Solution 2 to problem l2o57
Expressions |
Parameters |
Relevance |
Back to problem l2o57
Expressions
The solution is given through the following expressions:
b45=0
6 2
----*a14 *b12
25
b44=---------------
2
a10
b43=0
b42=0
b41=0
6
---*a14*b12
5
b40=-------------
a10
4
---*a14*b12
5
b39=-------------
a10
4
---*a14*b12
5
b38=-------------
a10
2
---*a14*b12
5
b37=-------------
a10
1 4
- -----*a14 *b12
125
b36=-------------------
3
a10 *a17
b35=0
3 3
----*a14 *b12
25
b34=---------------
2
a10 *a17
6 3
----*a14 *b12
25
b33=---------------
2
a10 *a17
1 3
- ----*a14 *b12
50
b32=------------------
2
a10 *a17
1 3
----*a14 *b12
25
b31=---------------
2
a10 *a17
b30=0
b29=0
2 2
---*a14 *b12
5
b28=--------------
a10*a17
4 2
---*a14 *b12
5
b27=--------------
a10*a17
2
a14 *b12
b26=----------
a10*a17
3 2
---*a14 *b12
5
b25=--------------
a10*a17
1 2
----*a14 *b12
10
b24=---------------
a10*a17
b23=0
2
---*a14*b12
7
b22=-------------
a17
12
----*a17*b12
5
b21=--------------
a10
6 2
- ----*a14 *b12
25
b20=------------------
2
a10
6 2
- ----*a14 *b12
25
b19=------------------
2
a10
2
---*a14*b12
5
b18=-------------
a10
6
- ---*a14*b12
5
b17=----------------
a10
4
---*a14*b12
5
b16=-------------
a10
2
- ---*a14*b12
5
b15=----------------
a10
b14=0
1
- ---*a14*b12
5
b13=----------------
a10
b11= - b12
b10= - 4*b12
1 4
----*a14 *b12
25
b9=---------------
3
a10 *a17
1 3
----*a14 *b12
10
b8=---------------
2
a10 *a17
2 3
---*a14 *b12
5
b7=--------------
2
a10 *a17
1 3
----*a14 *b12
10
b6=---------------
2
a10 *a17
1 2
---*a14 *b12
2
b5=--------------
a10*a17
3 2
----*a14 *b12
10
b4=---------------
a10*a17
1 2
----*a14 *b12
10
b3=---------------
a10*a17
1
----*a14*b12
28
b2=--------------
a17
a18=0
1 2
----*a14
10
a16=-----------
a10
a15=0
a13=a14
1
a12=---*a14
2
a11=0
a9=2*a17
a8=a17
5*a10*a17
a7=-----------
a14
5*a10*a17
a6=-----------
a14
3 2
- ----*a14
10
a5=--------------
a10
a4= - a14
1
a3= - ---*a14
2
1
a2= - ---*a10
4
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:
b12,a17,a14,a10
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 1 2 2
df(u,t)=( - ---*u *a10 *a14 - ---*u *a10*a14 *u - u *u *a10*a14
4 5x 2 3x 2x x
3 3 2 2
+ u *f(v,v)*a10*a14*a17 - ----*u *a14 *u + 5*f(v ,v )*a10 *a17
x 10 x x 2x
2
+ 5*f(v,v )*a10 *a17 + 2*f(v,v )*a10*a14*a17*u)/(a10*a14)
3x x
1 3 1 2 2
df(u,s)=(----*u *a10 *a14*b12 + ----*u *a10 *a14 *b12*u
28 7x 10 5x
3 2 2 1 2 2
+ ----*u *u *a10 *a14 *b12 + ---*u *u *a10 *a14 *b12
10 4x x 2 3x 2x
1 2 1 3 2
- ---*u *f(v,v)*a10 *a14*a17*b12 + ----*u *a10*a14 *b12*u
5 3x 10 3x
2 3 1 3 3
+ ---*u *u *a10*a14 *b12*u + ----*u *a10*a14 *b12
5 2x x 10 x
2 2 4 2
- ---*u *f(v ,v )*a10 *a14*a17*b12 + ---*u *f(v,v )*a10 *a14*a17*b12
5 x x x 5 x 2x
6 2 1 4 3
- ----*u *f(v,v)*a10*a14 *a17*b12*u + ----*u *a14 *b12*u
25 x 25 x
3 3
- 4*f(v ,v )*a10 *a17*b12 - f(v ,v )*a10 *a17*b12
2x 3x x 4x
6 2 3
- ---*f(v ,v )*a10 *a14*a17*b12*u + f(v,v )*a10 *a17*b12
5 x 2x 5x
2 2 12 2 2
+ ---*f(v,v )*a10 *a14*a17*b12*u + ----*f(v,v )*f(v,v)*a10 *a17 *b12
5 3x 5 x
6 2 2 3
- ----*f(v,v )*a10*a14 *a17*b12*u )/(a10 *a17)
25 x
1 2
df(v,t)=(---*u *v *a10*a14 + u *v *a10*a14 + v *a10 + v *a10*a14*u
2 2x x x 2x 5x 3x
1 2 2
+ v *f(v,v)*a10*a17 + ----*v *a14 *u )/a10
x 10 x
1 2 2 3 2 2
df(v,s)=(----*u *v *a10 *a14 *b12 + ---*u *v *a10 *a14 *b12
10 4x x 5 3x 2x
2 2 1 3
+ u *v *a10 *a14 *b12 + ----*u *v *a10*a14 *b12*u
2x 3x 25 2x x
1 2 3 4 2 2
- ----*u *v *a10*a14 *b12 + ---*u *v *a10 *a14 *b12
50 x x 5 x 4x
6 3 2 3
+ ----*u *v *a10*a14 *b12*u + ---*v *a10 *a14*b12
25 x 2x 7 7x
2 2 2 2 2
+ ---*v *a10 *a14 *b12*u + ---*v *f(v,v)*a10 *a14*a17*b12
5 5x 5 3x
3 3 2 4 2
+ ----*v *a10*a14 *b12*u + ---*v *f(v,v )*a10 *a14*a17*b12
25 3x 5 2x x
4 2 6 2
+ ---*v *f(v ,v )*a10 *a14*a17*b12 + ---*v *f(v,v )*a10 *a14*a17*b12
5 x x x 5 x 2x
6 2 1 4 3 3
+ ----*v *f(v,v)*a10*a14 *a17*b12*u - -----*v *a14 *b12*u )/(a10 *a17)
25 x 125 x
df(u,t)=( - 1/4*df(u,x,5)*a10**2*a14 - 1/2*df(u,x,3)*a10*a14**2*u - df(u,x,2)*df
(u,x)*a10*a14**2 + df(u,x)*f(v,v)*a10*a14*a17 - 3/10*df(u,x)*a14**3*u**2 + 5*f(
df(v,x),df(v,x,2))*a10**2*a17 + 5*f(v,df(v,x,3))*a10**2*a17 + 2*f(v,df(v,x))*a10
*a14*a17*u)/(a10*a14)$
df(u,s)=(1/28*df(u,x,7)*a10**3*a14*b12 + 1/10*df(u,x,5)*a10**2*a14**2*b12*u + 3/
10*df(u,x,4)*df(u,x)*a10**2*a14**2*b12 + 1/2*df(u,x,3)*df(u,x,2)*a10**2*a14**2*
b12 - 1/5*df(u,x,3)*f(v,v)*a10**2*a14*a17*b12 + 1/10*df(u,x,3)*a10*a14**3*b12*u
**2 + 2/5*df(u,x,2)*df(u,x)*a10*a14**3*b12*u + 1/10*df(u,x)**3*a10*a14**3*b12 -
2/5*df(u,x)*f(df(v,x),df(v,x))*a10**2*a14*a17*b12 + 4/5*df(u,x)*f(v,df(v,x,2))*
a10**2*a14*a17*b12 - 6/25*df(u,x)*f(v,v)*a10*a14**2*a17*b12*u + 1/25*df(u,x)*a14
**4*b12*u**3 - 4*f(df(v,x,2),df(v,x,3))*a10**3*a17*b12 - f(df(v,x),df(v,x,4))*
a10**3*a17*b12 - 6/5*f(df(v,x),df(v,x,2))*a10**2*a14*a17*b12*u + f(v,df(v,x,5))*
a10**3*a17*b12 + 2/5*f(v,df(v,x,3))*a10**2*a14*a17*b12*u + 12/5*f(v,df(v,x))*f(v
,v)*a10**2*a17**2*b12 - 6/25*f(v,df(v,x))*a10*a14**2*a17*b12*u**2)/(a10**3*a17)$
df(v,t)=(1/2*df(u,x,2)*df(v,x)*a10*a14 + df(u,x)*df(v,x,2)*a10*a14 + df(v,x,5)*
a10**2 + df(v,x,3)*a10*a14*u + df(v,x)*f(v,v)*a10*a17 + 1/10*df(v,x)*a14**2*u**2
)/a10$
df(v,s)=(1/10*df(u,x,4)*df(v,x)*a10**2*a14**2*b12 + 3/5*df(u,x,3)*df(v,x,2)*a10
**2*a14**2*b12 + df(u,x,2)*df(v,x,3)*a10**2*a14**2*b12 + 1/25*df(u,x,2)*df(v,x)*
a10*a14**3*b12*u - 1/50*df(u,x)**2*df(v,x)*a10*a14**3*b12 + 4/5*df(u,x)*df(v,x,4
)*a10**2*a14**2*b12 + 6/25*df(u,x)*df(v,x,2)*a10*a14**3*b12*u + 2/7*df(v,x,7)*
a10**3*a14*b12 + 2/5*df(v,x,5)*a10**2*a14**2*b12*u + 2/5*df(v,x,3)*f(v,v)*a10**2
*a14*a17*b12 + 3/25*df(v,x,3)*a10*a14**3*b12*u**2 + 4/5*df(v,x,2)*f(v,df(v,x))*
a10**2*a14*a17*b12 + 4/5*df(v,x)*f(df(v,x),df(v,x))*a10**2*a14*a17*b12 + 6/5*df(
v,x)*f(v,df(v,x,2))*a10**2*a14*a17*b12 + 6/25*df(v,x)*f(v,v)*a10*a14**2*a17*b12*
u - 1/125*df(v,x)*a14**4*b12*u**3)/(a10**3*a17)$