Solution 3 to problem l2o57
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
560 2
-----*a15 *a8*b2
27
b45=------------------
3
a13
280 2
-----*a15 *a8*b2
27
b44=------------------
3
a13
280 2
-----*a15 *a8*b2
27
b43=------------------
3
a13
112
-----*a15*a8*b2
9
b42=-----------------
2
a13
280
-----*a15*a8*b2
9
b41=-----------------
2
a13
56
----*a15*a8*b2
3
b40=----------------
2
a13
140
-----*a15*a8*b2
9
b39=-----------------
2
a13
112
-----*a15*a8*b2
9
b38=-----------------
2
a13
28
----*a15*a8*b2
9
b37=----------------
2
a13
140 3
-----*a15 *b2
27
b36=---------------
3
a13
140 3
-----*a15 *b2
9
b35=---------------
3
a13
70 2
----*a15 *b2
9
b34=--------------
2
a13
280 2
-----*a15 *b2
9
b33=---------------
2
a13
70 2
----*a15 *b2
3
b32=--------------
2
a13
280 2
-----*a15 *b2
9
b31=---------------
2
a13
280 2
-----*a15 *b2
9
b30=---------------
2
a13
140 2
-----*a15 *b2
9
b29=---------------
2
a13
14
----*a15*b2
3
b28=-------------
a13
14*a15*b2
b27=-----------
a13
70
----*a15*b2
3
b26=-------------
a13
70
----*a15*b2
3
b25=-------------
a13
14*a15*b2
b24=-----------
a13
14
----*a15*b2
3
b23=-------------
a13
b22=b2
560 2
-----*a15*a8 *b2
27
b21=------------------
3
a13
280 2
-----*a15 *a8*b2
9
b20=------------------
3
a13
280 2
-----*a15 *a8*b2
9
b19=------------------
3
a13
280
-----*a15*a8*b2
9
b18=-----------------
2
a13
560
-----*a15*a8*b2
9
b17=-----------------
2
a13
560
-----*a15*a8*b2
9
b16=-----------------
2
a13
140
-----*a15*a8*b2
3
b15=-----------------
2
a13
560
-----*a15*a8*b2
9
b14=-----------------
2
a13
140
-----*a15*a8*b2
9
b13=-----------------
2
a13
28
----*a8*b2
3
b12=------------
a13
28*a8*b2
b11=----------
a13
140
-----*a8*b2
3
b10=-------------
a13
140 3
-----*a15 *b2
27
b9=---------------
3
a13
70 2
----*a15 *b2
9
b8=--------------
2
a13
280 2
-----*a15 *b2
9
b7=---------------
2
a13
70 2
----*a15 *b2
9
b6=--------------
2
a13
70
----*a15*b2
3
b5=-------------
a13
14*a15*b2
b4=-----------
a13
14
----*a15*b2
3
b3=-------------
a13
2
a18=---*a8
3
1
a17=---*a8
3
1
a16=---*a15
2
1
a14=---*a13
2
a12=a13
1
a11=---*a13
2
3 2
----*a13
20
a10=-----------
a15
a9=2*a8
a13*a8
a7=--------
a15
2*a13*a8
a6=----------
a15
1
a5=---*a15
2
a4=a13
1
a3=---*a13
2
3 2
----*a13
20
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
3 2 1
df(u,t)=(----*u *a13 + ---*u *a13*a15*u + u *u *a13*a15 + u *f(v,v)*a15*a8
20 5x 2 3x 2x x x
1 2 2
+ ---*u *a15 *u + 2*f(v ,v )*a13*a8 + f(v,v )*a13*a8
2 x x 2x 3x
+ 2*f(v,v )*a15*a8*u)/a15
x
3 14 2 2
df(u,s)=(u *a13 *b2 + ----*u *a13 *a15*b2*u + 14*u *u *a13 *a15*b2
7x 3 5x 4x x
70 2 140
+ ----*u *u *a13 *a15*b2 + -----*u *f(v,v)*a13*a15*a8*b2
3 3x 2x 9 3x
70 2 2 280 2
+ ----*u *a13*a15 *b2*u + -----*u *u *a13*a15 *b2*u
9 3x 9 2x x
560 70 3 2
+ -----*u *f(v,v )*a13*a15*a8*b2 + ----*u *a13*a15 *b2
9 2x x 9 x
140 560
+ -----*u *f(v ,v )*a13*a15*a8*b2 + -----*u *f(v,v )*a13*a15*a8*b2
3 x x x 9 x 2x
280 2 140 3 3
+ -----*u *f(v,v)*a15 *a8*b2*u + -----*u *a15 *b2*u
9 x 27 x
140 2 2
+ -----*f(v ,v )*a13 *a8*b2 + 28*f(v ,v )*a13 *a8*b2
3 2x 3x x 4x
560 28 2
+ -----*f(v ,v )*a13*a15*a8*b2*u + ----*f(v,v )*a13 *a8*b2
9 x 2x 3 5x
280 560 2
+ -----*f(v,v )*a13*a15*a8*b2*u + -----*f(v,v )*f(v,v)*a15*a8 *b2
9 3x 27 x
280 2 2 3
+ -----*f(v,v )*a15 *a8*b2*u )/a13
9 x
1 2
df(v,t)=(---*u *a13*a15*v + u *v *a13*a15 + u *v *a13*a15 + u *a15 *u*v
2 3x 2x x x 2x x
3 2 1 1
+ ----*v *a13 + ---*v *a13*a15*u + ---*v *f(v,v)*a15*a8
20 5x 2 3x 3 x
1 2 2 2
+ ---*v *a15 *u + ---*f(v,v )*a15*a8*v)/a15
2 x 3 x
14 2 2
df(v,s)=(----*u *a13 *a15*b2*v + 14*u *v *a13 *a15*b2
3 5x 4x x
70 2 140 2
+ ----*u *v *a13 *a15*b2 + -----*u *a13*a15 *b2*u*v
3 3x 2x 9 3x
280 2 70 2
+ -----*u *u *a13*a15 *b2*v + ----*u *v *a13 *a15*b2
9 2x x 3 2x 3x
280 2 70 2 2
+ -----*u *v *a13*a15 *b2*u + ----*u *v *a13*a15 *b2
9 2x x 3 x x
2 280 2
+ 14*u *v *a13 *a15*b2 + -----*u *v *a13*a15 *b2*u
x 4x 9 x 2x
280 2 140 3 2 3
+ -----*u *f(v,v)*a15 *a8*b2*v + -----*u *a15 *b2*u *v + v *a13 *b2
27 x 9 x 7x
14 2 28
+ ----*v *a13 *a15*b2*u + ----*v *f(v,v)*a13*a15*a8*b2
3 5x 9 3x
70 2 2 112
+ ----*v *a13*a15 *b2*u + -----*v *f(v,v )*a13*a15*a8*b2
9 3x 9 2x x
140 56
+ -----*v *f(v ,v )*a13*a15*a8*b2 + ----*v *f(v,v )*a13*a15*a8*b2
9 x x x 3 x 2x
280 2 140 3 3
+ -----*v *f(v,v)*a15 *a8*b2*u + -----*v *a15 *b2*u
27 x 27 x
280 112
+ -----*f(v ,v )*a13*a15*a8*b2*v + -----*f(v,v )*a13*a15*a8*b2*v
9 x 2x 9 3x
560 2 3
+ -----*f(v,v )*a15 *a8*b2*u*v)/a13
27 x
df(u,t)=(3/20*df(u,x,5)*a13**2 + 1/2*df(u,x,3)*a13*a15*u + df(u,x,2)*df(u,x)*a13
*a15 + df(u,x)*f(v,v)*a15*a8 + 1/2*df(u,x)*a15**2*u**2 + 2*f(df(v,x),df(v,x,2))*
a13*a8 + 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 + 14/3*df(u,x,5)*a13**2*a15*b2*u + 14*df(u,x,4)*df(
u,x)*a13**2*a15*b2 + 70/3*df(u,x,3)*df(u,x,2)*a13**2*a15*b2 + 140/9*df(u,x,3)*f(
v,v)*a13*a15*a8*b2 + 70/9*df(u,x,3)*a13*a15**2*b2*u**2 + 280/9*df(u,x,2)*df(u,x)
*a13*a15**2*b2*u + 560/9*df(u,x,2)*f(v,df(v,x))*a13*a15*a8*b2 + 70/9*df(u,x)**3*
a13*a15**2*b2 + 140/3*df(u,x)*f(df(v,x),df(v,x))*a13*a15*a8*b2 + 560/9*df(u,x)*f
(v,df(v,x,2))*a13*a15*a8*b2 + 280/9*df(u,x)*f(v,v)*a15**2*a8*b2*u + 140/27*df(u,
x)*a15**3*b2*u**3 + 140/3*f(df(v,x,2),df(v,x,3))*a13**2*a8*b2 + 28*f(df(v,x),df(
v,x,4))*a13**2*a8*b2 + 560/9*f(df(v,x),df(v,x,2))*a13*a15*a8*b2*u + 28/3*f(v,df(
v,x,5))*a13**2*a8*b2 + 280/9*f(v,df(v,x,3))*a13*a15*a8*b2*u + 560/27*f(v,df(v,x)
)*f(v,v)*a15*a8**2*b2 + 280/9*f(v,df(v,x))*a15**2*a8*b2*u**2)/a13**3$
df(v,t)=(1/2*df(u,x,3)*a13*a15*v + 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 + 3/20*df(v,x,5)*a13**2 + 1/2*df(v,x,3)*a13*a15*u
+ 1/3*df(v,x)*f(v,v)*a15*a8 + 1/2*df(v,x)*a15**2*u**2 + 2/3*f(v,df(v,x))*a15*a8*
v)/a15$
df(v,s)=(14/3*df(u,x,5)*a13**2*a15*b2*v + 14*df(u,x,4)*df(v,x)*a13**2*a15*b2 +
70/3*df(u,x,3)*df(v,x,2)*a13**2*a15*b2 + 140/9*df(u,x,3)*a13*a15**2*b2*u*v + 280
/9*df(u,x,2)*df(u,x)*a13*a15**2*b2*v + 70/3*df(u,x,2)*df(v,x,3)*a13**2*a15*b2 +
280/9*df(u,x,2)*df(v,x)*a13*a15**2*b2*u + 70/3*df(u,x)**2*df(v,x)*a13*a15**2*b2
+ 14*df(u,x)*df(v,x,4)*a13**2*a15*b2 + 280/9*df(u,x)*df(v,x,2)*a13*a15**2*b2*u +
280/27*df(u,x)*f(v,v)*a15**2*a8*b2*v + 140/9*df(u,x)*a15**3*b2*u**2*v + df(v,x,
7)*a13**3*b2 + 14/3*df(v,x,5)*a13**2*a15*b2*u + 28/9*df(v,x,3)*f(v,v)*a13*a15*a8
*b2 + 70/9*df(v,x,3)*a13*a15**2*b2*u**2 + 112/9*df(v,x,2)*f(v,df(v,x))*a13*a15*
a8*b2 + 140/9*df(v,x)*f(df(v,x),df(v,x))*a13*a15*a8*b2 + 56/3*df(v,x)*f(v,df(v,x
,2))*a13*a15*a8*b2 + 280/27*df(v,x)*f(v,v)*a15**2*a8*b2*u + 140/27*df(v,x)*a15**
3*b2*u**3 + 280/9*f(df(v,x),df(v,x,2))*a13*a15*a8*b2*v + 112/9*f(v,df(v,x,3))*
a13*a15*a8*b2*v + 560/27*f(v,df(v,x))*a15**2*a8*b2*u*v)/a13**3$