Solution 5 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
3 2
---* a9 *b1
2
b73=-------------
2
a1
3 2
---* a9 *b1
8
b72=-------------
2
a1
b71=0
1 3
- ---* a9 *b1
2
b70=----------------
2
a1*a6
3 3
- ---* a9 *b1
4
b69=----------------
2
a1*a6
b68=0
b67=0
b66=0
2
- 2* a9 *b1
b65=--------------
a1*a6
b64=0
2
- a9 *b1
b63=------------
a1*a6
3 2
- ---* a9 *b1
2
b62=----------------
a1*a6
b61=0
b60=0
b59=0
- a9*b1
b58=-----------
a1
- a9*b1
b57=-----------
a1
b56=0
b55=0
b54=0
3 4
---* a9 *b1
8
b53=-------------
4
a6
b52=0
b51=0
3 3
---* a9 *b1
2
b50=-------------
3
a6
b49=0
b48=0
b47=0
b46=0
1 2
---* a9 *b1
2
b45=-------------
2
a6
2
- a9 *b1
b44=------------
2
a6
b43=0
b42=0
b41=0
b40=0
b39=0
- a9*b1
b38=-----------
a6
b37=0
b36=0
b35=0
b34=0
5 2
---* a9 *b1
2
b33=-------------
2
a1
15 2
----* a9 *b1
8
b32=--------------
2
a1
3* a9*a6*b1
b31=-------------
2
a1
5
---* a9*a6*b1
2
b30=---------------
2
a1
5
---* a9*a6*b1
2
b29=---------------
2
a1
b28=0
3 3
- ---* a9 *b1
2
b27=----------------
2
a1*a6
15 3
- ----* a9 *b1
4
b26=-----------------
2
a1*a6
3 2
- ---* a9 *b1
2
b25=----------------
a1*a6
3 2
- ---* a9 *b1
2
b24=----------------
a1*a6
2
- 3* a9 *b1
b23=--------------
a1*a6
b22=0
b21=0
a9*b1
b20=--------
a1
3* a9*b1
b19=----------
a1
4* a9*b1
b18=----------
a1
4* a9*b1
b17=----------
a1
6* a9*b1
b16=----------
a1
5
---* a9*b1
2
b15=------------
a1
a6*b1
b14=-------
a1
4*a6*b1
b13=---------
a1
3*a6*b1
b12=---------
a1
b11=0
15 4
----* a9 *b1
8
b10=--------------
4
a6
b9=0
b8=0
5 2
- ---* a9 *b1
2
b7=----------------
2
a6
2
- 10* a9 *b1
b6=---------------
2
a6
5 2
- ---* a9 *b1
2
b5=----------------
2
a6
b4=0
b3=0
b2=0
a21=0
a20= a9
1
a19=---* a9
2
a18=0
1 2
- ---* a9 *a1
2
a17=----------------
2
a6
a16=0
a15=0
- a9*a1
a14=-----------
a6
a13=0
a12=0
a11=0
a10=0
a9= a9
3
a8=---* a9
2
a7=a6
a5=0
3 2
- ---* a9 *a1
2
a4=----------------
2
a6
a3=0
a2=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, a9,a6,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) is a scalar function, v=v(t,x) is a vector
function of arbitrary dimension and f(..,..) is the scalar
product between two such vectors:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2 3 2 3 2 2 3
u =(u *a1*a6 + ---*u *f(v,v)* a9*a6 - ---*u * a9 *a1*u + f(v ,v )*a6
t 3x 2 x 2 x x x
3 2 2
+ f(v,v )*a6 + f(v,v )* a9*a6 *u)/a6
2x x
1 2 1 2 2
v =( - u *v * a9*a1*a6 + ---*v *f(v,v)* a9*a6 - ---*v * a9 *a1*u
t x x 2 x 2 x
2 2
+ f(v,v )* a9*a6 *v)/a6
x
2 4 5 4 5 2 2 2 2
u =(u *a1 *a6 *b1 + ---*u *f(v,v)* a9*a1*a6 *b1 - ---*u * a9 *a1 *a6 *b1*u
s 5x 2 3x 2 3x
2 2 2 4
- 10*u *u * a9 *a1 *a6 *b1*u + 6*u *f(v,v )* a9*a1*a6 *b1
2x x 2x x
5 3 2 2 2 4
- ---*u * a9 *a1 *a6 *b1 + 4*u *f(v ,v )* a9*a1*a6 *b1
2 x x x x
4 2 3
+ 4*u *f(v,v )* a9*a1*a6 *b1 - 3*u *f(v,v )* a9 *a1*a6 *b1*u
x 2x x x
15 2 2 4 15 3 2 2
+ ----*u *f(v,v) * a9 *a6 *b1 - ----*u *f(v,v)* a9 *a1*a6 *b1*u
8 x 4 x
15 4 2 4 5 5
+ ----*u * a9 *a1 *b1*u + 3*f(v ,v )*a1*a6 *b1 + 4*f(v ,v )*a1*a6 *b1
8 x 2x 2x x 3x
4 5 5
+ 3*f(v ,v )* a9*a1*a6 *b1*u + ---*f(v ,v )*f(v,v)* a9*a6 *b1
x 2x 2 x x
3 2 3 2 5
- ---*f(v ,v )* a9 *a1*a6 *b1*u + f(v,v )*a1*a6 *b1
2 x x 4x
4 5 5
+ f(v,v )* a9*a1*a6 *b1*u + ---*f(v,v )*f(v,v)* a9*a6 *b1
3x 2 2x
3 2 3 2 2 5
- ---*f(v,v )* a9 *a1*a6 *b1*u + 3*f(v,v ) * a9*a6 *b1
2 2x x
5 2 4 3 3 2 3 2
+ ---*f(v,v )*f(v,v)* a9 *a6 *b1*u - ---*f(v,v )* a9 *a1*a6 *b1*u )/(a1
2 x 2 x
4
*a6 )
2 3 2 2 2
v =( - u *v * a9*a1 *a6 *b1 - u *v * a9 *a1 *a6 *b1*u
s 3x x 2x x
1 2 2 2 2 3 2 3
+ ---*u *v * a9 *a1 *a6 *b1 - ---*u *v *f(v,v)* a9 *a1*a6 *b1
2 x x 2 x x
3 3 2 2 2 3
+ ---*u *v * a9 *a1 *a6*b1*u - u *f(v,v )* a9 *a1*a6 *b1*v
2 x x x x
4 4
- v *f(v ,v )* a9*a1*a6 *b1 - v *f(v,v )* a9*a1*a6 *b1
x x x x 2x
2 3 3 2 2 4
- 2*v *f(v,v )* a9 *a1*a6 *b1*u + ---*v *f(v,v) * a9 *a6 *b1
x x 8 x
3 3 2 2 3 4 2 4
- ---*v *f(v,v)* a9 *a1*a6 *b1*u + ---*v * a9 *a1 *b1*u
4 x 8 x
3 2 4 1 3 2 2 2
+ ---*f(v,v )*f(v,v)* a9 *a6 *b1*v - ---*f(v,v )* a9 *a1*a6 *b1*u *v)/(a1
2 x 2 x
4
*a6 )