Solution 13 to problem l1o35
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
b74=0
b73=0
b72=0
1 3
- ----*a2 *a8*b1
54
b71=-------------------
4
a1
1 2
- ---*a2 *a8*b1
3
b70=------------------
3
a1
b69=0
1 2
- ---*a2 *a8*b1
6
b68=------------------
3
a1
- a2*a8*b1
b67=-------------
2
a1
- a2*a8*b1
b66=-------------
2
a1
b65=0
b64=0
- a2*a8*b1
b63=-------------
2
a1
b62=0
1
- ---*a2*a8*b1
6
b61=-----------------
2
a1
- a8*b1
b60=----------
a1
- 3*a8*b1
b59=------------
a1
b58=0
b57=0
b56=0
b55=0
1 5
- -----*a2 *b1
486
b54=-----------------
5
a1
b53=0
5 4
- ----*a2 *b1
81
b52=----------------
4
a1
b51=0
b50=0
5 3
- ----*a2 *b1
18
b49=----------------
3
a1
5 3
- ----*a2 *b1
27
b48=----------------
3
a1
b47=0
b46=0
b45=0
b44=0
5 2
- ---*a2 *b1
9
b43=---------------
2
a1
5 2
- ----*a2 *b1
18
b42=----------------
2
a1
b41=0
b40=0
b39=0
b38=0
1
- ---*a2*b1
6
b37=--------------
a1
b36=0
b35=0
b34=0
b33=0
b32=0
b31=0
b30=0
b29=0
b28=0
2 2
---*a2 *a8*b1
9
b27=---------------
3
a1
1 2
---*a2 *a8*b1
3
b26=---------------
3
a1
2*a2*a8*b1
b25=------------
2
a1
2*a2*a8*b1
b24=------------
2
a1
6*a2*a8*b1
b23=------------
2
a1
a2*a8*b1
b22=----------
2
a1
a2*a8*b1
b21=----------
2
a1
6*a8*b1
b20=---------
a1
18*a8*b1
b19=----------
a1
12*a8*b1
b18=----------
a1
12*a8*b1
b17=----------
a1
8*a8*b1
b16=---------
a1
a8*b1
b15=-------
a1
6*a8*b1
b14=---------
a2
24*a8*b1
b13=----------
a2
18*a8*b1
b12=----------
a2
b11=0
5 4
----*a2 *b1
81
b10=-------------
4
a1
10 3
----*a2 *b1
9
b9=-------------
3
a1
10 3
----*a2 *b1
27
b8=-------------
3
a1
5 2
---*a2 *b1
3
b7=------------
2
a1
50 2
----*a2 *b1
9
b6=-------------
2
a1
10 2
----*a2 *b1
9
b5=-------------
2
a1
10
----*a2*b1
3
b4=------------
a1
5*a2*b1
b3=---------
a1
5
---*a2*b1
3
b2=-----------
a1
1
- ---*a2*a8
6
a21=--------------
a1
a20= - a8
a19=0
1 3
- ----*a2
54
a18=-------------
2
a1
a17=0
1 2
- ---*a2
6
a16=------------
a1
a15=0
a14=0
1
a13= - ---*a2
6
a12=0
a11=0
a10=0
a9=2*a8
a8=a8
6*a1*a8
a7=---------
a2
6*a1*a8
a6=---------
a2
a5=0
1 2
---*a2
3
a4=---------
a1
a3=a2
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,a8,a2,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 2 2 2 1 3 2
u =(u *a1 *a2 + u *a1*a2 *u + u *a1*a2 + u *f(v,v)*a1*a2*a8 + ---*u *a2 *u
t 3x 2x x x 3 x
2 2
+ 6*f(v ,v )*a1 *a8 + 6*f(v,v )*a1 *a8 + 2*f(v,v )*a1*a2*a8*u)/(a1*a2)
x x 2x x
1 2 1 2 2
v =( - ---*u *a1 *a2*v - ---*u *a1*a2 *u*v - f(v,v )*a1 *a8*v
t 6 2x 6 x x
1 1 3 3 2
- ---*f(v,v)*a1*a2*a8*u*v - ----*a2 *u *v)/a1
6 54
4 5 3 2 3 2
u =(u *a1 *a2*b1 + ---*u *a1 *a2 *b1*u + 5*u *u *a1 *a2 *b1
s 5x 3 4x 3x x
3 10 2 3 2 10 2 3 2
+ u *f(v,v)*a1 *a2*a8*b1 + ----*u *a1 *a2 *b1*u + ----*u *a1 *a2 *b1
3x 9 3x 3 2x
50 2 3 3
+ ----*u *u *a1 *a2 *b1*u + 8*u *f(v,v )*a1 *a2*a8*b1
9 2x x 2x x
2 2 10 4 3 5 3 2 3
+ u *f(v,v)*a1 *a2 *a8*b1*u + ----*u *a1*a2 *b1*u + ---*u *a1 *a2 *b1
2x 27 2x 3 x
2 2 2 10 2 4 2
+ u *f(v,v)*a1 *a2 *a8*b1 + ----*u *a1*a2 *b1*u
x 9 x
3 3
+ 12*u *f(v ,v )*a1 *a2*a8*b1 + 12*u *f(v,v )*a1 *a2*a8*b1
x x x x 2x
2 2 1 3 2
+ 6*u *f(v,v )*a1 *a2 *a8*b1*u + ---*u *f(v,v)*a1*a2 *a8*b1*u
x x 3 x
5 5 4 4 4
+ ----*u *a2 *b1*u + 18*f(v ,v )*a1 *a8*b1 + 24*f(v ,v )*a1 *a8*b1
81 x 2x 2x x 3x
3 2 2 2
+ 18*f(v ,v )*a1 *a2*a8*b1*u + 2*f(v ,v )*a1 *a2 *a8*b1*u
x 2x x x
4 3
+ 6*f(v,v )*a1 *a8*b1 + 6*f(v,v )*a1 *a2*a8*b1*u
4x 3x
2 2 2 2 3 3 4
+ 2*f(v,v )*a1 *a2 *a8*b1*u + ---*f(v,v )*a1*a2 *a8*b1*u )/(a1 *a2)
2x 9 x
1 4 5 3 2 5 3 2
v =( - ---*u *a1 *a2*b1*v - ----*u *a1 *a2 *b1*u*v - ---*u *u *a1 *a2 *b1*v
s 6 4x 18 3x 9 2x x
1 3 5 2 3 2
- ---*u *f(v,v)*a1 *a2*a8*b1*v - ----*u *a1 *a2 *b1*u *v
6 2x 27 2x
5 2 2 3 3
- ----*u *a1 *a2 *b1*u*v - u *f(v,v )*a1 *a2*a8*b1*v
18 x x x
1 2 2 5 4 3
- ---*u *f(v,v)*a1 *a2 *a8*b1*u*v - ----*u *a1*a2 *b1*u *v
6 x 81 x
4 3
- 3*f(v ,v )*a1 *a8*b1*v - f(v ,v )*a1 *a2*a8*b1*u*v
x 2x x x
4 3
- f(v,v )*a1 *a8*b1*v - f(v,v )*a1 *a2*a8*b1*u*v
3x 2x
1 2 2 2 1 3 3
- ---*f(v,v )*a1 *a2 *a8*b1*u *v - ----*f(v,v)*a1*a2 *a8*b1*u *v
3 x 54
1 5 5 5
- -----*a2 *b1*u *v)/a1
486