Problem l1o13

Unknowns | Inequalities | Equations | Solution 1 | Relevance | Back to overview

Unknowns

All solutions for the following 26 unknowns have to be determined: 
a0, a2, a3, a4, a5, b1, ..., b21
(where a4 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},                                         % 1st order system after
                                                % Galilei transformation
  {b1,b12},                                     % 3th order symmetry
  {a3,b8,b6,b7,b9,b11,b10},                     % v to occur in u_t or u_s
  {a5,b13,b14,b16,b15,b17,b21,b18}              % u to occur in v_t or v_s

Equations

All comma separated 35 expressions involving 98 terms have to vanish. All terms are products of one a- and one b-unknown. 
a2*b15,
a2*b17,
a2*b1,
a2*b2,
a2*b4,
a2*b5,
a3*b15,
a3*b14,
a2*b2 + a2*b3,
a0*b14 + 3*a5*b12,
a0*b15 + 3*a5*b12,
3*a2*b18 - a5*b5,
a3*b17 + a5*b19,
2*a3*b13 - a5*b6,
2*a3*b13 - a5*b7,
a3*b21 - a5*b11,
2*a0*b6 + a0*b7 - 6*a3*b1,
a0*b7 - 2*a3*b1 + 2*a3*b12,
a0*b13 - a5*b1 + a5*b12,
a0*b17 + a2*b14 + a5*b15,
2*a3*b16 + 2*a5*b20 - a5*b9,
2*a3*b13 - a3*b2 - a5*b7,
a3*b18 - 2*a3*b5 - a5*b10,
a0*b9 + 2*a2*b6 - 2*a3*b2 - 2*a5*b6,
3*a0*b18 + 3*a2*b16 + a5*b17 - a5*b4,
a0*b16 + 2*a2*b13 + a5*b14 - a5*b3,
a0*b16 + 2*a2*b13 + a5*b15 - a5*b2,
a2*b21 + 3*a3*b18 - a5*b10 + 2*a5*b21,
2*a3*b16 - 2*a3*b4 - 2*a5*b8 - a5*b9,
a2*b11 - a3*b10 + a3*b21 - 2*a5*b11,
4*a0*b11 + 2*a3*b19 + 2*a3*b20 - 2*a3*b8 - a3*b9,
2*a0*b10 + a2*b9 + 2*a3*b17 - 2*a3*b4 - 2*a5*b9,
a0*b9 + 2*a2*b7 + 2*a3*b15 - 2*a3*b2 - 2*a5*b7,
a0*b8 + a3*b14 - 2*a3*b3 - a5*b6 - a5*b7,
a0*b21 + a3*b16 + a5*b19 + a5*b20 - a5*b8

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 *(a0+a4) + f(v,v)*a3 + a2*u ,
  t    x

 v  = v *a4 + a5*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                            2
   u  = u  *b1 + u  *b2*u + u  *b3 + u *f(v,v)*b8 + u *b4*u  + f(v ,v )*b6
    s    3x       2x         x        x              x            x  x

                                               2                   2       4
         + f(v,v  )*b7 + f(v,v )* b9*u + f(v,v) *b11 + f(v,v)*b10*u  + b5*u
                2x            x

   v  = u  *b13*v + u *v *b14 + u *b16*u*v + v  *b12 + v  *b15*u + v *f(v,v)*b19
    s    2x          x  x        x            3x        2x          x

                   2                                         3
         + v *b17*u  + f(v,v )*b20*v + f(v,v)*b21*u*v + b18*u *v
            x               x