *t  - 4*u *t*x - u *x  + x
            t,x         t  x         t        x

-------- 22. Symmetry:

xi_t=0

xi_x=0

eta_u=u    + u *u
       t,x    t  x

-------- 23. Symmetry:

xi_t=0

xi_x=0

eta_u= - 2*u *t - x
            x

-------- 24. Symmetry:

xi_t=0

xi_x=0

eta_u=u
       x

-------- 25. Symmetry:

xi_t=0

xi_x=0

eta_u= - 4*u *t - 2*u *x + 1
            t        x

-------- 26. Symmetry:

xi_t=0

xi_x=0

               2                     2
eta_u= - 4*u *t  - 4*u *t*x - 2*t - x
            t         x

-------- 27. Symmetry:

xi_t=0

xi_x=0

eta_u=u
       t

-------- 28. Symmetry:

xi_t=0

xi_x=0

eta_u=1

-------- 

Further symmetries:

xi_t=0

xi_x=0

       c_94
eta_u=------
         u
        e

with c_94(t,x)
which still have to satisfy: 

0= - c_94  + c_94
         t       2x


{{ - c_94  + c_94  },
         t       2x

 {xi_t=0,

  xi_x=0,

               u                    u                2       u                4
  eta_u=( - 2*e *u   *u *c_100 - 8*e *u   *u *c_101*t  - 32*e *u   *u *c_102*t
                  t,x  t               t,x  t                   t,x  t

                u                5       u                3
          - 64*e *u   *u *c_103*t  - 32*e *u   *u *c_104*t
                   t,x  t                   t,x  t

               u                      u
          - 8*e *u   *u *c_105*t - 8*e *u   *u *c_107*t*x
                  t,x  t                 t,x  t

                u                3          u                5
          - 64*e *u   *u *c_108*t *x - 384*e *u   *u *c_109*t *x
                   t,x  t                      t,x  t

                 u                4         u                2
          - 160*e *u   *u *c_110*t *x - 48*e *u   *u *c_111*t *x
                    t,x  t                     t,x  t

               u                      u                 u             2
          - 2*e *u   *u *c_112*x + 2*e *u   *c_101*t - e *u   *c_101*x
                  t,x  t                 t,x               t,x

                u             3       u             2  2        u             4
          + 16*e *u   *c_102*t  - 24*e *u   *c_102*t *x  - 160*e *u   *c_103*t
                   t,x                   t,x                       t,x

                u             3  2       u             2       u               2
          - 80*e *u   *c_103*t *x  + 12*e *u   *c_104*t  - 12*e *u   *c_104*t*x
                   t,x                      t,x                   t,x

             u               u                    u             2
          + e *u   *c_105 + e *u   *c_107*x + 24*e *u   *c_108*t *x
                t,x             t,x                  t,x

               u               3        u             4
          - 8*e *u   *c_108*t*x  - 960*e *u   *c_109*t *x
                  t,x                      t,x

                 u             3  3       u             3
          - 160*e *u   *c_109*t *x  + 80*e *u   *c_110*t *x
                    t,x                      t,x

                u             2  3       u                     u             3
          - 40*e *u   *c_110*t *x  + 12*e *u   *c_111*t*x - 2*e *u   *c_111*x
                   t,x                      t,x                   t,x

               u                      u             3         u             2
          - 8*e *u   *c_115*t*x - 32*e *u   *c_116*t *x - 12*e *u   *c_117*t *x
                  t,x                    t,x                     t,x

             u                 u                 u             2
          - e *u   *c_118*x + e *u   *c_119 - 4*e *u   *c_120*t
                t,x               t,x               t,x

               u             3      u                 u
          - 8*e *u   *c_121*t  - 2*e *u   *c_122*t - e *u  *c_106
                  t,x                  t,x               3t

               u            2       u            4       u            6
          - 4*e *u  *c_107*t  - 16*e *u  *c_108*t  - 64*e *u  *c_109*t
                  3t                   3t                   3t

                u            5       u            3      u
          - 32*e *u  *c_110*t  - 16*e *u  *c_111*t  - 2*e *u  *c_112*t
                   3t                   3t                  3t

             u                  u              2       u              4
          - e *u    *c_100 - 4*e *u    *c_101*t  - 16*e *u    *c_102*t
                2t,x               2t,x                   2t,x

                u              5       u              3      u
          - 32*e *u    *c_103*t  - 16*e *u    *c_104*t  - 4*e *u    *c_105*t
                   2t,x                   2t,x                  2t,x

               u                       u              3
          - 4*e *u    *c_107*t*x - 32*e *u    *c_108*t *x
                  2t,x                    2t,x

                 u              5         u              4
          - 192*e *u    *c_109*t *x - 80*e *u    *c_110*t *x
                    2t,x                     2t,x

                u              2      u                    u
          - 24*e *u    *c_111*t *x - e *u    *c_112*x - 3*e *u  *u *c_106
                   2t,x                  2t,x                 2t  t

                u               2       u               4
          - 12*e *u  *u *c_107*t  - 48*e *u  *u *c_108*t
                   2t  t                   2t  t

                 u               6       u               5
          - 192*e *u  *u *c_109*t  - 96*e *u  *u *c_110*t
                    2t  t                   2t  t

                u               3      u                   u
          - 48*e *u  *u *c_111*t  - 6*e *u  *u *c_112*t - e *u  *u *c_100
                   2t  t                  2t  t               2t  x

               u               2       u               4       u               5
          - 4*e *u  *u *c_101*t  - 16*e *u  *u *c_102*t  - 32*e *u  *u *c_103*t
                  2t  x                   2t  x                   2t  x

                u               3      u                     u
          - 16*e *u  *u *c_104*t  - 4*e *u  *u *c_105*t - 4*e *u  *u *c_107*t*x
                   2t  x                  2t  x                 2t  x

                u               3          u               5
          - 32*e *u  *u *c_108*t *x - 192*e *u  *u *c_109*t *x
                   2t  x                      2t  x

                u               4         u               2
          - 80*e *u  *u *c_110*t *x - 24*e *u  *u *c_111*t *x
                   2t  x                     2t  x

             u                     u                     u            3
          - e *u  *u *c_112*x - 4*e *u  *c_101*t*x - 32*e *u  *c_102*t *x
                2t  x                 2t                    2t

                u            4         u            2        u
          - 80*e *u  *c_103*t *x - 24*e *u  *c_104*t *x - 2*e *u  *c_105*x
                   2t                     2t                    2t

             u            2       u            2  2        u            5
          - e *u  *c_107*x  - 24*e *u  *c_108*t *x  - 480*e *u  *c_109*t
                2t                   2t                       2t

                 u            4  2       u            3  2
          - 240*e *u  *c_109*t *x  - 80*e *u  *c_110*t *x
                    2t                      2t

                u              2    u                u            2
          - 12*e *u  *c_111*t*x  + e *u  *c_114 - 8*e *u  *c_115*t
                   2t                  2t               2t

                u            4      u            3      u
          - 16*e *u  *c_116*t  - 8*e *u  *c_117*t  - 2*e *u  *c_118*t
                   2t                  2t                  2t

             u   3            u   3        2       u   3        4
          - e *u  *c_106 - 4*e *u  *c_107*t  - 16*e *u  *c_108*t
                t                t                    t

                u   3        6       u   3        5       u   3        3
          - 64*e *u  *c_109*t  - 32*e *u  *c_110*t  - 16*e *u  *c_111*t
                   t                    t                    t

               u   3            u   2               u   2           2
          - 2*e *u  *c_112*t - e *u  *u *c_100 - 4*e *u  *u *c_101*t
                  t                t   x               t   x

                u   2           4       u   2           5
          - 16*e *u  *u *c_102*t  - 32*e *u  *u *c_103*t
                   t   x                   t   x

                u   2           3      u   2                 u   2
          - 16*e *u  *u *c_104*t  - 4*e *u  *u *c_105*t - 4*e *u  *u *c_107*t*x
                   t   x                  t   x                 t   x

                u   2           3          u   2           5
          - 32*e *u  *u *c_108*t *x - 192*e *u  *u *c_109*t *x
                   t   x                      t   x

                u   2           4         u   2           2
          - 80*e *u  *u *c_110*t *x - 24*e *u  *u *c_111*t *x
                   t   x                     t   x

             u   2                 u   2                 u   2        3
          - e *u  *u *c_112*x - 4*e *u  *c_101*t*x - 32*e *u  *c_102*t *x
                t   x                 t                     t

                u   2        4         u   2        2        u   2
          - 80*e *u  *c_103*t *x - 24*e *u  *c_104*t *x - 2*e *u  *c_105*x
                   t                      t                     t

             u   2        2       u   2        2  2        u   2        5
          - e *u  *c_107*x  - 24*e *u  *c_108*t *x  - 480*e *u  *c_109*t
                t                    t                        t

                 u   2        4  2       u   2        3  2
          - 240*e *u  *c_109*t *x  - 80*e *u  *c_110*t *x
                    t                       t

                u   2          2    u   2            u   2        2
          - 12*e *u  *c_111*t*x  + e *u  *c_114 - 8*e *u  *c_115*t
                   t                   t                t

                u   2        4      u   2        3      u   2
          - 16*e *u  *c_116*t  - 8*e *u  *c_117*t  - 2*e *u  *c_118*t
                   t                   t                   t

               u                  u              2       u              3
          + 2*e *u *u *c_101*t - e *u *u *c_101*x  + 16*e *u *u *c_102*t
                  t  x               t  x                   t  x

                u              2  2        u              4
          - 24*e *u *u *c_102*t *x  - 160*e *u *u *c_103*t
                   t  x                       t  x

                u              3  2       u              2
          - 80*e *u *u *c_103*t *x  + 12*e *u *u *c_104*t
                   t  x                      t  x

                u                2    u                u
          - 12*e *u *u *c_104*t*x  + e *u *u *c_105 + e *u *u *c_107*x
                   t  x                  t  x             t  x

                u              2        u                3
          + 24*e *u *u *c_108*t *x - 8*e *u *u *c_108*t*x
                   t  x                    t  x

                 u              4          u              3  3
          - 960*e *u *u *c_109*t *x - 160*e *u *u *c_109*t *x
                    t  x                      t  x

                u              3         u              2  3
          + 80*e *u *u *c_110*t *x - 40*e *u *u *c_110*t *x
                   t  x                     t  x

                u                      u              3      u
          + 12*e *u *u *c_111*t*x - 2*e *u *u *c_111*x  - 8*e *u *u *c_115*t*x
                   t  x                   t  x                  t  x

                u              3         u              2      u
          - 32*e *u *u *c_116*t *x - 12*e *u *u *c_117*t *x - e *u *u *c_118*x
                   t  x                     t  x                  t  x

             u                  u              2      u              3
          + e *u *u *c_119 - 4*e *u *u *c_120*t  - 8*e *u *u *c_121*t
                t  x               t  x                  t  x

               u                    u                  u           2
          - 2*e *u *u *c_122*t + 2*e *u *c_101*x + 48*e *u *c_102*t *x
                  t  x                 t                  t

               u             3        u           3         u           2  3
          - 8*e *u *c_102*t*x  - 240*e *u *c_103*t *x - 40*e *u *c_103*t *x
                  t                      t                     t

                u                   u           3       u             2
          + 24*e *u *c_104*t*x - 2*e *u *c_104*x  + 24*e *u *c_108*t*x
                   t                   t                   t

             u           4        u           4        u           3  2
          - e *u *c_108*x  - 720*e *u *c_109*t  - 720*e *u *c_109*t *x
                t                    t                    t

                u           2  4        u           3        u           2  2
          - 60*e *u *c_109*t *x  + 360*e *u *c_110*t  + 120*e *u *c_110*t *x
                   t                       t                    t

                u             4      u           2      u           2
          - 10*e *u *c_110*t*x  + 6*e *u *c_111*x  - 2*e *u *c_115*x
                   t                    t                  t

                u           3       u           2  2      u             2
          - 48*e *u *c_116*t  - 24*e *u *c_116*t *x  - 6*e *u *c_117*t*x
                   t                   t                     t

               u                    u           2      u
          - 4*e *u *c_120*t*x - 12*e *u *c_121*t *x - e *u *c_122*x
                  t                    t                  t

             u               u           2      u                  u           2
          + e *u *c_125 - 4*e *u *c_126*t  - 4*e *u *c_127*t + 84*e *u *c_102*t
                t               t                  t                  x

                u             2    u           4        u           3
          + 36*e *u *c_102*t*x  - e *u *c_102*x  - 120*e *u *c_103*t
                   x                  x                    x

                 u           2  2       u             4      u           2
          - 120*e *u *c_103*t *x  - 10*e *u *c_103*t*x  + 9*e *u *c_104*x
                    x                      x                    x

                u                   u           3        u           3
          - 24*e *u *c_108*t*x + 6*e *u *c_108*x  - 720*e *u *c_109*t *x
                   x                   x                    x

                 u           2  3       u             5        u           2
          - 240*e *u *c_109*t *x  - 12*e *u *c_109*t*x  + 420*e *u *c_110*t *x
                    x                      x                      x

                u             3    u           5      u
          + 60*e *u *c_110*t*x  - e *u *c_110*x  - 6*e *u *c_111*x
                   x                  x                  x

               u                  u           2        u             3
          + 2*e *u *c_115*x - 48*e *u *c_116*t *x - 8*e *u *c_116*t*x
                  x                  x                    x

               u                 u           3    u           2
          + 6*e *u *c_117*t*x - e *u *c_117*x  - e *u *c_120*x
                  x                 x                x

                u           2      u             2    u
          - 12*e *u *c_121*t  - 6*e *u *c_121*t*x  + e *u *c_123
                   x                  x                  x

               u                 u                   u
          - 2*e *u *c_124*t - 4*e *u *c_126*t*x - 2*e *u *c_127*x
                  x                 x                   x

                u                u        3       u        2
          + 48*e *c_102*t*x + 8*e *c_102*x  - 60*e *c_103*t *x

                u          3    u        5      u               u
          - 20*e *c_103*t*x  - e *c_103*x  - 9*e *c_104*x - 30*e *c_108*t

                u        2        u        3        u        2  2
          - 15*e *c_108*x  - 120*e *c_109*t  - 180*e *c_109*t *x

                u          4    u        6        u        2        u          2
          - 30*e *c_109*t*x  - e *c_109*x  + 120*e *c_110*t  + 120*e *c_110*t*x

                u        4      u          u             u        2
          + 10*e *c_110*x  + 3*e *c_111 - e *c_115 - 12*e *c_116*t

                u          2    u        4      u              u        2
          - 12*e *c_116*t*x  - e *c_116*x  + 6*e *c_117*t + 3*e *c_117*x

             u              u              u        3    u
          + e *c_120*x - 6*e *c_121*t*x - e *c_121*x  - e *c_124*x

               u            u        2    u          u                u
          - 2*e *c_126*t - e *c_126*x  + e *c_127 + e *c_128 + c_94)/e },

 {c_94,

  c_105,

  c_104,

  c_103,

  c_102,

  c_101,

  c_100,

  c_112,

  c_111,

  c_110,

  c_109,

  c_108,

  c_107,

  c_106,

  c_118,

  c_117,

  c_116,

  c_115,

  c_114,

  c_122,

  c_121,

  c_120,

  c_119,

  c_124,

  c_123,

  c_127,

  c_126,

  c_125,

  c_128}}

nodepnd {u}$



comment
-------------------------------------------------------
The following example includes the Karpman equations
for three unknown functions in 4 variables. 

If point symmetries are to be computed for a single
equation or a system of equations of higher than first
order then there is the option to formulate at first
preliminary conditions for each equation, have CRACK
solving these conditions before the full set of conditions
is formulated and solved. This strategy is adopted if a
lisp flag prelim_ has the value t. The default value
is nil. 

Similarly, if a system of equations is to be investigated
and a flag individual_ has the value t then symmetry
conditions are formulated and investigated for each
individual equation successively. The default value is nil.

It is advantageous to split a large set of conditions
into smaller sets to be investigated successively if
each set is sufficiently overdetermined to be solvable
quickly. Then any substitutions are done in the smaller
set and the next set of conditions is shorter. For
example, for the Karpman equations below the speedup for
prelim_:=t and individual_:=t is a factor of 10.
(Time ~ 1 min.);

write"-------------------------------------------------------";


-------------------------------------------------------


lisp(prelim_:=t)$


lisp(individual_:=t)$



depend r,x,y,z,t;


depend f,x,y,z,t;


depend v,x,y,z,t;



on time$


Time: 10461 ms  plus GC time: 264 ms

liepde({

first 
solve(
        {df(r,t) + w1*df(r,z)
         + s1*(df(r,x)*df(f,x)+df(r,y)*df(f,y)+r*df(f,x,2)/2+r*df(f,y,2)/2) 
         + s2*(df(r,z)*df(f,z)+r*df(f,z,2)/2),
 
         df(f,t) + w1*df(f,z) 
         - (s1*(df(r,x,2)/r+df(r,y,2)/r-df(f,x)**2-df(f,y)**2) +
            s2*(df(r,z,2)/r-df(f,z)**2))/2 + a1*v,
       
         df(v,t,2) - w2**2*(df(v,x,2)+df(v,y,2)+df(v,z,2))
         - 2*a2*r*(df(r,x,2)+df(r,y,2)+df(r,z,2))
         - 2*a2*(df(r,x)**2+df(r,y)**2+df(r,z)**2)},
         
        {df(v,x,2), df(r,x,2), df(f,x,2)}         
 
     )         
         , {r,f,v}, {x,y,z,t}},

       {"point"}, 
       
       {},{})$



The PDE-system under investigation is :

                 2       2     2          2     2          2     2
v  =( - 4*f *a2*r  - 2*f  *a2*r *s1 - 2*f  *a2*r *s1 - 2*f  *a2*r *s2
 2x        t            x                y                z

                 2          2             2
      - 4*f *a2*r *w1 - 2*r  *a2*s1 - 2*r  *a2*s1 - 2*r  *a2*r*s1
           z               x             y             2z

                            2                           2            2
      + 2*r  *a2*r*s2 - 2*r  *a2*s1 + v  *s1 - v  *s1*w2  - v  *s1*w2
           2z              z           2t       2y           2z

                 2          2
      - 4*a1*a2*r *v)/(s1*w2 )

                2          2          2
r  =(2*f *r + f  *r*s1 + f  *r*s1 + f  *r*s2 + 2*f *r*w1 - r  *s1 - r  *s2
 2x     t      x          y          z            z         2y       2z

      + 2*a1*r*v)/s1

f  =
 2x

  - 2*f *r *s1 - f  *r*s1 - 2*f *r *s1 - f  *r*s2 - 2*f *r *s2 - 2*r  - 2*r *w1
       x  x       2y           y  y       2z           z  z         t      z
--------------------------------------------------------------------------------
                                      r*s1

for the function(s) : 

r(z,y,t,x), f(z,y,t,x), v(z,y,t,x)




=============== Initializations
time for initializations: 37 ms    GC time : 0 ms


=============== Preconditions for the 1. equation
time to formulate conditions: 273 ms    GC time : 0 ms
CRACK needed :  841 ms    GC time : 58 ms


=============== Preconditions for the 2. equation


=============== Preconditions for the 3. equation
time to formulate conditions: 127 ms    GC time : 0 ms
CRACK needed :  256 ms    GC time : 0 ms


=============== Full conditions for the 1. equation
time to formulate conditions: 54 ms    GC time : 0 ms
CRACK needed :  1321 ms    GC time : 43 ms


=============== Full conditions for the 2. equation
time to formulate conditions: 33 ms    GC time : 0 ms
CRACK needed :  67 ms    GC time : 0 ms


=============== Full conditions for the 3. equation
time to formulate conditions: 39 ms    GC time : 0 ms
CRACK needed :  77 ms    GC time : 21 ms

The symmetries are:

-------- 1. Symmetry:

xi_x=0

xi_y=0

xi_z=0

xi_t=0

eta_r=0

eta_f=1

eta_v=0

-------- 2. Symmetry:

xi_x=0

xi_y=0

xi_z=0

xi_t=0

eta_r=0

          2
eta_f= - t

       2*t
eta_v=-----
       a1

-------- 3. Symmetry:

xi_x=0

xi_y=0

xi_z=0

xi_t=0

eta_r=0

eta_f= - t

       1
eta_v=----
       a1

-------- 4. Symmetry:

xi_x=0

      1
xi_y=----
      s2

xi_z=0

xi_t=0

eta_r=0

eta_f=0

eta_v=0

-------- 5. Symmetry:

      1
xi_x=----
      s2

xi_y=0

xi_z=0

xi_t=0

eta_r=0

eta_f=0

eta_v=0

-------- 6. Symmetry:

      y
xi_x=----
      s2

       - x
xi_y=------
       s2

xi_z=0

xi_t=0

eta_r=0

eta_f=0

eta_v=0

-------- 7. Symmetry:

xi_x=0

xi_y=0

      1
xi_z=----
      s1

xi_t=0

eta_r=0

eta_f=0

eta_v=0

-------- 8. Symmetry:

xi_x=0

xi_y=0

xi_z=0

xi_t=1

eta_r=0

eta_f=0

eta_v=0

-------- 


{{},

        c_211 + c_212*y
 {xi_x=-----------------,
              s2

         - c_212*x + c_219
  xi_y=--------------------,
                s2

        c_208
  xi_z=-------,
         s1

  xi_t=c_221,

  eta_r=0,

                       2
  eta_f=c_222 - c_223*t  - c_224*t,

         2*c_223*t + c_224
  eta_v=-------------------},
                a1

 {c_222,

  c_223,

  c_224,

  c_219,

  c_211,

  c_212,

  c_208,

  c_221}}

Time: 3318 ms  plus GC time: 122 ms


off time$


nodepnd {r,f,v}$



comment
-------------------------------------------------------
In the following example a system of two equations (by
V.Sokolov) is investigated concerning a special ansatz for
4th order symmetries. The ansatz for the symmetries includes
two unknown functions f,g. Because x is the second variable
in the list of variables {t,x}, the name u!`2 stands for
df(u,x).
Because higher order symmetries are investigated we have
to set prelim_:=nil. The symmetries to be calculated are
lengthy and therefore conditions are not very overdetermined.
In that case CRACK can take long to solve a single 
subset of conditions. The complete set of conditions would
have been more overdetermined and easier to solve. Therefore
the advantage of first formulating all conditions and then
solving them together with one CRACK call is that having
more equations, the chance of finding short integrable
equations among then is higher, i.e. CRACK has more freedom
in optimizing the computation. Therefore individual_:=nil
is more appropriate in this example.

Because 4th order conditions are to be computed the
`binding stack size' is increased.
(Time ~ 5 min.);

write"-------------------------------------------------------";


-------------------------------------------------------


lisp(prelim_:=nil)$


lisp(individual_:=nil)$


lisp(if getd 'set_bndstk_size then set_bndstk_size(7000))$



nodepnd {u,v}$


depend  u,x,t;


depend  v,x,t;



des:={df(u,t)=+df(u,x,2) + (u + v)*df(u,x) + u*df(v,x),
      df(v,t)=-df(v,x,2) + (u + v)*df(v,x) + v*df(u,x)
     }$



nodepnd {f,g}$


depend f,t,x,u,v,u!`2,v!`2,u!`2!`2,v!`2!`2,u!`2!`2!`2,v!`2!`2!`2$


depend g,t,x,u,v,u!`2,v!`2,u!`2!`2,v!`2!`2,u!`2!`2!`2,v!`2!`2!`2$


liepde({des,{u,v},{t,x}},
       {xi_t=0,
        xi_x=0,
        eta_u=+df(u,x,4)+f,
        eta_v=-df(v,x,4)+g
       },
       {f,g},{}
      )$



The PDE-system under investigation is :

u =u   + u *u + u *v + v *u
 t  2x    x      x      x

v =u *v - v   + v *u + v *v
 t  x      2x    x      x

for the function(s) : 

u(x,t), v(x,t)



The symmetries are:

-------- 1. Symmetry:

xi_t=0

xi_x=0

eta_u=u   + 4*u  *t + 2*u  *u + 2*u  *v + 5*u  *u  + 3*u  *v  + 6*u  *t*u
       4x      3x        3x        3x        2x  x      2x  x      2x

                3       2    9             3       2                 2
 + 6*u  *t*v + ---*u  *u  + ---*u  *u*v + ---*u  *v  + 2*u  *x + 6*u  *t
      2x        2   2x       2   2x        2   2x         2x        x

       2      9    2                             9
 + 3*u  *u + ---*u  *v + 2*u *v   + 6*u *v *t + ---*u *v *u + 3*u *v *v
      x       2   x         x  2x      x  x      2   x  x        x  x

           2                         2    1      3    9      2      9        2
 + 3*u *t*u  + 12*u *t*u*v + 3*u *t*v  + ---*u *u  + ---*u *u *v + ---*u *u*v
      x            x            x         2   x       2   x         2   x

               1      3                                     2
 + 2*u *u*x + ---*u *v  + 2*u *v*x + 8*u  + v  *u + 6*v *t*u  + 6*v *t*u*v
      x        2   x         x          x    3x        x           x

    3      3    9      2      3        2               2
 + ---*v *u  + ---*v *u *v + ---*v *u*v  + 2*v *u*x + u  + 3*u*v
    2   x       2   x         2   x           x

                                                             9
eta_v=u  *v + 2*u  *v  + 3*u *v   - 6*u *v *t - 3*u *v *u - ---*u *v *v
       3x        2x  x      x  2x      x  x        x  x      2   x  x

                        2    3      2      9        2    3      3
 + 6*u *t*u*v + 6*u *t*v  + ---*u *u *v + ---*u *u*v  + ---*u *v  + 2*u *v*x
      x            x         2   x         2   x         2   x         x

 - v   + 4*v  *t + 2*v  *u + 2*v  *v + 5*v  *v  - 6*v  *t*u - 6*v  *t*v
    4x      3x        3x        3x        2x  x      2x          2x

    3       2    9             3       2                 2      9    2
 - ---*v  *u  - ---*v  *u*v - ---*v  *v  - 2*v  *x - 6*v  *t - ---*v  *u
    2   2x       2   2x        2   2x         2x        x       2   x

       2             2                         2    1      3    9      2
 - 3*v  *v + 3*v *t*u  + 12*v *t*u*v + 3*v *t*v  + ---*v *u  + ---*v *u *v
      x         x            x            x         2   x       2   x

    9        2               1      3                       2
 + ---*v *u*v  + 2*v *u*x + ---*v *v  + 2*v *v*x + 3*u*v + v
    2   x           x        2   x         x

-------- 2. Symmetry:

xi_t=0

xi_x=0

                                                               3       2
eta_u=u   + 2*u  *u + 2*u  *v + 4*u   + 5*u  *u  + 3*u  *v  + ---*u  *u
       4x      3x        3x        3x      2x  x      2x  x    2   2x

    9                       3       2                 2      9    2         2
 + ---*u  *u*v + 6*u  *u + ---*u  *v  + 6*u  *v + 3*u  *u + ---*u  *v + 6*u
    2   2x          2x      2   2x         2x        x       2   x         x

               9                                   1      3    9      2
 + 2*u *v   + ---*u *v *u + 3*u *v *v + 6*u *v  + ---*u *u  + ---*u *u *v
      x  2x    2   x  x        x  x        x  x    2   x       2   x

         2    9        2                1      3         2            3      3
 + 3*u *u  + ---*u *u*v  + 12*u *u*v + ---*u *v  + 3*u *v  + v  *u + ---*v *u
      x       2   x            x        2   x         x       3x      2   x

    9      2           2    3        2
 + ---*v *u *v + 6*v *u  + ---*v *u*v  + 6*v *u*v
    2   x           x       2   x           x

                                                 9
eta_v=u  *v + 2*u  *v  + 3*u *v   - 3*u *v *u - ---*u *v *v - 6*u *v
       3x        2x  x      x  2x      x  x      2   x  x        x  x

    3      2      9        2               3      3         2
 + ---*u *u *v + ---*u *u*v  + 6*u *u*v + ---*u *v  + 6*u *v  - v   + 2*v  *u
    2   x         2   x           x        2   x         x       4x      3x

                                 3       2    9                       3       2
 + 2*v  *v + 4*v   + 5*v  *v  - ---*v  *u  - ---*v  *u*v - 6*v  *u - ---*v  *v
      3x        3x      2x  x    2   2x       2   2x          2x      2   2x

              9    2         2         2    1      3    9      2           2
 - 6*v  *v - ---*v  *u - 3*v  *v - 6*v   + ---*v *u  + ---*v *u *v + 3*v *u
      2x      2   x         x         x     2   x       2   x           x

    9        2                1      3         2
 + ---*v *u*v  + 12*v *u*v + ---*v *v  + 3*v *v
    2   x            x        2   x         x

-------- 3. Symmetry:

xi_t=0

xi_x=0

                                                                 3       2
eta_u=u   + 2*u  *u + 2*u  *v + 5*u  *u  + 3*u  *v  + 2*u  *t + ---*u  *u
       4x      3x        3x        2x  x      2x  x      2x      2   2x

    9             3       2       2      9    2                 9
 + ---*u  *u*v + ---*u  *v  + 3*u  *u + ---*u  *v + 2*u *v   + ---*u *v *u
    2   2x        2   2x         x       2   x         x  2x    2   x  x

                                      1      3    9      2      9        2
 + 3*u *v *v + 2*u *t*u + 2*u *t*v + ---*u *u  + ---*u *u *v + ---*u *u*v
      x  x        x          x        2   x       2   x         2   x

    1      3                              3      3    9      2      3        2
 + ---*u *v  + u *x + v  *u + 2*v *t*u + ---*v *u  + ---*v *u *v + ---*v *u*v
    2   x       x      3x        x        2   x       2   x         2   x

 + u

                                                 9
eta_v=u  *v + 2*u  *v  + 3*u *v   - 3*u *v *u - ---*u *v *v + 2*u *t*v
       3x        2x  x      x  2x      x  x      2   x  x        x

    3      2      9        2    3      3
 + ---*u *u *v + ---*u *u*v  + ---*u *v  - v   + 2*v  *u + 2*v  *v + 5*v  *v
    2   x         2   x         2   x       4x      3x        3x        2x  x

              3       2    9             3       2    9    2         2
 - 2*v  *t - ---*v  *u  - ---*v  *u*v - ---*v  *v  - ---*v  *u - 3*v  *v
      2x      2   2x       2   2x        2   2x       2   x         x

                          1      3    9      2      9        2    1      3
 + 2*v *t*u + 2*v *t*v + ---*v *u  + ---*v *u *v + ---*v *u*v  + ---*v *v
      x          x        2   x       2   x         2   x         2   x

 + v *x + v
    x

-------- 4. Symmetry:

xi_t=0

xi_x=0

                                                       3       2    9
eta_u=u   + 2*u  *u + 2*u  *v + 5*u  *u  + 3*u  *v  + ---*u  *u  + ---*u  *u*v
       4x      3x        3x        2x  x      2x  x    2   2x       2   2x

    3       2             2      9    2                 9
 + ---*u  *v  + u   + 3*u  *u + ---*u  *v + 2*u *v   + ---*u *v *u + 3*u *v *v
    2   2x       2x      x       2   x         x  2x    2   x  x        x  x

    1      3    9      2      9        2           1      3
 + ---*u *u  + ---*u *u *v + ---*u *u*v  + u *u + ---*u *v  + u *v + v  *u
    2   x       2   x         2   x         x      2   x       x      3x

    3      3    9      2      3        2
 + ---*v *u  + ---*v *u *v + ---*v *u*v  + v *u
    2   x       2   x         2   x         x

                                                 9             3      2
eta_v=u  *v + 2*u  *v  + 3*u *v   - 3*u *v *u - ---*u *v *v + ---*u *u *v
       3x        2x  x      x  2x      x  x      2   x  x      2   x

    9        2    3      3
 + ---*u *u*v  + ---*u *v  + u *v - v   + 2*v  *u + 2*v  *v + 5*v  *v
    2   x         2   x       x      4x      3x        3x        2x  x

    3       2    9             3       2          9    2         2      1      3
 - ---*v  *u  - ---*v  *u*v - ---*v  *v  - v   - ---*v  *u - 3*v  *v + ---*v *u
    2   2x       2   2x        2   2x       2x    2   x         x       2   x

    9      2      9        2           1      3
 + ---*v *u *v + ---*v *u*v  + v *u + ---*v *v  + v *v
    2   x         2   x         x      2   x       x

-------- 5. Symmetry:

xi_t=0

xi_x=0

                                                       3       2    9
eta_u=u   + 2*u  *u + 2*u  *v + 5*u  *u  + 3*u  *v  + ---*u  *u  + ---*u  *u*v
       4x      3x        3x        2x  x      2x  x    2   2x       2   2x

    3       2       2      9    2                 9
 + ---*u  *v  + 3*u  *u + ---*u  *v + 2*u *v   + ---*u *v *u + 3*u *v *v
    2   2x         x       2   x         x  2x    2   x  x        x  x

    1      3    9      2      9        2    1      3                 3      3
 + ---*u *u  + ---*u *u *v + ---*u *u*v  + ---*u *v  + u  + v  *u + ---*v *u
    2   x       2   x         2   x         2   x       x    3x      2   x

    9      2      3        2
 + ---*v *u *v + ---*v *u*v
    2   x         2   x

                                                 9             3      2
eta_v=u  *v + 2*u  *v  + 3*u *v   - 3*u *v *u - ---*u *v *v + ---*u *u *v
       3x        2x  x      x  2x      x  x      2   x  x      2   x

    9        2    3      3                                         3       2
 + ---*u *u*v  + ---*u *v  - v   + 2*v  *u + 2*v  *v + 5*v  *v  - ---*v  *u
    2   x         2   x       4x      3x        3x        2x  x    2   2x

    9             3       2    9    2         2      1      3    9      2
 - ---*v  *u*v - ---*v  *v  - ---*v  *u - 3*v  *v + ---*v *u  + ---*v *u *v
    2   2x        2   2x       2   x         x       2   x       2   x

    9        2    1      3
 + ---*v *u*v  + ---*v *v  + v
    2   x         2   x       x

-------- 


{{},

 {xi_t=0,

  xi_x=0,

  eta_u=(2*u   + 8*u  *c_326 + 8*u  *c_328*t + 4*u  *u + 4*u  *v + 10*u  *u
            4x      3x            3x              3x        3x         2x  x

          + 6*u  *v  + 12*u  *c_326*u + 12*u  *c_326*v + 12*u  *c_328*t*u
               2x  x       2x               2x               2x

          + 12*u  *c_328*t*v + 4*u  *c_328*x + 2*u  *c_332 + 4*u  *c_333*t
                2x                2x              2x            2x

                   2                      2        2              2
          + 3*u  *u  + 9*u  *u*v + 3*u  *v  + 12*u  *c_326 + 12*u  *c_328*t
               2x         2x          2x          x              x

                2         2
          + 6*u  *u + 9*u  *v + 4*u *v   + 12*u *v *c_326 + 12*u *v *c_328*t
               x         x         x  2x       x  x             x  x

                                                2
          + 9*u *v *u + 6*u *v *v + 6*u *c_326*u  + 24*u *c_326*u*v
               x  x        x  x        x                x

                        2                 2
          + 6*u *c_326*v  + 6*u *c_328*t*u  + 24*u *c_328*t*u*v
               x               x                  x

                          2
          + 6*u *c_328*t*v  + 4*u *c_328*u*x + 4*u *c_328*v*x + 16*u *c_328
               x                 x                x                 x

          + 2*u *c_332*u + 2*u *c_332*v + 4*u *c_333*t*u + 4*u *c_333*t*v
               x              x              x                x

                                            3         2             2       3
          + 2*u *c_333*x + 2*u *c_334 + u *u  + 9*u *u *v + 9*u *u*v  + u *v
               x              x          x         x           x         x

                                   2                                    2
          + 2*v  *u + 12*v *c_326*u  + 12*v *c_326*u*v + 12*v *c_328*t*u
               3x         x                x                 x

          + 12*v *c_328*t*u*v + 4*v *c_328*u*x + 2*v *c_332*u + 4*v *c_333*t*u
                x                  x                x              x

                  3         2             2            2
          + 3*v *u  + 9*v *u *v + 3*v *u*v  + 2*c_328*u  + 6*c_328*u*v
               x         x           x

          + 2*c_333*u)/2,

  eta_v=(2*u  *v + 4*u  *v  + 6*u *v   - 12*u *v *c_326 - 12*u *v *c_328*t
            3x        2x  x      x  2x       x  x             x  x

                                                                   2
          - 6*u *v *u - 9*u *v *v + 12*u *c_326*u*v + 12*u *c_326*v
               x  x        x  x         x                 x

                                               2
          + 12*u *c_328*t*u*v + 12*u *c_328*t*v  + 4*u *c_328*v*x + 2*u *c_332*v
                x                   x                 x                x

                                   2             2         3
          + 4*u *c_333*t*v + 3*u *u *v + 9*u *u*v  + 3*u *v  - 2*v
               x                x           x           x         4x

          + 8*v  *c_326 + 8*v  *c_328*t + 4*v  *u + 4*v  *v + 10*v  *v
               3x            3x              3x        3x         2x  x

          - 12*v  *c_326*u - 12*v  *c_326*v - 12*v  *c_328*t*u
                2x               2x               2x

          - 12*v  *c_328*t*v - 4*v  *c_328*x - 2*v  *c_332 - 4*v  *c_333*t
                2x                2x              2x            2x

                   2                      2        2              2
          - 3*v  *u  - 9*v  *u*v - 3*v  *v  - 12*v  *c_326 - 12*v  *c_328*t
               2x         2x          2x          x              x

                2         2                 2                                 2
          - 9*v  *u - 6*v  *v + 6*v *c_326*u  + 24*v *c_326*u*v + 6*v *c_326*v
               x         x         x                x                x

                          2                                     2
          + 6*v *c_328*t*u  + 24*v *c_328*t*u*v + 6*v *c_328*t*v
               x                  x                  x

          + 4*v *c_328*u*x + 4*v *c_328*v*x + 2*v *c_332*u + 2*v *c_332*v
               x                x                x              x

                                                                              3
          + 4*v *c_333*t*u + 4*v *c_333*t*v + 2*v *c_333*x + 2*v *c_334 + v *u
               x                x                x              x          x

                  2             2       3                          2
          + 9*v *u *v + 9*v *u*v  + v *v  + 6*c_328*u*v + 2*c_328*v  + 2*c_333*v
               x           x         x

         )/2},

 {c_328,

  c_326,

  c_333,

  c_332,

  c_334}}

nodepnd {f,g}$



end$


Time for test: 23535 ms, plus GC time: 689 ms
