Problem l112o12

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

Unknowns

All solutions for the following 26 unknowns have to be determined: 
a0, a2, a3, a4, a5, a6, a7, a8, b1, .., b19

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 problem after
                                                     % Galilei transformation
   {b1,b5,b12},                                      % 2nd order symmetry
   {a3,a5,a4,b6,b4,b5,b9,b7,b11,b10,b8},             % u_t or u_s to involve a v
   {a7,b13,b14,b18,b15}                              % v_t or v_s to involve a u

Equations

All comma separated 36 expressions involving 159 terms have to vanish. All terms are products of one a- and one b-unknown. 
a2*b14,
a2*b1,
a2*b2,
a2*b3,
a0*b14 + 2*a7*b12,
2*a2*b15 - a7*b3,
a3*b14 + 4*a8*b12,
a4*b14 + 2*a7*b16,
a3*b13 - a7*b5,
a5*b14 + 2*a8*b16,
a0*b13 - a7*b1 + a7*b12,
a3*b13 - a7*b4 + 2*a8*b12,
a0*b5 - a3*b1 + a3*b12,
a4*b15 - a4*b3 - a7*b8,
a5*b18 - a7*b11 + 2*a8*b19,
2*a0*b15 + 2*a2*b13 + a7*b14 - a7*b2,
2*a3*b15 + 2*a4*b13 + 2*a7*b17 - a7*b7,
2*a0*b4 + a0*b5 - 3*a3*b1 + a3*b12,
a3*b18 + 4*a5*b13 - a7*b9 + 2*a8*b17,
a4*b18 + 2*a5*b15 - a7*b10 + 4*a7*b19,
a4*b11 - a5*b10 + 4*a5*b19 - 6*a8*b11,
a2*b18 + 2*a4*b15 + 2*a7*b18 - a7*b8 - 2*a8*b15,
2*a3*b15 + 2*a4*b13 - a4*b2 - 2*a7*b6 - a7*b7,
a3*b18 + 4*a5*b13 - a5*b2 - a7*b9 - 2*a8*b6,
a0*b18 + a4*b13 + a7*b16 + a7*b17 - a7*b6 - 2*a8*b13,
a0*b7 + 2*a2*b4 + a3*b14 - a3*b2 - 2*a4*b1 - 2*a7*b4,
a2*b11 + a4*b19 + 2*a5*b18 - a5*b8 - 3*a7*b11 - 2*a8*b10,
2*a0*b6 + 2*a3*b13 + a3*b14 - a3*b2 - 4*a4*b1 - 2*a7*b4 - 2*a7*b5,
a0*b7 + 2*a2*b5 + a3*b14 - a3*b2 - 2*a4*b1 + 2*a4*b12 - 2*a7*b5,
2*a0*b8 + a2*b7 + 2*a3*b15 - 3*a3*b3 + 2*a4*b14 - 2*a4*b2 - 2*a7*b7,
2*a0*b9 + 2*a3*b16 + 2*a3*b17 - a3*b7 - 8*a5*b1 - 4*a8*b4 - 4*a8*b5,
a2*b10 + 2*a4*b18 - a4*b8 + 4*a5*b15 - 3*a5*b3 - 4*a7*b10 - 2*a8*b8,
a0*b9 + a3*b16 + a3*b17 - a3*b6 + a4*b4 - 4*a5*b1 - 2*a8*b4 - 2*a8*b5,
a0*b9 + a3*b16 + a3*b17 - a3*b6 + a4*b5 - 4*a5*b1 + 4*a5*b12 - 4*a8*b5,
6*a0*b11 - a3*b10 + 6*a3*b19 + a4*b9 + 4*a5*b16 + 4*a5*b17 - 4*a5*b6 - a5*b7 - 
6*a8*b9,
2*a0*b10 + a2*b9 + 2*a3*b18 - a3*b8 + a4*b16 + a4*b17 - a4*b6 + 2*a5*b14 - 2*a5
*b2 - 2*a7*b9 - 2*a8*b7

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 + u *a6 + f(v,v )*a3 + f(v,v) *a5 + f(v,v)*a4*u + a2*u ,
  t    x       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: 
 u  = u  *b1 + u *f(v,v)*b6 + u *b2*u + f(v ,v )*b4 + f(v,v  )*b5
  s    2x       x              x           x  x            2x

                                                  3             2
       + f(v,v )*f(v,v)*b9 + f(v,v )*b7*u + f(v,v) *b11 + f(v,v) *b10*u
              x                   x

                    2       3
       + f(v,v)*b8*u  + b3*u ,


 v  = u *b13*v + v  *b12 + v *f(v,v)*b16 + v *b14*u + f(v,v )*b17*v
  s    x          2x        x               x              x

               2                               2
       + f(v,v) *b19*v + f(v,v)*b18*u*v + b15*u *v