Bi-linear algebraic problem l112o13

Problem l112o13

Unknowns

All solutions for the following 50 unknowns have to be determined:

a0, a2, a3, a4, a5, a6, a7, a8, b1, .., b42

(where a6 does not occur in the equations as it can be gauged to zero by a Galilei transformation).

Inequalities

Each of the following lists represents one inequality which states that not all unknowns in this list may vanish. These inequalities filter out solutions which are trivial for the application.

   {a0,a3},                                          % 1st order system after
                                                     % Galilei transformation
   {b1,b7,b24},                                      % 3rd order symmetry
   {a3,a5,a4,b8,b9,b18,b12,b6,b15,b10,b7,b16,
    b11,b17,b21,b19,b13,b23,b22,b20,b14},            % u_t or u_s to involve v
   {a7,b25,b26,b35,b28,b27,b36,b29,b37,
    b41,b38,b30}                                     % v_t or v_s to involve u

Equations

All comma separated 80 expressions involving 430 terms have to vanish. All terms are products of one a- and one b-unknown.

a2*b27,
a2*b29,
a2*b1,
a2*b2,
a2*b4,
a2*b5,
a2*b2 + a2*b3,
a0*b6 - 3*a3*b1,
a0*b26 + 3*a7*b24,
a0*b27 + 3*a7*b24,
3*a2*b30 - a7*b5,
a3*b27 + 6*a8*b24,
a4*b27 + 2*a7*b31,
a3*b26 + 6*a8*b24,
a3*b25 - a7*b7,
a5*b27 + 2*a8*b31,
a5*b36 + 4*a8*b39,
a0*b7 - a3*b1 + a3*b24,
a0*b25 - a7*b1 + a7*b24,
a0*b29 + a2*b26 + a7*b27,
a2*b36 + 2*a4*b29 + 2*a7*b36,
3*a3*b25 - a7*b6 + 6*a8*b24,
a5*b41 - a7*b23 + 4*a8*b42,
a0*b6 + a0*b7 - 4*a3*b1 + a3*b24,
a0*b36 + a4*b26 + 2*a7*b31 + a7*b32,
3*a0*b30 + 3*a2*b28 + a7*b29 - a7*b4,
a0*b28 + 2*a2*b25 + a7*b26 - a7*b3,
a0*b28 + 2*a2*b25 - a7*b2 + a7*b27,
a3*b29 + a4*b26 + a7*b32 + 2*a8*b27,
a3*b28 + 2*a4*b25 - a7*b11 + 2*a7*b34,
2*a2*b8 - a3*b28 + a4*b3 + a7*b9,
a2*b14 - 2*a4*b30 + 3*a4*b5 + 2*a7*b14,
a3*b36 + 4*a5*b26 + 4*a8*b31 + 4*a8*b32,
a4*b36 + 2*a5*b29 + 4*a7*b39 + 2*a8*b36,
a3*b35 + 4*a5*b25 - a7*b16 + 2*a8*b34,
a4*b23 - a5*b22 + 4*a5*b42 - 8*a8*b23,
a0*b9 + a3*b26 - 2*a3*b3 - 6*a4*b1 - 2*a7*b6,
a3*b28 + 2*a4*b25 - a7*b10 + 2*a7*b33 + 2*a8*b27,
a2*b37 + 3*a3*b30 + 2*a4*b28 - a7*b13 + 2*a7*b37,
2*a2*b38 + 3*a4*b30 - a7*b14 + 2*a7*b38 - 2*a8*b30,
a3*b28 - a4*b2 + 2*a4*b25 - a7*b11 - 2*a7*b8,
a3*b35 + 4*a5*b25 - a7*b15 + 2*a8*b31 + 2*a8*b34,
a3*b37 + 8*a5*b25 - a7*b17 + 4*a8*b33 + 4*a8*b34,
a2*b41 + 2*a4*b38 + 3*a5*b30 - a7*b20 + 4*a7*b41,
a3*b35 - a5*b2 + 4*a5*b25 - a7*b16 - 2*a8*b8,
a3*b41 + 4*a5*b35 + a5*b37 - a7*b21 + 4*a8*b40,
a4*b41 + 2*a5*b38 - a7*b22 + 6*a7*b42 + 2*a8*b41,
a3*b41 - a5*b12 + 4*a5*b35 - a7*b21 - 4*a8*b18,
a0*b41 + a4*b35 + a5*b28 - a7*b18 + a7*b39 + a7*b40,
a0*b14 + a3*b30 - 2*a3*b5 + a4*b29 - a4*b4 - a7*b13,
a0*b13 + a2*b10 + a3*b29 - a3*b4 - 2*a4*b2 - 2*a7*b10,
a0*b35 + a4*b25 + a7*b31 + a7*b34 - a7*b8 - 2*a8*b25,
a2*b12 - 3*a3*b30 - 2*a4*b28 + 2*a4*b4 + 2*a7*b12 + a7*b13,
a4*b14 - a4*b38 - 2*a5*b30 + 2*a5*b5 + 2*a7*b20 + a8*b14,
a2*b23 + a4*b42 - a5*b20 + 2*a5*b41 - 4*a7*b23 - 3*a8*b22,
2*a0*b10 + a0*b11 + 2*a2*b6 - 3*a3*b2 + a3*b27 - 6*a4*b1 - 2*a7*b6,
a0*b13 + a2*b11 + a3*b29 - a3*b4 - 2*a4*b2 + 2*a4*b27 - 2*a7*b11,
a0*b11 + 2*a2*b7 - a3*b2 + a3*b27 - 2*a4*b1 + 2*a4*b24 - 2*a7*b7,
a0*b9 + a3*b26 + a3*b27 - 2*a3*b3 - 6*a4*b1 - a7*b6 - 3*a7*b7,
2*a0*b8 - a3*b2 + 2*a3*b25 + a3*b26 - 6*a4*b1 - a7*b6 - 3*a7*b7,
a0*b37 + a3*b28 + 4*a4*b25 + a7*b32 + 2*a7*b33 + 2*a7*b34 - a7*b9,
2*a3*b38 + 2*a4*b35 + a4*b37 + 4*a5*b28 - a7*b19 + 4*a7*b40 + 2*a8*b37,
a2*b22 - a4*b20 + 2*a4*b41 - 3*a5*b14 + 4*a5*b38 - 6*a7*b22 - 4*a8*b20,
a0*b16 + a3*b31 + a3*b34 - a3*b8 + a4*b7 - 4*a5*b1 + 4*a5*b24 - 4*a8*b7,
2*a0*b38 + 2*a2*b35 + 2*a4*b28 - a7*b12 + 2*a7*b35 + a7*b36 + a7*b37 - 2*a8*b28
,
2*a3*b38 - a4*b12 + 2*a4*b35 + 4*a5*b28 - 2*a5*b4 - 4*a7*b18 - a7*b19 - 2*a8*
b12,
2*a0*b15 + 2*a0*b17 - a3*b10 + 2*a3*b32 + 2*a3*b33 - a3*b9 - 24*a5*b1 - 6*a8*b6
 - 6*a8*b7,
8*a0*b23 - a3*b22 + 8*a3*b42 + a4*b21 - 4*a5*b18 - a5*b19 + 4*a5*b39 + 4*a5*b40
 - 8*a8*b21,
2*a0*b16 + 2*a0*b17 - a3*b11 + 2*a3*b31 + 2*a3*b32 + 2*a3*b34 - a3*b9 - 24*a5*
b1 - 4*a8*b6 - 12*a8*b7,
2*a0*b15 + a0*b16 + a3*b31 + 2*a3*b33 + a3*b34 - 3*a3*b8 + a4*b6 - 12*a5*b1 - 2
*a8*b6 - 6*a8*b7,
a0*b21 - a3*b18 + a3*b39 + a3*b40 + a4*b16 - a5*b11 + 4*a5*b31 + 4*a5*b34 - 4*
a5*b8 - 6*a8*b16,
a0*b21 - a3*b18 + a3*b39 + a3*b40 + a4*b15 - a5*b10 + 4*a5*b33 - 4*a5*b8 - 4*a8
*b15 - 2*a8*b16,
a0*b12 + a3*b28 + a3*b29 - a3*b4 - 2*a4*b2 + a4*b26 - 2*a4*b3 - a7*b10 - a7*b11
 - a7*b9,
4*a0*b21 - a3*b19 + 4*a3*b39 + 4*a3*b40 + a4*b17 + 4*a5*b32 - 8*a5*b8 - 4*a5*b9
 - 4*a8*b15 - 4*a8*b16 - 8*a8*b17,
a0*b19 + a2*b17 - a3*b13 + a3*b36 + a3*b37 + a4*b32 - a4*b9 - 4*a5*b2 - 2*a7*
b17 - 2*a8*b10 - 2*a8*b11,
a0*b19 + 2*a2*b15 - a3*b12 + a3*b36 + a3*b37 + 2*a4*b33 - 2*a4*b8 - 4*a5*b2 - 4
*a7*b15 - 2*a8*b10 - 2*a8*b11,
4*a0*b20 + a2*b19 - 3*a3*b14 + 4*a3*b38 - 2*a4*b12 - a4*b13 + 2*a4*b36 + 2*a4*
b37 + 4*a5*b29 - 4*a5*b4 - 4*a7*b19 - 4*a8*b13,
a0*b19 + 2*a2*b16 - a3*b12 + a3*b36 + a3*b37 + 2*a4*b31 + 2*a4*b34 - 2*a4*b8 - 
4*a5*b2 + 4*a5*b27 - 4*a7*b16 - 4*a8*b11,
4*a0*b18 - a3*b12 + 4*a3*b35 + a3*b36 + a3*b37 - 4*a4*b8 + 4*a5*b26 - 8*a5*b3 -
 2*a7*b15 - 2*a7*b16 - 2*a7*b17 - 4*a8*b9,
3*a0*b22 + a2*b21 - a3*b20 + 3*a3*b41 - a4*b18 + a4*b39 + a4*b40 - 2*a5*b12 - 
a5*b13 + 2*a5*b36 + 2*a5*b37 - 3*a7*b21 - 3*a8*b19

Relevance

In the following evolutionary system u=u(t,x) is a scalar function, v=v(t,x) is a vector function and f(..,..) stands for the scalar product of both vector arguments of f.

                                      2                        2
 u  = u *(a0+a6) + f(v,v )*a3 + f(v,v) *a5 + f(v,v)*a4*u + a2*u ,
  t    x                x

 v  = v *a6 + f(v,v)*a8*v + a7*u*v
  t    x

For any solution of the above algebraic conditions for the a's and b's this system has the following symmetry:

                                            2
 u  = u  *b1 + u  *f(v,v)*b8 + u  *b2*u + u  *b3 + u *f(v,v )*b9
  s    3x       2x              2x         x        x      x

                  2                                2
       + u *f(v,v) *b18 + u *f(v,v)*b12*u + u *b4*u  + f(v ,v  )*b6
          x                x                 x            x  2x

       + f(v ,v )*f(v,v)*b15 + f(v ,v )*b10*u + f(v,v  )*b7
            x  x                  x  x               3x

                                                       2
       + f(v,v  )*f(v,v)*b16 + f(v,v  )*b11*u + f(v,v ) *b17
              2x                    2x               x

                       2                                           2
       + f(v,v )*f(v,v) *b21 + f(v,v )*f(v,v)*b19*u + f(v,v )*b13*u
              x                     x                      x

               4             3               2      2               3       4
       + f(v,v) *b23 + f(v,v) *b22*u + f(v,v) *b20*u  + f(v,v)*b14*u  + b5*u ,

 v  = u  *b25*v + u *v *b26 + u *f(v,v)*b35*v + u *b28*u*v + v  *b24
  s    2x          x  x        x                 x            3x

                                                                2
       + v  *f(v,v)*b31 + v  *b27*u + v *f(v,v )*b32 + v *f(v,v) *b39
          2x               2x          x      x         x

                                   2
       + v *f(v,v)*b36*u + v *b29*u  + f(v ,v )*b33*v + f(v,v  )*b34*v
          x                 x             x  x               2x

                                                        3
       + f(v,v )*f(v,v)*b40*v + f(v,v )*b37*u*v + f(v,v) *b42*v
              x                      x

               2                       2          3
       + f(v,v) *b41*u*v + f(v,v)*b38*u *v + b30*u *v