Solution 1 to problem l2o57
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
b45=2*b43
b44=b43
2
---*a2*b43
3
b42=------------
a11
2*a2*b43
b41=----------
a11
2
---*a2*b43
3
b40=------------
a11
2
---*a2*b43
3
b39=------------
a11
b38=0
b37=0
5
---*a11*b43
3
b36=-------------
a7
5*a11*b43
b35=-----------
a7
b34=0
b33=0
5
---*a2*b43
3
b32=------------
a7
10
----*a2*b43
3
b31=-------------
a7
20
----*a2*b43
3
b30=-------------
a7
10
----*a2*b43
3
b29=-------------
a7
b28=0
b27=0
b26=0
b25=0
2 2
---*a2 *b43
3
b24=-------------
a11*a7
2 2
---*a2 *b43
3
b23=-------------
a11*a7
b22=0
a7*b43
b21=--------
a11
b20=5*b43
b19=5*b43
10
----*a2*b43
3
b18=-------------
a11
10*a2*b43
b17=-----------
a11
20
----*a2*b43
3
b16=-------------
a11
20
----*a2*b43
3
b15=-------------
a11
20
----*a2*b43
3
b14=-------------
a11
5
---*a2*b43
3
b13=------------
a11
2 2
---*a2 *b43
3
b12=-------------
2
a11
10 2
----*a2 *b43
3
b11=--------------
2
a11
20 2
----*a2 *b43
3
b10=--------------
2
a11
35
----*a11*b43
3
b9=--------------
a7
35
----*a2*b43
3
b8=-------------
a7
140
-----*a2*b43
3
b7=--------------
a7
35
----*a2*b43
3
b6=-------------
a7
70 2
----*a2 *b43
3
b5=--------------
a11*a7
2
14*a2 *b43
b4=------------
a11*a7
14 2
----*a2 *b43
3
b3=--------------
a11*a7
2 3
---*a2 *b43
3
b2=-------------
2
a11 *a7
a11*a7
a18=--------
a2
1
---*a11*a7
2
a17=------------
a2
3 2
---*a11
2
a16=----------
a2
2
3*a11
a15=--------
a2
a14=0
a13=0
a12=a11
a10=0
3*a11*a7
a9=----------
a2
3
---*a11*a7
2
a8=------------
a2
a6=3*a7
15 2
----*a11
2
a5=-----------
a2
a4=10*a11
a3=5*a11
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:
b43,a7,a11,a2
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 3
df(u,t)=(u *a2 + 5*u *a11*a2*u + 10*u *u *a11*a2 + ---*u *f(v,v)*a11*a7
5x 3x 2x x 2 x
15 2 2
+ ----*u *a11 *u + 3*f(v ,v )*a2*a7 + f(v,v )*a2*a7
2 x x 2x 3x
+ 3*f(v,v )*a11*a7*u)/a2
x
2 3 14 2 2
df(u,s)=(---*u *a2 *b43 + ----*u *a11*a2 *b43*u + 14*u *u *a11*a2 *b43
3 7x 3 5x 4x x
70 2 5
+ ----*u *u *a11*a2 *b43 + ---*u *f(v,v)*a11*a2*a7*b43
3 3x 2x 3 3x
35 2 2 140 2
+ ----*u *a11 *a2*b43*u + -----*u *u *a11 *a2*b43*u
3 3x 3 2x x
20 35 3 2
+ ----*u *f(v,v )*a11*a2*a7*b43 + ----*u *a11 *a2*b43
3 2x x 3 x
20 20
+ ----*u *f(v ,v )*a11*a2*a7*b43 + ----*u *f(v,v )*a11*a2*a7*b43
3 x x x 3 x 2x
2 35 3 3
+ 5*u *f(v,v)*a11 *a7*b43*u + ----*u *a11 *b43*u
x 3 x
20 2 10 2
+ ----*f(v ,v )*a2 *a7*b43 + ----*f(v ,v )*a2 *a7*b43
3 2x 3x 3 x 4x
2 2
+ 10*f(v ,v )*a11*a2*a7*b43*u + ---*f(v,v )*a2 *a7*b43
x 2x 3 5x
10 2
+ ----*f(v,v )*a11*a2*a7*b43*u + f(v,v )*f(v,v)*a11*a7 *b43
3 3x x
2 2 2
+ 5*f(v,v )*a11 *a7*b43*u )/(a11 *a7)
x
2 1
df(v,t)=(u *a11*a2*v + u *v *a11*a2 + 3*u *a11 *u*v + ---*v *f(v,v)*a11*a7
3x 2x x x 2 x
3 2 2
+ ---*v *a11 *u + f(v,v )*a11*a7*v)/a2
2 x x
2 2 2 2 10
df(v,s)=(---*u *a2 *b43*v + ---*u *v *a2 *b43 + ----*u *a11*a2*b43*u*v
3 5x 3 4x x 3 3x
20 10
+ ----*u *u *a11*a2*b43*v + ----*u *v *a11*a2*b43*u
3 2x x 3 2x x
5 2 2 2
+ ---*u *v *a11*a2*b43 + u *f(v,v)*a11*a7*b43*v + 5*u *a11 *b43*u *v
3 x x x x
2 2
+ ---*v *f(v ,v )*a2*a7*b43 + ---*v *f(v,v )*a2*a7*b43
3 x x x 3 x 2x
5 2 3
+ v *f(v,v)*a11*a7*b43*u + ---*v *a11 *b43*u
x 3 x
2
+ 2*f(v ,v )*a2*a7*b43*v + ---*f(v,v )*a2*a7*b43*v
x 2x 3 3x
+ 2*f(v,v )*a11*a7*b43*u*v)/(a11*a7)
x
df(u,t)=(df(u,x,5)*a2**2 + 5*df(u,x,3)*a11*a2*u + 10*df(u,x,2)*df(u,x)*a11*a2 +
3/2*df(u,x)*f(v,v)*a11*a7 + 15/2*df(u,x)*a11**2*u**2 + 3*f(df(v,x),df(v,x,2))*a2
*a7 + f(v,df(v,x,3))*a2*a7 + 3*f(v,df(v,x))*a11*a7*u)/a2$
df(u,s)=(2/3*df(u,x,7)*a2**3*b43 + 14/3*df(u,x,5)*a11*a2**2*b43*u + 14*df(u,x,4)
*df(u,x)*a11*a2**2*b43 + 70/3*df(u,x,3)*df(u,x,2)*a11*a2**2*b43 + 5/3*df(u,x,3)*
f(v,v)*a11*a2*a7*b43 + 35/3*df(u,x,3)*a11**2*a2*b43*u**2 + 140/3*df(u,x,2)*df(u,
x)*a11**2*a2*b43*u + 20/3*df(u,x,2)*f(v,df(v,x))*a11*a2*a7*b43 + 35/3*df(u,x)**3
*a11**2*a2*b43 + 20/3*df(u,x)*f(df(v,x),df(v,x))*a11*a2*a7*b43 + 20/3*df(u,x)*f(
v,df(v,x,2))*a11*a2*a7*b43 + 5*df(u,x)*f(v,v)*a11**2*a7*b43*u + 35/3*df(u,x)*a11
**3*b43*u**3 + 20/3*f(df(v,x,2),df(v,x,3))*a2**2*a7*b43 + 10/3*f(df(v,x),df(v,x,
4))*a2**2*a7*b43 + 10*f(df(v,x),df(v,x,2))*a11*a2*a7*b43*u + 2/3*f(v,df(v,x,5))*
a2**2*a7*b43 + 10/3*f(v,df(v,x,3))*a11*a2*a7*b43*u + f(v,df(v,x))*f(v,v)*a11*a7
**2*b43 + 5*f(v,df(v,x))*a11**2*a7*b43*u**2)/(a11**2*a7)$
df(v,t)=(df(u,x,3)*a11*a2*v + df(u,x,2)*df(v,x)*a11*a2 + 3*df(u,x)*a11**2*u*v +
1/2*df(v,x)*f(v,v)*a11*a7 + 3/2*df(v,x)*a11**2*u**2 + f(v,df(v,x))*a11*a7*v)/a2$
df(v,s)=(2/3*df(u,x,5)*a2**2*b43*v + 2/3*df(u,x,4)*df(v,x)*a2**2*b43 + 10/3*df(u
,x,3)*a11*a2*b43*u*v + 20/3*df(u,x,2)*df(u,x)*a11*a2*b43*v + 10/3*df(u,x,2)*df(v
,x)*a11*a2*b43*u + 5/3*df(u,x)**2*df(v,x)*a11*a2*b43 + df(u,x)*f(v,v)*a11*a7*b43
*v + 5*df(u,x)*a11**2*b43*u**2*v + 2/3*df(v,x)*f(df(v,x),df(v,x))*a2*a7*b43 + 2/
3*df(v,x)*f(v,df(v,x,2))*a2*a7*b43 + df(v,x)*f(v,v)*a11*a7*b43*u + 5/3*df(v,x)*
a11**2*b43*u**3 + 2*f(df(v,x),df(v,x,2))*a2*a7*b43*v + 2/3*f(v,df(v,x,3))*a2*a7*
b43*v + 2*f(v,df(v,x))*a11*a7*b43*u*v)/(a11*a7)$