Bi-linear algebraic problem l21o23

Problem l21o23

Unknowns

All solutions for the following 21 unknowns have to be determined:

a1, ..., a9, b1, ..., b12

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.

{a1,a4,a7},             
{a3,a4,a6,a5},          
{a8},                   
{b1,b4,b8},             
{b1,b2,b3,b4,b5,b6,b7}, 
{b8,b9,b10,b11,b12}

Equations

All comma separated 34 expressions involving 169 terms have to vanish. All terms are products of one a- and one b-unknown.

a2*b10,
a2*b2,
2*a7*b10 - 3*a8*b8,
a1*b2 - 3*a2*b1,
2*a7*b11 - 3*a9*b8,
a5*b10 + 2*a8*b11,
a4*b9 - a8*b4,
a6*b10 + 2*a9*b11,
2*a2*b9 + a8*b10 - a8*b2,
a4*b10 - 2*a7*b12 + 6*a9*b8,
2*a5*b9 + 2*a8*b12 - a8*b6,
4*a6*b9 - a8*b7 + 2*a9*b12,
a1*b10 - a7*b10 - 2*a7*b9 + 3*a8*b8,
a1*b9 - a7*b9 - a8*b1 + a8*b8,
a3*b10 - 2*a7*b11 - 2*a7*b12 + 6*a9*b8,
a5*b2 - 2*a5*b9 + 2*a8*b5 + a8*b6,
a1*b4 - a4*b1 + a4*b8 - a7*b4,
a2*b6 + 2*a5*b10 - 2*a5*b2 - 2*a8*b6,
a6*b2 - 4*a6*b9 + a8*b7 + 2*a9*b5,
a5*b9 + a8*b11 + a8*b12 - a8*b5 - 2*a9*b9,
2*a3*b9 + a4*b9 - 2*a7*b12 - a8*b3 + 6*a9*b8,
a1*b3 + 2*a1*b4 - 2*a3*b1 + 2*a3*b8 - 3*a4*b1 - a7*b3,
a5*b7 + 4*a6*b11 + 4*a6*b12 - 4*a6*b5 - a6*b6 - 6*a9*b7,
4*a1*b5 + 2*a3*b9 + a4*b10 + 2*a4*b9 - 6*a5*b1 - a8*b3 - 3*a8*b4,
2*a1*b5 + 2*a1*b6 + 2*a3*b10 - a3*b2 + 2*a3*b9 - 6*a5*b1 - 2*a8*b3,
3*a1*b3 + a1*b4 - 6*a3*b1 - 4*a4*b1 + a4*b8 - a7*b3 - a7*b4,
a1*b6 + 2*a2*b4 + a4*b10 - a4*b2 - 2*a5*b1 + 2*a5*b8 - a7*b6 - 2*a8*b4,
a2*b7 + a5*b11 + a5*b12 - a5*b5 + 2*a6*b10 - 2*a6*b2 - 2*a8*b7 - 2*a9*b6,
2*a1*b5 + 2*a1*b6 + 2*a4*b10 - a4*b2 + 2*a4*b9 - 6*a5*b1 - 2*a7*b5 - a8*b3 - 3*
a8*b4,
3*a1*b6 + 2*a2*b3 + 2*a3*b10 - 2*a3*b2 + a4*b10 - a4*b2 - 6*a5*b1 - a7*b6 - 2*
a8*b3,
6*a1*b7 + 4*a3*b11 + 4*a3*b12 - a3*b6 + 2*a4*b11 + 2*a4*b12 - 24*a6*b1 - 6*a9*
b3 - 6*a9*b4,
a1*b7 + a4*b11 + a4*b12 - a4*b5 + a5*b4 - 4*a6*b1 + 4*a6*b8 - a7*b7 - 4*a9*b4,
6*a1*b7 + 2*a3*b12 + 6*a4*b11 + 4*a4*b12 - a4*b6 - 24*a6*b1 - 2*a7*b7 - 4*a9*b3
 - 12*a9*b4,
3*a1*b7 + 2*a3*b11 - 2*a3*b5 + a4*b11 + 3*a4*b12 - a4*b5 + a5*b3 - 12*a6*b1 - 
a7*b7 - 2*a9*b3 - 6*a9*b4

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  *a1 + f(v ,v )*a3 + f(v,v  )*a4 + f(v,v) *a6 + f(v,v)*a5*u + a2*u ,
  t    2x         x  x            2x

 v  = v  *a7 + f(v,v)*a9*v + a8*u*v
  t    2x

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

 u  = u  *b1 + u *f(v,v)*b5 + u *b2*u + f(v ,v  )*b3 + f(v,v  )*b4
  t    3x       x              x           x  2x            3x

       + f(v,v )*f(v,v)*b7 + f(v,v )*b6*u,
              x                   x

 v  = u *b9*v + v  *b8 + v *f(v,v)*b11 + v *b10*u + f(v,v )*b12*v
  t    x         3x       x               x              x