Solution 1 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
5 2 2
- ----*a13 *a7 *b36
27
b73=----------------------
4
a12
5 2 2
- ----*a13 *a7 *b36
27
b72=----------------------
4
a12
b71=0
b70=0
b69=0
b68=0
b67=0
b66=0
b65=0
b64=0
5 2
- ---*a13 *a7*b36
9
b63=--------------------
3
a12
5 2
---*a13 *a7*b36
9
b62=-----------------
3
a12
b61=0
10
- ----*a13*a7*b36
9
b60=--------------------
2
a12
25
- ----*a13*a7*b36
9
b59=--------------------
2
a12
10
- ----*a13*a7*b36
9
b58=--------------------
2
a12
5
- ---*a13*a7*b36
3
b57=-------------------
2
a12
5
- ---*a13*a7*b36
3
b56=-------------------
2
a12
b55=0
b54=0
b53=0
b52=0
b51=0
b50=0
b49=0
b48=0
b47=0
b46=0
5 2
---*a13 *b36
9
b45=--------------
2
a12
b44=0
10 2
----*a13 *b36
9
b43=---------------
2
a12
b42=0
b41=0
5
---*a13*b36
3
b40=-------------
a12
10
----*a13*b36
3
b39=--------------
a12
25
----*a13*b36
9
b38=--------------
a12
10
----*a13*b36
9
b37=--------------
a12
b35=0
b34=0
b33=0
10 2 2
- ----*a13 *a7 *b36
27
b32=----------------------
4
a12
5 2
- ----*a13*a7 *b36
27
b31=---------------------
3
a12
40 2
- ----*a13*a7 *b36
27
b30=---------------------
3
a12
25 2
- ----*a13*a7 *b36
27
b29=---------------------
3
a12
b28=0
b27=0
b26=0
b25=0
b24=0
b23=0
10 2
----*a13 *a7*b36
9
b22=------------------
3
a12
b21=0
b20=0
b19=0
10
----*a13*a7*b36
3
b18=-----------------
2
a12
5
---*a13*a7*b36
9
b17=----------------
2
a12
10
----*a13*a7*b36
9
b16=-----------------
2
a12
10
----*a13*a7*b36
9
b15=-----------------
2
a12
10
----*a7*b36
9
b14=-------------
a12
5
---*a7*b36
9
b13=------------
a12
10
----*a7*b36
9
b12=-------------
a12
b11=0
b10=0
b9=0
b8=0
5 2
- ----*a13 *b36
27
b7=------------------
2
a12
b6=0
b5=0
b4=0
5
- ---*a13*b36
9
b3=----------------
a12
b2=0
1
b1= - ---*b36
9
a21=0
- a13*a7
a20=-----------
a12
a19=0
a18=0
a17=0
a16=0
a15=0
a14=a13
1 2
- ---*a13*a7
3
a11=----------------
2
a12
a10=0
a9=0
a13*a7
a8=--------
a12
a6=0
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,a7,a13,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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2 1 2 2
u *f(v,v)*a12*a13*a7 + f(v,v )*a12 *a7 - ---*f(v,v) *a13*a7
x 2x 3
u =---------------------------------------------------------------
t 2
a12
2
u *a12*a13*v + u *v *a12*a13 + v *a12 - f(v,v )*a13*a7*v
2x x x 3x x
v =-------------------------------------------------------------
t a12
1 4 5 3
u =( - ---*u *a12 *b36 - ---*u *u *a12 *a13*b36
s 9 5x 9 3x x
10 2 10 2
+ ----*u *f(v,v)*a12 *a13*a7*b36 + ----*u *f(v,v )*a12 *a13*a7*b36
9 3x 9 2x x
5 3 2 2 10 2 2
- ----*u *a12 *a13 *b36 + ----*u *f(v,v)*a12*a13 *a7*b36
27 x 9 x
5 2 10 2
+ ---*u *f(v ,v )*a12 *a13*a7*b36 + ----*u *f(v,v )*a12 *a13*a7*b36
9 x x x 3 x 2x
10 2 2 2 10 3
- ----*u *f(v,v) *a13 *a7 *b36 + ----*f(v ,v )*a12 *a7*b36
27 x 9 2x 2x
5 3 25 2
+ ---*f(v ,v )*a12 *a7*b36 - ----*f(v ,v )*f(v,v)*a12*a13*a7 *b36
9 x 3x 27 x x
10 3 40 2
+ ----*f(v,v )*a12 *a7*b36 - ----*f(v,v )*f(v,v)*a12*a13*a7 *b36
9 4x 27 2x
5 2 2 4
- ----*f(v,v ) *a12*a13*a7 *b36)/a12
27 x
10 3 25 3
v =(----*u *a12 *a13*b36*v + ----*u *v *a12 *a13*b36
s 9 4x 9 3x x
10 2 2 10 3
+ ----*u *u *a12 *a13 *b36*v + ----*u *v *a12 *a13*b36
9 2x x 3 2x 2x
5 2 2 2 5 3
+ ---*u *v *a12 *a13 *b36 + ---*u *v *a12 *a13*b36
9 x x 3 x 3x
5 2 5 2
+ ---*u *v *f(v,v)*a12*a13 *a7*b36 - ---*u *f(v,v )*a12*a13 *a7*b36*v
9 x x 9 x x
4 5 2
+ v *a12 *b36 - ---*v *f(v,v )*a12 *a13*a7*b36
5x 3 2x x
5 2 10 2
- ---*v *f(v ,v )*a12 *a13*a7*b36 - ----*v *f(v,v )*a12 *a13*a7*b36
3 x x x 9 x 2x
5 2 2 2 25 2
- ----*v *f(v,v) *a13 *a7 *b36 - ----*f(v ,v )*a12 *a13*a7*b36*v
27 x 9 x 2x
10 2 5 2 2
- ----*f(v,v )*a12 *a13*a7*b36*v - ----*f(v,v )*f(v,v)*a13 *a7 *b36*v)/
9 3x 27 x
4
a12