Solution 2 to problem v2l1o23

Expressions | Parameters | Relevance | Back to problem v2l1o23

Expressions

The solution is given through the following expressions:


b32=0


b31=0


b30=0


      3
     ---*a5*b1
      2
b27=-----------
        a1


      3
     ---*a5*b1
      2
b26=-----------
        a1


b24=0


b21=0


b13=0


b12=0


     3
    ---*a5*b1
     2
b9=-----------
       a1


     3
    ---*a5*b1
     2
b7=-----------
       a1


b6=0


b4=0


b3=0


a16=0


a15=0


a14= - a5


a13=0


a6=0


a4=0


a3=0

Parameters

Apart from the condition that they must not vanish to give a non-trivial solution and a non-singular solution with non-vanishing denominators, the following parameters are free:

 b1,a5,a1

Relevance for the application:

The solution given above tells us that the system {u_s, v_s} is a higher order symmetry for the lower order system {u_t,v_t} where u=u(t,x), v=v(t,x) are vector functions of arbitrary dimension and f(..,..) is the scalar product between two such vectors:

/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\

df(u,t)=u  *a1 + f(v,u)*a5*u
         2x

                      3                     3
         u  *a1*b1 + ---*u *f(v,u)*a5*b1 + ---*f(v,u )*a5*b1*u
          3x          2   x                 2       x
df(u,s)=-------------------------------------------------------
                                  a1

df(v,t)= - v  *a1 - f(v,u)*a5*v
            2x

                      3                     3
         v  *a1*b1 + ---*v *f(v,u)*a5*b1 + ---*f(u,v )*a5*b1*v
          3x          2   x                 2       x
df(v,s)=-------------------------------------------------------
                                  a1