Problem l2o35

Unknowns | Inequalities | Equations | Solution 1 | Solution 2 | Solution 3 | Solution 4 | Relevance | Back to overview

Unknowns

All solutions for the following 23 unknowns have to be determined: 
a1, ..., a6, b1, ..., b17

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. 
{a3},     
{a1,a4},
{a5,a6},
{b1, b9},
{b1,b2,b3,b4,b5,b6,b7,b8},
{ b9,b10,b11,b12,b13,b14,b15,b16,b17}

Equations

All comma separated 50 expressions involving 218 terms have to vanish. All terms are products of one a- and one b-unknown. (only in the first two equations an non-vanishing overall factor a3 has been dropped.) 
2*b16 + 2*b17 - b8,
b16 + b17 - b7,
3*a4*b13 - 5*a6* b9,
3*a1*b2 - 5*a2*b1,
3*a1*b3 - 10*a2*b1,
2*a1*b4 - a2*b2,
4*a1*b4 - a2*b3,
a3*b13 - 6*a4*b16,
2*a3*b15 - a6*b8,
a3*b14 - a5*b8,
a3*b10 - a5*b6,
a3*b14 - 2*a5*b7,
2*a2*b15 - a6*b15 - a6*b4,
2*a2*b14 - 2*a5*b4 - a6*b14,
a3*b12 - 6*a4*b16 - 3*a4*b17,
a3*b11 - 3*a4*b17 - a6*b6,
3*a3*b10 - 3*a4*b17 - a5*b5,
a2*b13 - 6*a4*b15 + 2*a6*b13,
a1*b2 + a1*b3 - 5*a2*b1,
2*a5*b16 + a6*b16 - a6*b7,
3*a3*b11 - 6*a4*b16 - 9*a4*b17 - a6*b5,
4*a3*b10 - 3*a4*b17 - a5*b5 - a5*b6,
3*a4*b12 + 3*a4*b13 - 5*a5* b9 - 10*a6* b9,
a2*b12 - 3*a4*b14 - 6*a4*b15 + 2*a6*b12,
a1*b10 - a4*b10 + a5* b9 - a5*b1,
3*a2*b10 - 3*a4*b14 + a5*b11 - a5*b3,
a2*b14 + a5*b15 - a5*b4 - a6*b14,
3*a1*b5 + a1*b6 - 10*a3*b1 - a4*b6,
4*a1*b5 + 3*a1*b6 - 15*a3*b1 - a4*b5,
3*a1*b8 + a2*b5 - 3*a3*b2 - a6*b5,
a2*b8 + a3*b15 - a3*b4 - a6*b8,
a1*b6 + a3* b9 - a3*b1 - a4*b6,
a3*b14 + 2*a5*b17 - 2*a5*b7 - a5*b8 + a6*b17,
a1*b5 + 3*a1*b6 + a3* b9 - 6*a3*b1 - a4*b5,
a1*b13 - 3*a4*b11 - 3*a4*b12 - a4*b13 + 10*a5* b9 + 10*a6* b9,
a1*b12 - 3*a4*b10 - 3*a4*b11 - a4*b12 + 10*a5* b9 + 5*a6* b9,
a2*b12 - 3*a4*b14 - 6*a4*b15 + 3*a5*b13 - a6*b12 + 3*a6*b13,
a1*b11 - 3*a4*b10 - a4*b11 + 5*a5* b9 + a6* b9 - a6*b1,
a1*b14 + a2*b10 - a4*b14 + a5*b13 - a5*b2 - a6*b10,
a1*b14 + 4*a2*b10 - 4*a4*b14 + a5*b12 - a5*b2 - a5*b3,
6*a1*b7 + 3*a1*b8 + a3*b11 - a3*b3 - 2*a5*b5 - a6*b5,
a1*b8 + a2*b6 + a3*b13 - a3*b2 - a4*b8 - a6*b6,
6*a1*b7 + 2*a3*b10 + a3*b11 - a3*b2 - a5*b5 - 3*a5*b6 - a6*b6,
3*a2*b11 - 9*a4*b14 - 6*a4*b15 + 2*a5*b12 + a6*b11 + a6*b12 - a6*b3,
6*a1*b7 + 9*a1*b8 + a2*b5 + a3*b12 - 3*a3*b3 - 2*a5*b5 - 3*a6*b5,
2*a1*b15 + a2*b11 - 3*a4*b14 - 2*a4*b15 + 3*a5*b13 - a6*b11 + a6*b13 - a6*b2,
6*a1*b7 + 3*a1*b8 + a3*b11 + a3*b12 - a3*b3 - a5*b5 - 3*a5*b6 - 3*a6*b6,
2*a2*b7 + a2*b8 + 2*a3*b14 + 2*a3*b15 - 2*a3*b4 - 2*a5*b8 - 2*a6*b7 - a6*b8,
4*a1*b8 + a2*b5 + a2*b6 + a3*b13 - 4*a3*b2 - a4*b8 - a6*b5 - a6*b6,
2*a1*b7 + 3*a1*b8 + a2*b6 + a3*b12 + a3*b13 - a3*b3 - 2*a4*b7 - 2*a5*b6 - 3*a6*b6

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. 
   u  = u  *a1 + u *a2*u + f(v,v )*a3
    t    3x       x             x

   v  = u *a5*v + v  *a4 + v *a6*u
    t    x         3x       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  *b2*u + u  *u *b3 + u *f(v,v)*b7 + u *b4*u  + f(v ,v  )*b5
    t    5x       3x         2x  x       x              x            x  2x

         + f(v,v  )*b6 + f(v,v )*b8*u
                3x            x

   v  = u  *b10*v + u  *v *b11 + u *v  *b12 + u *b14*u*v + v  * b9 + v  *b13*u
    t    3x          2x  x        x  2x        x            5x        3x

                                   2
         + v *f(v,v)*b16 + v *b15*u  + f(v,v )*b17*v
            x               x               x