Solution 12 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
2
2*a20 *b1
b73=-----------
2
a1
2
a20 *b1
b72=---------
2
a1
b71=0
1 2
---*a20*a2 *b1
3
b70=----------------
3
a1
1 2
---*a20*a2 *b1
3
b69=----------------
3
a1
1 2
---*a20*a2 *b1
3
b68=----------------
3
a1
a20*a2*b1
b67=-----------
2
a1
a20*a2*b1
b66=-----------
2
a1
2*a20*a2*b1
b65=-------------
2
a1
b64=0
2*a20*a2*b1
b63=-------------
2
a1
a20*a2*b1
b62=-----------
2
a1
1
---*a20*a2*b1
2
b61=---------------
2
a1
a20*b1
b60=--------
a1
3*a20*b1
b59=----------
a1
2*a20*b1
b58=----------
a1
2*a20*b1
b57=----------
a1
b56=0
b55=0
b54=0
1 4
----*a2 *b1
81
b53=-------------
4
a1
2 4
----*a2 *b1
81
b52=-------------
4
a1
b51=0
2 3
---*a2 *b1
9
b50=------------
3
a1
2 3
---*a2 *b1
9
b49=------------
3
a1
1 3
---*a2 *b1
9
b48=------------
3
a1
b47=0
b46=0
1 2
---*a2 *b1
3
b45=------------
2
a1
4 2
---*a2 *b1
9
b44=------------
2
a1
5 2
---*a2 *b1
9
b43=------------
2
a1
2 2
---*a2 *b1
9
b42=------------
2
a1
b41=0
b40=0
b39=0
1
---*a2*b1
3
b38=-----------
a1
1
---*a2*b1
6
b37=-----------
a1
b36=0
b35=0
b34=0
2
12*a20 *b1
b33=------------
2
a1
2
3*a20 *b1
b32=-----------
2
a1
2
36*a20 *b1
b31=------------
a1*a2
2
18*a20 *b1
b30=------------
a1*a2
2
18*a20 *b1
b29=------------
a1*a2
b28=0
8 2
---*a20*a2 *b1
9
b27=----------------
3
a1
4 2
---*a20*a2 *b1
3
b26=----------------
3
a1
4*a20*a2*b1
b25=-------------
2
a1
4*a20*a2*b1
b24=-------------
2
a1
16*a20*a2*b1
b23=--------------
2
a1
4*a20*a2*b1
b22=-------------
2
a1
4*a20*a2*b1
b21=-------------
2
a1
8*a20*b1
b20=----------
a1
24*a20*b1
b19=-----------
a1
20*a20*b1
b18=-----------
a1
20*a20*b1
b17=-----------
a1
20*a20*b1
b16=-----------
a1
4*a20*b1
b15=----------
a1
6*a20*b1
b14=----------
a2
24*a20*b1
b13=-----------
a2
18*a20*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
a21=0
a19=a20
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=4*a20
a8=2*a20
6*a1*a20
a7=----------
a2
6*a1*a20
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,a20,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
u =(u *a1 *a2 + u *a1*a2 *u + u *a1*a2 + 2*u *f(v,v)*a1*a20*a2
t 3x 2x x x
1 3 2 2 2
+ ---*u *a2 *u + 6*f(v ,v )*a1 *a20 + 6*f(v,v )*a1 *a20
3 x x x 2x
+ 4*f(v,v )*a1*a20*a2*u)/(a1*a2)
x
1 1 1 2
v =(---*u *a1*a2*v + ---*u *v *a1*a2 + ---*u *a2 *u*v + v *f(v,v)*a1*a20
t 6 2x 3 x x 9 x x
1 2 2
+ ---*v *a2 *u + f(v,v )*a1*a20*v)/a1
9 x x
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
+ 4*u *f(v,v)*a1 *a20*a2*b1 + ----*u *a1 *a2 *b1*u
3x 9 3x
10 2 3 2 50 2 3
+ ----*u *a1 *a2 *b1 + ----*u *u *a1 *a2 *b1*u
3 2x 9 2x x
3 2 2
+ 20*u *f(v,v )*a1 *a20*a2*b1 + 4*u *f(v,v)*a1 *a20*a2 *b1*u
2x x 2x
10 4 3 5 3 2 3 2 2 2
+ ----*u *a1*a2 *b1*u + ---*u *a1 *a2 *b1 + 4*u *f(v,v)*a1 *a20*a2 *b1
27 2x 3 x x
10 2 4 2 3
+ ----*u *a1*a2 *b1*u + 20*u *f(v ,v )*a1 *a20*a2*b1
9 x x x x
3 2 2
+ 20*u *f(v,v )*a1 *a20*a2*b1 + 16*u *f(v,v )*a1 *a20*a2 *b1*u
x 2x x x
2 2 2 4 3 2
+ 3*u *f(v,v) *a1 *a20 *a2*b1 + ---*u *f(v,v)*a1*a20*a2 *b1*u
x 3 x
5 5 4 4 4
+ ----*u *a2 *b1*u + 18*f(v ,v )*a1 *a20*b1 + 24*f(v ,v )*a1 *a20*b1
81 x 2x 2x x 3x
3 3 2
+ 24*f(v ,v )*a1 *a20*a2*b1*u + 18*f(v ,v )*f(v,v)*a1 *a20 *b1
x 2x x x
2 2 2 4
+ 4*f(v ,v )*a1 *a20*a2 *b1*u + 6*f(v,v )*a1 *a20*b1
x x 4x
3 3 2
+ 8*f(v,v )*a1 *a20*a2*b1*u + 18*f(v,v )*f(v,v)*a1 *a20 *b1
3x 2x
2 2 2 2 3 2
+ 4*f(v,v )*a1 *a20*a2 *b1*u + 36*f(v,v ) *a1 *a20 *b1
2x x
2 2 8 3 3 4
+ 12*f(v,v )*f(v,v)*a1 *a20 *a2*b1*u + ---*f(v,v )*a1*a20*a2 *b1*u )/(a1
x 9 x
*a2)
1 3 1 3 2 2 2
v =(---*u *a1 *a2*b1*v + ---*u *v *a1 *a2*b1 + ---*u *a1 *a2 *b1*u*v
s 6 4x 3 3x x 9 3x
5 2 2 4 2 2
+ ---*u *u *a1 *a2 *b1*v + ---*u *v *a1 *a2 *b1*u
9 2x x 9 2x x
1 2 1 3 2
+ ---*u *f(v,v)*a1 *a20*a2*b1*v + ---*u *a1*a2 *b1*u *v
2 2x 9 2x
1 2 2 2 2 2 3
+ ---*u *v *a1 *a2 *b1 + ---*u *a1*a2 *b1*u*v
3 x x 9 x
2 2 3 2
+ u *v *f(v,v)*a1 *a20*a2*b1 + ---*u *v *a1*a2 *b1*u
x x 9 x x
2 1 2
+ 2*u *f(v,v )*a1 *a20*a2*b1*v + ---*u *f(v,v)*a1*a20*a2 *b1*u*v
x x 3 x
2 4 3 3 3
+ ----*u *a2 *b1*u *v + 2*v *f(v ,v )*a1 *a20*b1 + 2*v *f(v,v )*a1 *a20*b1
81 x x x x x 2x
2 2 2 2
+ 2*v *f(v,v )*a1 *a20*a2*b1*u + v *f(v,v) *a1 *a20 *b1
x x x
1 2 2 1 4 4
+ ---*v *f(v,v)*a1*a20*a2 *b1*u + ----*v *a2 *b1*u
3 x 81 x
3 2
+ 3*f(v ,v )*a1 *a20*b1*v + f(v ,v )*a1 *a20*a2*b1*u*v
x 2x x x
3 2
+ f(v,v )*a1 *a20*b1*v + f(v,v )*a1 *a20*a2*b1*u*v
3x 2x
2 2 1 2 2 4
+ 2*f(v,v )*f(v,v)*a1 *a20 *b1*v + ---*f(v,v )*a1*a20*a2 *b1*u *v)/a1
x 3 x