Problem l2o24

Unknowns | Inequalities | Equations | Relevance | Back to overview

Unknowns

All solutions for the following 18 unknowns have to be determined: 
a1, ..., a5, b1, ..., b13

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},                  
{a3},                     
{a5},                     
{b1,b8},                  
{b1,b2,b3,b4,b5,b6,b7},   
{b8,b9,b10,b11,b12,b13}

Equations

All comma separated 26 expressions involving 77 terms have to vanish. All terms are products of one a- and one b-unknown.
(Comment: These equations are generated in stages and if in one of the first stages a simple condition, like b11=0 results then it is utilized instantly. This is the case here and therefore b11 does not occur in the following conditions. The condition b8=0 resulted only at the last stage of formulating all conditions, therefore it is not utilized yet and b8 occurs as one of the expressions below.) 
b8,
2*b13 - b7,
a2*b2,
a2*b4,
a4*b10 - 3*a5*b8,
a2*b10 - 2*a4*b12,
a1*b2 - 4*a2*b1,
a1*b3 - 3*a2*b1,
2*a2*b12 - a5*b4,
a3*b10 - 2*a4*b13,
2*a3*b9 - a5*b6,
3*a1*b4 - a2*b2 - a2*b3,
2*a3*b9 - 2*a4*b13 - a5*b5,
2*a3*b12 + 2*a5*b13 - a5*b7,
a3*b2 - 2*a3*b9 + a5*b6,
a1*b10 - a4*b10 - 2*a4*b9 + 4*a5*b8,
2*a1*b12 + 2*a2*b9 - 2*a4*b12 - a5*b2,
a1*b9 - a4*b9 - a5*b1 + a5*b8,
2*a2*b9 - 2*a4*b12 + a5*b10 - a5*b3,
2*a1*b5 + a1*b6 - 6*a3*b1 - a4*b6,
a1*b5 + a1*b6 - 4*a3*b1 - a4*b5,
a1*b7 + a2*b5 - a3*b2 - a5*b5,
a1*b6 - 2*a3*b1 + 2*a3*b8 - a4*b6,
a2*b7 + 2*a3*b12 - 3*a3*b4 - 2*a5*b7,
2*a1*b7 + a3*b10 - 2*a3*b3 - a5*b5 - a5*b6,
a1*b7 + a2*b6 - a3*b2 - a4*b7 - a5*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. 
                               2
 u  = u  *a1 + f(v,v)*a3 + a2*u ,
  t    2x

 v  = v  *a4 + a5*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: 
                            2
 u  = u  *b1 + u  *b2*u + u  *b3 + f(v ,v )*b5 + f(v,v  )*b6 + f(v,v)*b7*u
  t    4x       2x         x          x  x            2x

             3
       + b4*u ,

                                                                      2
 v  = u  *b9*v + u *v *b10 + v  *b8 + v  *b11*u + f(v,v)*b13*v + b12*u *v
  t    2x         x  x        4x       2x