Solution 11 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
b73=0
1
---*a19*b59
2
b72=-------------
a1
b71=0
2 2
----*a2 *b59
27
b70=--------------
2
a1
4 2
----*a2 *b59
27
b69=--------------
2
a1
1 2
---*a2 *b59
9
b68=-------------
2
a1
1
---*a2*b59
3
b67=------------
a1
1
---*a2*b59
3
b66=------------
a1
2
---*a2*b59
3
b65=------------
a1
b64=0
4
---*a2*b59
9
b63=------------
a1
5
---*a2*b59
9
b62=------------
a1
1
---*a2*b59
6
b61=------------
a1
1
b60=---*b59
3
2
b58=---*b59
3
2
b57=---*b59
3
b56=0
b55=0
b54=0
2 4
-----*a2 *b59
243
b53=---------------
3
a1 *a19
4 4
-----*a2 *b59
243
b52=---------------
3
a1 *a19
b51=0
4 3
----*a2 *b59
27
b50=--------------
2
a1 *a19
4 3
----*a2 *b59
27
b49=--------------
2
a1 *a19
2 3
----*a2 *b59
27
b48=--------------
2
a1 *a19
b47=0
b46=0
2 2
---*a2 *b59
9
b45=-------------
a1*a19
8 2
----*a2 *b59
27
b44=--------------
a1*a19
10 2
----*a2 *b59
27
b43=--------------
a1*a19
4 2
----*a2 *b59
27
b42=--------------
a1*a19
b41=0
b40=0
b39=0
2
---*a2*b59
9
b38=------------
a19
1
---*a2*b59
9
b37=------------
a19
b36=0
b35=0
b34=0
2*a19*b59
b33=-----------
a1
1
---*a19*b59
2
b32=-------------
a1
6*a19*b59
b31=-----------
a2
3*a19*b59
b30=-----------
a2
3*a19*b59
b29=-----------
a2
b28=0
8 2
----*a2 *b59
27
b27=--------------
2
a1
4 2
---*a2 *b59
9
b26=-------------
2
a1
4
---*a2*b59
3
b25=------------
a1
4
---*a2*b59
3
b24=------------
a1
16
----*a2*b59
3
b23=-------------
a1
4
---*a2*b59
3
b22=------------
a1
4
---*a2*b59
3
b21=------------
a1
8
b20=---*b59
3
b19=8*b59
20
b18=----*b59
3
20
b17=----*b59
3
20
b16=----*b59
3
4
b15=---*b59
3
2*a1*b59
b14=----------
a2
8*a1*b59
b13=----------
a2
6*a1*b59
b12=----------
a2
b11=0
10 4
-----*a2 *b59
243
b10=---------------
3
a1 *a19
20 3
----*a2 *b59
27
b9=--------------
2
a1 *a19
20 3
----*a2 *b59
81
b8=--------------
2
a1 *a19
10 2
----*a2 *b59
9
b7=--------------
a1*a19
100 2
-----*a2 *b59
27
b6=---------------
a1*a19
20 2
----*a2 *b59
27
b5=--------------
a1*a19
20
----*a2*b59
9
b4=-------------
a19
10
----*a2*b59
3
b3=-------------
a19
10
----*a2*b59
9
b2=-------------
a19
2
---*a1*b59
3
b1=------------
a19
a21=0
a20=0
a18=0
1 2
---*a2
9
a17=---------
a1
1 2
---*a2
9
a16=---------
a1
a15=0
1
a14=---*a2
3
1
a13=---*a2
6
a12=0
a11=0
a10=0
a9=2*a19
a8=a19
3*a1*a19
a7=----------
a2
3*a1*a19
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:
b59,a19,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*a19*a2 + ---*u *a2 *u
t 3x 2x x x 3 x
2 2
+ 3*f(v ,v )*a1 *a19 + 3*f(v,v )*a1 *a19 + 2*f(v,v )*a1*a19*a2*u)/(a1*a2)
x x 2x x
1 1 1 2
v =(---*u *a1*a2*v + ---*u *v *a1*a2 + ---*u *a2 *u*v + v *f(v,v)*a1*a19
t 6 2x 3 x x 9 x x
1 2 2
+ ---*v *a2 *u )/a1
9 x
2 4 10 3 2 10 3 2
u =(---*u *a1 *a2*b59 + ----*u *a1 *a2 *b59*u + ----*u *u *a1 *a2 *b59
s 3 5x 9 4x 3 3x x
4 3 20 2 3 2
+ ---*u *f(v,v)*a1 *a19*a2*b59 + ----*u *a1 *a2 *b59*u
3 3x 27 3x
20 2 3 2 100 2 3
+ ----*u *a1 *a2 *b59 + -----*u *u *a1 *a2 *b59*u
9 2x 27 2x x
20 3 4 2 2
+ ----*u *f(v,v )*a1 *a19*a2*b59 + ---*u *f(v,v)*a1 *a19*a2 *b59*u
3 2x x 3 2x
20 4 3 10 3 2 3
+ ----*u *a1*a2 *b59*u + ----*u *a1 *a2 *b59
81 2x 9 x
4 2 2 2 20 2 4 2
+ ---*u *f(v,v)*a1 *a19*a2 *b59 + ----*u *a1*a2 *b59*u
3 x 27 x
20 3 20 3
+ ----*u *f(v ,v )*a1 *a19*a2*b59 + ----*u *f(v,v )*a1 *a19*a2*b59
3 x x x 3 x 2x
16 2 2 1 2 2 2
+ ----*u *f(v,v )*a1 *a19*a2 *b59*u + ---*u *f(v,v) *a1 *a19 *a2*b59
3 x x 2 x
4 3 2 10 5 4
+ ---*u *f(v,v)*a1*a19*a2 *b59*u + -----*u *a2 *b59*u
9 x 243 x
4 4
+ 6*f(v ,v )*a1 *a19*b59 + 8*f(v ,v )*a1 *a19*b59
2x 2x x 3x
3 3 2
+ 8*f(v ,v )*a1 *a19*a2*b59*u + 3*f(v ,v )*f(v,v)*a1 *a19 *b59
x 2x x x
4 2 2 2 4
+ ---*f(v ,v )*a1 *a19*a2 *b59*u + 2*f(v,v )*a1 *a19*b59
3 x x 4x
8 3 3 2
+ ---*f(v,v )*a1 *a19*a2*b59*u + 3*f(v,v )*f(v,v)*a1 *a19 *b59
3 3x 2x
4 2 2 2 2 3 2
+ ---*f(v,v )*a1 *a19*a2 *b59*u + 6*f(v,v ) *a1 *a19 *b59
3 2x x
2 2 8 3 3 3
+ 2*f(v,v )*f(v,v)*a1 *a19 *a2*b59*u + ----*f(v,v )*a1*a19*a2 *b59*u )/(a1
x 27 x
*a19*a2)
1 3 2 3 4 2 2
v =(---*u *a1 *a2*b59*v + ---*u *v *a1 *a2*b59 + ----*u *a1 *a2 *b59*u*v
s 9 4x 9 3x x 27 3x
10 2 2 8 2 2
+ ----*u *u *a1 *a2 *b59*v + ----*u *v *a1 *a2 *b59*u
27 2x x 27 2x x
1 2 2 3 2
+ ---*u *f(v,v)*a1 *a19*a2*b59*v + ----*u *a1*a2 *b59*u *v
6 2x 27 2x
2 2 2 2 4 2 3
+ ---*u *v *a1 *a2 *b59 + ----*u *a1*a2 *b59*u*v
9 x x 27 x
5 2 4 3 2
+ ---*u *v *f(v,v)*a1 *a19*a2*b59 + ----*u *v *a1*a2 *b59*u
9 x x 27 x x
4 2 1 2
+ ---*u *f(v,v )*a1 *a19*a2*b59*v + ---*u *f(v,v)*a1*a19*a2 *b59*u*v
9 x x 9 x
4 4 3 2 3
+ -----*u *a2 *b59*u *v + ---*v *f(v ,v )*a1 *a19*b59
243 x 3 x x x
2 3 2 2
+ ---*v *f(v,v )*a1 *a19*b59 + ---*v *f(v,v )*a1 *a19*a2*b59*u
3 x 2x 3 x x
1 2 2 2 4 2 2
+ ---*v *f(v,v) *a1 *a19 *b59 + ----*v *f(v,v)*a1*a19*a2 *b59*u
2 x 27 x
2 4 4 3
+ -----*v *a2 *b59*u + f(v ,v )*a1 *a19*b59*v
243 x x 2x
1 2 1 3
+ ---*f(v ,v )*a1 *a19*a2*b59*u*v + ---*f(v,v )*a1 *a19*b59*v
3 x x 3 3x
1 2 2 2 2 3
+ ---*f(v,v )*a1 *a19*a2*b59*u*v + ----*f(v,v )*a1*a19*a2 *b59*u *v)/(a1
3 2x 27 x
*a19)