Solution 1 to problem v1l05o35
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
b27=0
b26=0
5 4
----*a2 *b1
81
b25=-------------
4
a1
b24=0
b23=0
10 3
----*a2 *b1
27
b22=-------------
3
a1
20 3
----*a2 *b1
9
b21=-------------
3
a1
10 3
----*a2 *b1
27
b20=-------------
3
a1
b19=0
b18=0
40 2
----*a2 *b1
9
b17=-------------
2
a1
20 2
----*a2 *b1
9
b16=-------------
2
a1
20 2
----*a2 *b1
3
b15=-------------
2
a1
40 2
----*a2 *b1
9
b14=-------------
2
a1
40 2
----*a2 *b1
9
b13=-------------
2
a1
20 2
----*a2 *b1
3
b12=-------------
2
a1
10 2
----*a2 *b1
9
b11=-------------
2
a1
b10=0
10
----*a2*b1
3
b9=------------
a1
5
---*a2*b1
3
b8=-----------
a1
10
----*a2*b1
3
b7=------------
a1
10*a2*b1
b6=----------
a1
20
----*a2*b1
3
b5=------------
a1
20
----*a2*b1
3
b4=------------
a1
20
----*a2*b1
3
b3=------------
a1
5
---*a2*b1
3
b2=-----------
a1
a8=0
a7=0
1 2
---*a2
3
a6=---------
a1
a5=0
a4=a2
a3=2*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,a2,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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2 1 2 2
df(v,t)=(v *a1 + v *f(v,v)*a1*a2 + 2*v *f(v,v )*a1*a2 + ---*v *f(v,v) *a2
3x 2x x x 3 x
+ f(v ,v )*a1*a2*v)/a1
x x
4 5 3 20 3
df(v,s)=(v *a1 *b1 + ---*v *f(v,v)*a1 *a2*b1 + ----*v *f(v,v )*a1 *a2*b1
5x 3 4x 3 3x x
10 2 2 2 20 3
+ ----*v *f(v,v) *a1 *a2 *b1 + ----*v *f(v ,v )*a1 *a2*b1
9 3x 3 2x x x
20 3 20 2 2
+ ----*v *f(v,v )*a1 *a2*b1 + ----*v *f(v,v )*f(v,v)*a1 *a2 *b1
3 2x 2x 3 2x x
10 3 3 3
+ ----*v *f(v,v) *a1*a2 *b1 + 10*v *f(v ,v )*a1 *a2*b1
27 2x x x 2x
40 2 2 10 3
+ ----*v *f(v ,v )*f(v,v)*a1 *a2 *b1 + ----*v *f(v,v )*a1 *a2*b1
9 x x x 3 x 3x
40 2 2 20 2 2 2
+ ----*v *f(v,v )*f(v,v)*a1 *a2 *b1 + ----*v *f(v,v ) *a1 *a2 *b1
9 x 2x 3 x x
20 2 3 5 4 4
+ ----*v *f(v,v )*f(v,v) *a1*a2 *b1 + ----*v *f(v,v) *a2 *b1
9 x x 81 x
5 3 10 3
+ ---*f(v ,v )*a1 *a2*b1*v + ----*f(v ,v )*a1 *a2*b1*v
3 2x 2x 3 x 3x
20 2 2
+ ----*f(v ,v )*f(v,v)*a1 *a2 *b1*v
9 x 2x
40 2 2
+ ----*f(v ,v )*f(v,v )*a1 *a2 *b1*v
9 x x x
10 2 3 4
+ ----*f(v ,v )*f(v,v) *a1*a2 *b1*v)/a1
27 x x