Solution 7 to problem l1o35
Expressions |
Parameters |
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Back to problem l1o35
Expressions
The solution is given through the following expressions:
4
2*a19 *b1
b74=-----------
3
a11*a1
2
2*a19 *b1
b73=-----------
2
a1
1 2
- ---*a19 *b1
2
b72=----------------
2
a1
7
4*a19 *b1
b71=-----------
3 4
a11 *a1
b70=0
5
2*a19 *b1
b69=-----------
2 3
a11 *a1
b68=0
b67=0
b66=0
3
8*a19 *b1
b65=-----------
2
a11*a1
b64=0
b63=0
3
2*a19 *b1
b62=-----------
2
a11*a1
3
- 2*a19 *b1
b61=--------------
2
a11*a1
b60=0
b59=0
2*a19*b1
b58=----------
a1
2*a19*b1
b57=----------
a1
b56=0
b55=0
b54=0
8
6*a19 *b1
b53=-----------
4 4
a11 *a1
b52=0
b51=0
6
12*a19 *b1
b50=------------
3 3
a11 *a1
b49=0
b48=0
b47=0
b46=0
4
2*a19 *b1
b45=-----------
2 2
a11 *a1
4
- 4*a19 *b1
b44=--------------
2 2
a11 *a1
b43=0
b42=0
b41=0
b40=0
b39=0
2
- 2*a19 *b1
b38=--------------
a11*a1
b37=0
b36=0
- a11*a19*b1
b35=---------------
2
a1
4
- 2*a19 *b1
b34=--------------
3
a11*a1
2
10*a19 *b1
b33=------------
2
a1
11 2
----*a19 *b1
2
b32=--------------
2
a1
6*a11*b1
b31=----------
a1
5*a11*b1
b30=----------
a1
5*a11*b1
b29=----------
a1
b28=0
5
12*a19 *b1
b27=------------
2 3
a11 *a1
5
30*a19 *b1
b26=------------
2 3
a11 *a1
3
6*a19 *b1
b25=-----------
2
a11*a1
3
6*a19 *b1
b24=-----------
2
a11*a1
3
12*a19 *b1
b23=------------
2
a11*a1
b22=0
b21=0
- 2*a19*b1
b20=-------------
a1
- 6*a19*b1
b19=-------------
a1
- 8*a19*b1
b18=-------------
a1
- 8*a19*b1
b17=-------------
a1
- 12*a19*b1
b16=--------------
a1
- 5*a19*b1
b15=-------------
a1
- a11*b1
b14=-----------
a19
- 4*a11*b1
b13=-------------
a19
- 3*a11*b1
b12=-------------
a19
b11=0
8
30*a19 *b1
b10=------------
4 4
a11 *a1
b9=0
b8=0
4
- 10*a19 *b1
b7=---------------
2 2
a11 *a1
4
- 40*a19 *b1
b6=---------------
2 2
a11 *a1
4
- 10*a19 *b1
b5=---------------
2 2
a11 *a1
b4=0
b3=0
b2=0
3
- 2*a19
a21=-----------
a11*a1
a20=0
a18=0
4
- 2*a19
a17=-----------
2
a11 *a1
a16=0
a15=0
2
- 2*a19
a14=-----------
a11
a13=0
a12=0
a10=0
a9= - 2*a19
a8= - 3*a19
- a11*a1
a7=-----------
a19
- a11*a1
a6=-----------
a19
a5=0
4
- 6*a19
a4=-----------
2
a11 *a1
a3=0
a2=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:
b1,a11,a19,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 5 2
u =(u *a11 *a1 *a19 - 3*u *f(v,v)*a11 *a1*a19 - 6*u *a19 *u
t 3x x x
3 2 3 2 2 2
- f(v ,v )*a11 *a1 - f(v,v )*a11 *a1 - 2*f(v,v )*a11 *a1*a19 *u
x x 2x x
2 3 2
+ f(v,v) *a11 *a1*a19)/(a11 *a1*a19)
2 2 4 2
v =( - 2*u *v *a11*a1*a19 + v *f(v,v)*a11 *a1*a19 - 2*v *a19 *u
t x x x x
3 2
- 2*f(v,v)*a11*a19 *u*v)/(a11 *a1)
4 4 4 3 2
u =(u *a11 *a1 *a19*b1 - 5*u *f(v,v)*a11 *a1 *a19 *b1
s 5x 3x
2 2 5 2 2 2 5
- 10*u *a11 *a1 *a19 *b1*u - 40*u *u *a11 *a1 *a19 *b1*u
3x 2x x
4 3 2 3 2 2 5
- 12*u *f(v,v )*a11 *a1 *a19 *b1 - 10*u *a11 *a1 *a19 *b1
2x x x
4 3 2 4 3 2
- 8*u *f(v ,v )*a11 *a1 *a19 *b1 - 8*u *f(v,v )*a11 *a1 *a19 *b1
x x x x 2x
3 2 4 11 2 4 2 3
+ 12*u *f(v,v )*a11 *a1 *a19 *b1*u + ----*u *f(v,v) *a11 *a1 *a19 *b1
x x 2 x
2 6 2 9 4
+ 30*u *f(v,v)*a11 *a1*a19 *b1*u + 30*u *a19 *b1*u
x x
5 4 5 4
- 3*f(v ,v )*a11 *a1 *b1 - 4*f(v ,v )*a11 *a1 *b1
2x 2x x 3x
4 3 2 5 3
- 6*f(v ,v )*a11 *a1 *a19 *b1*u + 5*f(v ,v )*f(v,v)*a11 *a1 *a19*b1
x 2x x x
3 2 4 2 5 4
+ 6*f(v ,v )*a11 *a1 *a19 *b1*u - f(v,v )*a11 *a1 *b1
x x 4x
4 3 2 5 3
- 2*f(v,v )*a11 *a1 *a19 *b1*u + 5*f(v,v )*f(v,v)*a11 *a1 *a19*b1
3x 2x
3 2 4 2 2 5 3
+ 6*f(v,v )*a11 *a1 *a19 *b1*u + 6*f(v,v ) *a11 *a1 *a19*b1
2x x
4 2 3 2 6 3
+ 10*f(v,v )*f(v,v)*a11 *a1 *a19 *b1*u + 12*f(v,v )*a11 *a1*a19 *b1*u
x x
3 5 2 2 2 3 5 2 4 4
- f(v,v) *a11 *a1 *a19 *b1 - 2*f(v,v) *a11 *a1*a19 *b1*u )/(a11 *a1 *a19)
3 3 2 2 2 4
v =( - 2*u *v *a11 *a1 *a19 *b1 - 4*u *v *a11 *a1 *a19 *b1*u
s 3x x 2x x
3 2 3 2 2 2 4
- 2*u *f(v,v)*a11 *a1 *a19 *b1*v + 2*u *v *a11 *a1 *a19 *b1
2x x x
3 2 3 6 2
+ 2*u *v *f(v,v)*a11 *a1 *a19 *b1 + 12*u *v *a11*a1*a19 *b1*u
x x x x
4 3 4 3
+ 2*v *f(v ,v )*a11 *a1 *a19*b1 + 2*v *f(v,v )*a11 *a1 *a19*b1
x x x x 2x
3 2 3 1 2 4 2 2
+ 8*v *f(v,v )*a11 *a1 *a19 *b1*u - ---*v *f(v,v) *a11 *a1 *a19 *b1
x x 2 x
2 5 2 8 4
+ 2*v *f(v,v)*a11 *a1*a19 *b1*u + 6*v *a19 *b1*u
x x
4 2 2 2 3 4
+ 2*f(v,v )*f(v,v)*a11 *a1 *a19 *b1*v + 2*f(v,v) *a11 *a1*a19 *b1*u*v
x
7 3 4 4
+ 4*f(v,v)*a11*a19 *b1*u *v)/(a11 *a1 )