Solution 1 to problem v2l1o23

Expressions | Parameters | Relevance | Back to problem v2l1o23

Expressions

The solution is given through the following expressions:


b32=0


b31=0


     3*a4*b1
b30=---------
       a1


      - 3*a4*b1
b27=------------
         a1


      - 3*a4*b1
b26=------------
         a1


b24=0


b21=0


b13=0


b12=0


     - 3*a4*b1
b9=------------
        a1


     - 3*a4*b1
b7=------------
        a1


    3*a4*b1
b6=---------
      a1


b4=0


b3=0


a16=0


a15= - a4


a14=2*a4


a13=0


a6=0


a5= - 2*a4


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,a4,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 the same arbitrary dimension and f(..,..) is the scalar product between two such vectors:

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

df(u,t)=u  *a1 + f(u,u)*a4*v - 2*f(v,u)*a4*u
         2x

         u  *a1*b1 - 3*u *f(v,u)*a4*b1 + 3*f(u,u )*a4*b1*v - 3*f(v,u )*a4*b1*u
          3x            x                       x                   x
df(u,s)=-----------------------------------------------------------------------
                                          a1

df(v,t)= - v  *a1 + 2*f(v,u)*a4*v - f(v,v)*a4*u
            2x

         v  *a1*b1 - 3*v *f(v,u)*a4*b1 - 3*f(u,v )*a4*b1*v + 3*f(v,v )*a4*b1*u
          3x            x                       x                   x
df(v,s)=-----------------------------------------------------------------------
                                          a1