Solution 14 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
b73=0
b72=0
b71=0
b70=0
b69=0
b68=0
b67=0
b66=0
b65=0
b64=0
b63=0
b62=0
b61=0
b60=0
b59=0
b58=0
5
---*a14*a6*b36
9
b57=----------------
2
a12
b56=0
b55=0
b54=0
b53=0
b52=0
b51=0
b50=0
b49=0
b48=0
b47=0
b46=0
5 2
---*a14 *b36
6
b45=--------------
2
a12
b44=0
b43=0
b42=0
b41=0
5
---*a14*b36
3
b40=-------------
a12
5
---*a14*b36
3
b39=-------------
a12
5
---*a14*b36
3
b38=-------------
a12
b37=0
b35=0
b34=0
b33=0
b32=0
b31=0
b30=0
b29=0
b28=0
b27=0
b26=0
b25=0
b24=0
b23=0
b22=0
b21=0
b20=0
b19=0
b18=0
5
---*a14*a6*b36
3
b17=----------------
2
a12
b16=0
b15=0
b14=0
10
----*a6*b36
3
b13=-------------
a12
5
---*a6*b36
3
b12=------------
a12
b11=0
b10=0
b9=0
b8=0
5 2
----*a14 *b36
18
b7=---------------
2
a12
b6=0
b5=0
5
---*a14*b36
6
b4=-------------
a12
5
---*a14*b36
3
b3=-------------
a12
b2=0
b1=b36
a21=0
a20=0
a19=0
a18=0
a17=0
a16=0
a15=0
a13=0
a11=0
a10=0
a9=0
a8=0
a7=0
a5=0
a4=0
1
a3=---*a14
2
a2=0
a1=a12
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:
b36,a6,a14,a12
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
1 2
u =u *a12 + ---*u *a14 + f(v ,v )*a6
t 3x 2 x x x
v =u *v *a14 + v *a12
t x x 3x
2 5 5 2
u =(u *a12 *b36 + ---*u *u *a12*a14*b36 + ---*u *a12*a14*b36
s 5x 3 3x x 6 2x
5 3 2 5
+ ----*u *a14 *b36 + ---*u *f(v ,v )*a14*a6*b36
18 x 3 x x x
5 10 2
+ ---*f(v ,v )*a12*a6*b36 + ----*f(v ,v )*a12*a6*b36)/a12
3 2x 2x 3 x 3x
5 5 5 2 2
v =(---*u *v *a12*a14*b36 + ---*u *v *a12*a14*b36 + ---*u *v *a14 *b36
s 3 3x x 3 2x 2x 6 x x
5 2 5 2
+ ---*u *v *a12*a14*b36 + v *a12 *b36 + ---*v *f(v ,v )*a14*a6*b36)/a12
3 x 3x 5x 9 x x x