Solution 15 to problem l1o35
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
b74=0
b73=0
5 2
---*a19 *b36
6
b72=--------------
2
a12
b71=0
b70=0
5
---*a17*a19*b36
3
b69=-----------------
2
a12
b68=0
b67=0
b66=0
b65=0
b64=0
b63=0
b62=0
b61=0
b60=0
b59=0
10
----*a19*b36
3
b58=--------------
a12
5
---*a19*b36
3
b57=-------------
a12
10
----*a19*b36
3
b56=--------------
a12
5
---*a19*b36
3
b55=-------------
a12
b54=0
5 2
---*a17 *b36
6
b53=--------------
2
a12
b52=0
b51=0
b50=0
b49=0
b48=0
5
---*a17*b36
3
b47=-------------
a12
10
----*a17*b36
3
b46=--------------
a12
5
---*a17*b36
3
b45=-------------
a12
10
----*a17*b36
3
b44=--------------
a12
b43=0
b42=0
b41=0
b40=0
b39=0
b38=0
b37=0
b35=0
b34=0
b33=0
5 2
---*a19 *b36
6
b32=--------------
2
a12
b31=0
b30=0
b29=0
b28=0
b27=0
5
---*a17*a19*b36
3
b26=-----------------
2
a12
b25=0
b24=0
b23=0
b22=0
b21=0
b20=0
b19=0
10
----*a19*b36
3
b18=--------------
a12
5
---*a19*b36
3
b17=-------------
a12
10
----*a19*b36
3
b16=--------------
a12
5
---*a19*b36
3
b15=-------------
a12
b14=0
b13=0
b12=0
b11=0
5 2
---*a17 *b36
6
b10=--------------
2
a12
b9=0
b8=0
5
---*a17*b36
3
b7=-------------
a12
20
----*a17*b36
3
b6=--------------
a12
5
---*a17*b36
3
b5=-------------
a12
b4=0
b3=0
b2=0
b1=b36
a21=0
a20=0
a18=0
a16=0
a15=0
a14=0
a13=0
a11=0
a10=0
a9=0
a8=a19
a7=0
a6=0
a5=0
a4=a17
a3=0
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,a19,a17,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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2
u =u *a12 + u *f(v,v)*a19 + u *a17*u
t 3x x x
2
v =v *a12 + v *f(v,v)*a19 + v *a17*u
t 3x x x
2 5 5 2
u =(u *a12 *b36 + ---*u *f(v,v)*a12*a19*b36 + ---*u *a12*a17*b36*u
s 5x 3 3x 3 3x
20 10
+ ----*u *u *a12*a17*b36*u + ----*u *f(v,v )*a12*a19*b36
3 2x x 3 2x x
5 3 5
+ ---*u *a12*a17*b36 + ---*u *f(v ,v )*a12*a19*b36
3 x 3 x x x
10 5 2 2
+ ----*u *f(v,v )*a12*a19*b36 + ---*u *f(v,v) *a19 *b36
3 x 2x 6 x
5 2 5 2 4 2
+ ---*u *f(v,v)*a17*a19*b36*u + ---*u *a17 *b36*u )/a12
3 x 6 x
10 5 2
v =(----*u *v *a12*a17*b36*u + ---*u *v *a12*a17*b36
s 3 2x x 3 x x
10 2 5
+ ----*u *v *a12*a17*b36*u + v *a12 *b36 + ---*v *f(v,v)*a12*a19*b36
3 x 2x 5x 3 3x
5 2 10
+ ---*v *a12*a17*b36*u + ----*v *f(v,v )*a12*a19*b36
3 3x 3 2x x
5 10
+ ---*v *f(v ,v )*a12*a19*b36 + ----*v *f(v,v )*a12*a19*b36
3 x x x 3 x 2x
5 2 2 5 2
+ ---*v *f(v,v) *a19 *b36 + ---*v *f(v,v)*a17*a19*b36*u
6 x 3 x
5 2 4 2
+ ---*v *a17 *b36*u )/a12
6 x