Solution 2 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
10 2
----*a19 *b36
9
b73=---------------
2
a12
5 2
---*a19 *b36
9
b72=--------------
2
a12
b71=0
b70=0
b69=0
b68=0
b67=0
b66=0
b65=0
b64=0
20
----*a13*a19*b36
9
b63=------------------
2
a12
20
----*a13*a19*b36
9
b62=------------------
2
a12
10
----*a13*a19*b36
9
b61=------------------
2
a12
10
----*a19*b36
9
b60=--------------
a12
10
----*a19*b36
3
b59=--------------
a12
35
----*a19*b36
9
b58=--------------
a12
35
----*a19*b36
9
b57=--------------
a12
5*a19*b36
b56=-----------
a12
5
---*a19*b36
3
b55=-------------
a12
b54=0
b53=0
b52=0
b51=0
b50=0
b49=0
b48=0
b47=0
b46=0
20 2
----*a13 *b36
9
b45=---------------
2
a12
b44=0
20 2
----*a13 *b36
9
b43=---------------
2
a12
b42=0
b41=0
10
----*a13*b36
3
b40=--------------
a12
5*a13*b36
b39=-----------
a12
35
----*a13*b36
9
b38=--------------
a12
10
----*a13*b36
9
b37=--------------
a12
10 3
- ----*a19 *b36
27
b35=------------------
2
a12 *a13
b34=0
b33=0
5 2
- ---*a19 *b36
3
b32=-----------------
2
a12
5 2
- ---*a19 *b36
2
b31=-----------------
a12*a13
10 2
- ----*a19 *b36
3
b30=------------------
a12*a13
5 2
- ---*a19 *b36
6
b29=-----------------
a12*a13
b28=0
b27=0
b26=0
b25=0
b24=0
b23=0
20
- ----*a13*a19*b36
9
b22=---------------------
2
a12
b21=0
b20=0
b19=0
40
- ----*a19*b36
9
b18=-----------------
a12
5
---*a19*b36
9
b17=-------------
a12
10
- ----*a19*b36
3
b16=-----------------
a12
5
- ---*a19*b36
3
b15=----------------
a12
10
- ----*a19*b36
9
b14=-----------------
a13
5
---*a19*b36
9
b13=-------------
a13
5
- ---*a19*b36
6
b12=----------------
a13
b11=0
b10=0
b9=0
b8=0
20 2
- ----*a13 *b36
27
b7=------------------
2
a12
b6=0
b5=0
5
- ---*a13*b36
6
b4=----------------
a12
10
- ----*a13*b36
9
b3=-----------------
a12
b2=0
1
b1= - ---*b36
9
a21=0
a20=a19
a18=0
a17=0
a16=0
a15=0
a14=2*a13
1 2
- ---*a19
2
a11=-------------
a13
a10=0
a9=0
a8= - a19
- a12*a19
a7=------------
a13
1
---*a12*a19
2
a6=-------------
a13
a5=0
a4=0
a3=0
a2=0
a1=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:
b36,a13,a19,a12
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
1
u =( - u *f(v,v)*a13*a19 + ---*f(v ,v )*a12*a19 - f(v,v )*a12*a19
t x 2 x x 2x
1 2 2
- ---*f(v,v) *a19 )/a13
2
v =u *a13*v + 2*u *v *a13 + v *a12 + v *f(v,v)*a19 + f(v,v )*a19*v
t 2x x x 3x x x
1 2 10 2
u =( - ---*u *a12 *a13*b36 - ----*u *u *a12*a13 *b36
s 9 5x 9 3x x
5 5 2 2
- ---*u *f(v,v)*a12*a13*a19*b36 - ---*u *a12*a13 *b36
3 3x 6 2x
10 20 3 3
- ----*u *f(v,v )*a12*a13*a19*b36 - ----*u *a13 *b36
3 2x x 27 x
20 2 2 5
- ----*u *f(v,v)*a13 *a19*b36 + ---*u *f(v ,v )*a12*a13*a19*b36
9 x 9 x x x
40 5 2 2
- ----*u *f(v,v )*a12*a13*a19*b36 - ---*u *f(v,v) *a13*a19 *b36
9 x 2x 3 x
5 2 5 2
- ---*f(v ,v )*a12 *a19*b36 + ---*f(v ,v )*a12 *a19*b36
6 2x 2x 9 x 3x
5 2 10 2
- ---*f(v ,v )*f(v,v)*a12*a19 *b36 - ----*f(v,v )*a12 *a19*b36
6 x x 9 4x
10 2 5 2 2
- ----*f(v,v )*f(v,v)*a12*a19 *b36 - ---*f(v,v ) *a12*a19 *b36
3 2x 2 x
10 3 3 2
- ----*f(v,v) *a19 *b36)/(a12 *a13)
27
10 35 20 2
v =(----*u *a12*a13*b36*v + ----*u *v *a12*a13*b36 + ----*u *u *a13 *b36*v
s 9 4x 9 3x x 9 2x x
10
+ 5*u *v *a12*a13*b36 + ----*u *f(v,v)*a13*a19*b36*v
2x 2x 9 2x
20 2 2 10
+ ----*u *v *a13 *b36 + ----*u *v *a12*a13*b36
9 x x 3 x 3x
20 20
+ ----*u *v *f(v,v)*a13*a19*b36 + ----*u *f(v,v )*a13*a19*b36*v
9 x x 9 x x
2 5
+ v *a12 *b36 + ---*v *f(v,v)*a12*a19*b36 + 5*v *f(v,v )*a12*a19*b36
5x 3 3x 2x x
35 35
+ ----*v *f(v ,v )*a12*a19*b36 + ----*v *f(v,v )*a12*a19*b36
9 x x x 9 x 2x
5 2 2 10
+ ---*v *f(v,v) *a19 *b36 + ----*f(v ,v )*a12*a19*b36*v
9 x 3 x 2x
10 10 2 2
+ ----*f(v,v )*a12*a19*b36*v + ----*f(v,v )*f(v,v)*a19 *b36*v)/a12
9 3x 9 x