Solution 4 to problem l2o57
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
5103 2
- ------*a15 *a8*b2
4
b45=----------------------
3
a13
5103 2
- ------*a15 *a8*b2
4
b44=----------------------
3
a13
5103 2
- ------*a15 *a8*b2
8
b43=----------------------
3
a13
2835
- ------*a15*a8*b2
8
b42=---------------------
2
a13
1701
- ------*a15*a8*b2
2
b41=---------------------
2
a13
7371
- ------*a15*a8*b2
8
b40=---------------------
2
a13
2835
- ------*a15*a8*b2
4
b39=---------------------
2
a13
5103
- ------*a15*a8*b2
8
b38=---------------------
2
a13
1701
- ------*a15*a8*b2
8
b37=---------------------
2
a13
1701 3
- ------*a15 *b2
4
b36=-------------------
3
a13
5103 3
- ------*a15 *b2
8
b35=-------------------
3
a13
1701 2
- ------*a15 *b2
2
b34=-------------------
2
a13
5103 2
- ------*a15 *b2
2
b33=-------------------
2
a13
8505 2
- ------*a15 *b2
8
b32=-------------------
2
a13
2
- 1701*a15 *b2
b31=-----------------
2
a13
5103 2
- ------*a15 *b2
8
b30=-------------------
2
a13
1701 2
- ------*a15 *b2
4
b29=-------------------
2
a13
567
- -----*a15*b2
2
b28=-----------------
a13
2835
- ------*a15*b2
4
b27=------------------
a13
2079
- ------*a15*b2
2
b26=------------------
a13
1701
- ------*a15*b2
2
b25=------------------
a13
1449
- ------*a15*b2
4
b24=------------------
a13
- 63*a15*b2
b23=--------------
a13
b22= - 27*b2
5103 2
- ------*a15*a8 *b2
4
b21=----------------------
3
a13
5103 2
- ------*a15 *a8*b2
4
b20=----------------------
3
a13
5103 2
- ------*a15 *a8*b2
4
b19=----------------------
3
a13
1701
- ------*a15*a8*b2
2
b18=---------------------
2
a13
1701
- ------*a15*a8*b2
2
b17=---------------------
2
a13
6237
- ------*a15*a8*b2
4
b16=---------------------
2
a13
2835
- ------*a15*a8*b2
4
b15=---------------------
2
a13
5103
- ------*a15*a8*b2
4
b14=---------------------
2
a13
567
- -----*a15*a8*b2
2
b13=--------------------
2
a13
- 126*a8*b2
b12=--------------
a13
315
- -----*a8*b2
2
b11=----------------
a13
- 126*a8*b2
b10=--------------
a13
1701 3
------*a15 *b2
2
b9=----------------
3
a13
2835 2
------*a15 *b2
8
b8=----------------
2
a13
5103 2
------*a15 *b2
4
b7=----------------
2
a13
567 2
-----*a15 *b2
2
b6=---------------
2
a13
189*a15*b2
b5=------------
a13
441
-----*a15*b2
4
b4=--------------
a13
63
----*a15*b2
2
b3=-------------
a13
1
a18=---*a8
2
1
a17=---*a8
2
a16=a15
2
a14=---*a13
3
7
a12=---*a13
9
2
a11=---*a13
9
4 2
----*a13
45
a10=-----------
a15
a9=2*a8
4
---*a13*a8
9
a7=------------
a15
2
---*a13*a8
3
a6=------------
a15
a5= - a15
5
a4= - ---*a13
9
2
a3= - ---*a13
9
4 2
- -----*a13
405
a2=---------------
a15
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:
b2,a13,a15,a8
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
4 2 2 5
df(u,t)=( - -----*u *a13 - ---*u *a13*a15*u - ---*u *u *a13*a15
405 5x 9 3x 9 2x x
2 2 2
+ u *f(v,v)*a15*a8 - u *a15 *u + ---*f(v ,v )*a13*a8
x x 3 x 2x
4
+ ---*f(v,v )*a13*a8 + 2*f(v,v )*a15*a8*u)/a15
9 3x x
3 63 2 441 2
df(u,s)=(u *a13 *b2 + ----*u *a13 *a15*b2*u + -----*u *u *a13 *a15*b2
7x 2 5x 4 4x x
2 567
+ 189*u *u *a13 *a15*b2 - -----*u *f(v,v)*a13*a15*a8*b2
3x 2x 2 3x
567 2 2 5103 2
+ -----*u *a13*a15 *b2*u + ------*u *u *a13*a15 *b2*u
2 3x 4 2x x
5103 2835 3 2
- ------*u *f(v,v )*a13*a15*a8*b2 + ------*u *a13*a15 *b2
4 2x x 8 x
2835 6237
- ------*u *f(v ,v )*a13*a15*a8*b2 - ------*u *f(v,v )*a13*a15*a8*b2
4 x x x 4 x 2x
5103 2 1701 3 3
- ------*u *f(v,v)*a15 *a8*b2*u + ------*u *a15 *b2*u
4 x 2 x
2 315 2
- 126*f(v ,v )*a13 *a8*b2 - -----*f(v ,v )*a13 *a8*b2
2x 3x 2 x 4x
1701 2
- ------*f(v ,v )*a13*a15*a8*b2*u - 126*f(v,v )*a13 *a8*b2
2 x 2x 5x
1701 5103 2
- ------*f(v,v )*a13*a15*a8*b2*u - ------*f(v,v )*f(v,v)*a15*a8 *b2
2 3x 4 x
5103 2 2 3
- ------*f(v,v )*a15 *a8*b2*u )/a13
4 x
2 7 2
df(v,t)=(---*u *a13*a15*v + ---*u *v *a13*a15 + u *v *a13*a15 + u *a15 *u*v
9 3x 9 2x x x 2x x
4 2 2 1
+ ----*v *a13 + ---*v *a13*a15*u + ---*v *f(v,v)*a15*a8
45 5x 3 3x 2 x
2 2 1
+ v *a15 *u + ---*f(v,v )*a15*a8*v)/a15
x 2 x
2 1449 2
df(v,s)=( - 63*u *a13 *a15*b2*v - ------*u *v *a13 *a15*b2
5x 4 4x x
1701 2 1701 2
- ------*u *v *a13 *a15*b2 - ------*u *a13*a15 *b2*u*v
2 3x 2x 4 3x
5103 2 2079 2
- ------*u *u *a13*a15 *b2*v - ------*u *v *a13 *a15*b2
8 2x x 2 2x 3x
2 8505 2 2
- 1701*u *v *a13*a15 *b2*u - ------*u *v *a13*a15 *b2
2x x 8 x x
2835 2 5103 2
- ------*u *v *a13 *a15*b2 - ------*u *v *a13*a15 *b2*u
4 x 4x 2 x 2x
5103 2 5103 3 2
- ------*u *f(v,v)*a15 *a8*b2*v - ------*u *a15 *b2*u *v
8 x 8 x
3 567 2
- 27*v *a13 *b2 - -----*v *a13 *a15*b2*u
7x 2 5x
1701 1701 2 2
- ------*v *f(v,v)*a13*a15*a8*b2 - ------*v *a13*a15 *b2*u
8 3x 2 3x
5103 2835
- ------*v *f(v,v )*a13*a15*a8*b2 - ------*v *f(v ,v )*a13*a15*a8*b2
8 2x x 4 x x x
7371 5103 2
- ------*v *f(v,v )*a13*a15*a8*b2 - ------*v *f(v,v)*a15 *a8*b2*u
8 x 2x 4 x
1701 3 3 1701
- ------*v *a15 *b2*u - ------*f(v ,v )*a13*a15*a8*b2*v
4 x 2 x 2x
2835 5103 2
- ------*f(v,v )*a13*a15*a8*b2*v - ------*f(v,v )*a15 *a8*b2*u*v)/
8 3x 4 x
3
a13
df(u,t)=( - 4/405*df(u,x,5)*a13**2 - 2/9*df(u,x,3)*a13*a15*u - 5/9*df(u,x,2)*df(
u,x)*a13*a15 + df(u,x)*f(v,v)*a15*a8 - df(u,x)*a15**2*u**2 + 2/3*f(df(v,x),df(v,
x,2))*a13*a8 + 4/9*f(v,df(v,x,3))*a13*a8 + 2*f(v,df(v,x))*a15*a8*u)/a15$
df(u,s)=(df(u,x,7)*a13**3*b2 + 63/2*df(u,x,5)*a13**2*a15*b2*u + 441/4*df(u,x,4)*
df(u,x)*a13**2*a15*b2 + 189*df(u,x,3)*df(u,x,2)*a13**2*a15*b2 - 567/2*df(u,x,3)*
f(v,v)*a13*a15*a8*b2 + 567/2*df(u,x,3)*a13*a15**2*b2*u**2 + 5103/4*df(u,x,2)*df(
u,x)*a13*a15**2*b2*u - 5103/4*df(u,x,2)*f(v,df(v,x))*a13*a15*a8*b2 + 2835/8*df(u
,x)**3*a13*a15**2*b2 - 2835/4*df(u,x)*f(df(v,x),df(v,x))*a13*a15*a8*b2 - 6237/4*
df(u,x)*f(v,df(v,x,2))*a13*a15*a8*b2 - 5103/4*df(u,x)*f(v,v)*a15**2*a8*b2*u +
1701/2*df(u,x)*a15**3*b2*u**3 - 126*f(df(v,x,2),df(v,x,3))*a13**2*a8*b2 - 315/2*
f(df(v,x),df(v,x,4))*a13**2*a8*b2 - 1701/2*f(df(v,x),df(v,x,2))*a13*a15*a8*b2*u
- 126*f(v,df(v,x,5))*a13**2*a8*b2 - 1701/2*f(v,df(v,x,3))*a13*a15*a8*b2*u - 5103
/4*f(v,df(v,x))*f(v,v)*a15*a8**2*b2 - 5103/4*f(v,df(v,x))*a15**2*a8*b2*u**2)/a13
**3$
df(v,t)=(2/9*df(u,x,3)*a13*a15*v + 7/9*df(u,x,2)*df(v,x)*a13*a15 + df(u,x)*df(v,
x,2)*a13*a15 + df(u,x)*a15**2*u*v + 4/45*df(v,x,5)*a13**2 + 2/3*df(v,x,3)*a13*
a15*u + 1/2*df(v,x)*f(v,v)*a15*a8 + df(v,x)*a15**2*u**2 + 1/2*f(v,df(v,x))*a15*
a8*v)/a15$
df(v,s)=( - 63*df(u,x,5)*a13**2*a15*b2*v - 1449/4*df(u,x,4)*df(v,x)*a13**2*a15*
b2 - 1701/2*df(u,x,3)*df(v,x,2)*a13**2*a15*b2 - 1701/4*df(u,x,3)*a13*a15**2*b2*u
*v - 5103/8*df(u,x,2)*df(u,x)*a13*a15**2*b2*v - 2079/2*df(u,x,2)*df(v,x,3)*a13**
2*a15*b2 - 1701*df(u,x,2)*df(v,x)*a13*a15**2*b2*u - 8505/8*df(u,x)**2*df(v,x)*
a13*a15**2*b2 - 2835/4*df(u,x)*df(v,x,4)*a13**2*a15*b2 - 5103/2*df(u,x)*df(v,x,2
)*a13*a15**2*b2*u - 5103/8*df(u,x)*f(v,v)*a15**2*a8*b2*v - 5103/8*df(u,x)*a15**3
*b2*u**2*v - 27*df(v,x,7)*a13**3*b2 - 567/2*df(v,x,5)*a13**2*a15*b2*u - 1701/8*
df(v,x,3)*f(v,v)*a13*a15*a8*b2 - 1701/2*df(v,x,3)*a13*a15**2*b2*u**2 - 5103/8*df
(v,x,2)*f(v,df(v,x))*a13*a15*a8*b2 - 2835/4*df(v,x)*f(df(v,x),df(v,x))*a13*a15*
a8*b2 - 7371/8*df(v,x)*f(v,df(v,x,2))*a13*a15*a8*b2 - 5103/4*df(v,x)*f(v,v)*a15
**2*a8*b2*u - 1701/4*df(v,x)*a15**3*b2*u**3 - 1701/2*f(df(v,x),df(v,x,2))*a13*
a15*a8*b2*v - 2835/8*f(v,df(v,x,3))*a13*a15*a8*b2*v - 5103/4*f(v,df(v,x))*a15**2
*a8*b2*u*v)/a13**3$