Solution 1 to problem v1l1o35
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
20 2
----*a3 *b1
9
b9=-------------
2
a1
10 2
----*a3 *b1
9
b8=-------------
2
a1
5
---*a3*b1
3
b7=-----------
a1
10
----*a3*b1
3
b6=------------
a1
5*a3*b1
b5=---------
a1
10
----*a3*b1
3
b4=------------
a1
5*a3*b1
b3=---------
a1
5
---*a3*b1
3
b2=-----------
a1
a2=a3
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,a3,a1
Relevance for the application:
The solution given above tells us that v_s
is a higher order symmetry for v_t
where v=v(t,x) is a vector
function of arbitrary dimension and f(..,..) is the scalar
product between two such vectors:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
df(v,t)=v *a1 + v *f(v,v)*a3 + f(v,v )*a3*v
3x x x
2 5
df(v,s)=(v *a1 *b1 + ---*v *f(v,v)*a1*a3*b1 + 5*v *f(v,v )*a1*a3*b1
5x 3 3x 2x x
10
+ ----*v *f(v ,v )*a1*a3*b1 + 5*v *f(v,v )*a1*a3*b1
3 x x x x 2x
10 2 2 10
+ ----*v *f(v,v) *a3 *b1 + ----*f(v ,v )*a1*a3*b1*v
9 x 3 x 2x
5 20 2 2
+ ---*f(v,v )*a1*a3*b1*v + ----*f(v,v )*f(v,v)*a3 *b1*v)/a1
3 3x 9 x