Solution 4 to problem v2l05o23
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
b172=0
b171=0
b170=0
b169=0
b168=0
b167=0
b166=0
b165=0
b164=0
b163=0
b162=0
b161=0
- 3*a51*b1
b160=-------------
a1
b158=0
b157=0
b156=0
b155=0
3 2
---*a51 *b1
2
b154=-------------
2
a1
b153=0
b152=0
b151=0
b150=0
b149=0
b148=0
b147=0
b146=0
b145=0
b144=0
b143=0
b142=0
b141=0
3*a10*b1
b140=----------
a1
b139=0
b138=0
2
- 3*a1*a10*b1 + 3*a51 *b1
b137=----------------------------
2
a1
b136=0
2 2
3*a1*a10 *b1 - 3*a10*a51 *b1
b135=------------------------------
2
a1 *a51
2 3 4
- 2*a1 *a10 *b1 + 2*a10*a51 *b1
b134=----------------------------------
2 3
a1 *a51
b133=0
2 2
6*a1*a10 *b1 - 6*a10*a51 *b1
b132=------------------------------
2
a1*a51
- 3*a10*b1
b131=-------------
a1
2 2 2 4
3*a1 *a10 *b1 - 9*a1*a10*a51 *b1 + 3*a51 *b1
b130=----------------------------------------------
2 2
a1 *a51
b129=0
b128=0
b127=0
b125=0
3*a10*b1
b124=----------
a51
3*a10*b1
b123=----------
a51
2
3*a1*a10*b1 - 3*a51 *b1
b122=-------------------------
a1*a51
2
3*a1*a10*b1 - 3*a51 *b1
b121=-------------------------
a1*a51
b119=0
b118=0
b117=0
b116=0
b114=0
b113=0
b112=0
b111=0
b110=0
b109=0
b108=0
b107=0
b106=0
b105=0
b104=0
b103=0
b102=0
b101=0
b100=0
b99=0
b98=0
b97=0
b95=0
b94=0
b93=0
b91=0
b90=0
b86=0
b85=0
b84=0
b83=0
b82=0
b81=0
b79=0
b78=0
b75=0
b74=0
b73=0
b72=0
b71=0
b70=0
b69=0
b68=0
b67=0
b66=0
3 2
---*a51 *b1
2
b65=-------------
2
a1
b64=0
b63=0
b62=0
b61=0
b60=0
b59=0
b58=0
b57=0
b56=0
b55=0
b54=0
b53=0
b52=0
b51=0
b50=0
b49=0
b48=0
2 3 4
2*a1 *a10 *b1 - 2*a10*a51 *b1
b47=-------------------------------
2 3
a1 *a51
b46=0
b45=0
b44=0
- 3*a10*b1
b43=-------------
a1
2 2
6*a1*a10 *b1 - 6*a10*a51 *b1
b42=------------------------------
2
a1*a51
2 2 2 4
3*a1 *a10 *b1 - 9*a1*a10*a51 *b1 + 3*a51 *b1
b41=----------------------------------------------
2 2
a1 *a51
b40=0
b38=0
b37=0
b36=0
b34=0
- 3*a10*b1
b32=-------------
a51
2
- 3*a1*a10*b1 + 3*a51 *b1
b31=----------------------------
a1*a51
- 3*a10*b1
b30=-------------
a51
2
- 3*a1*a10*b1 + 3*a51 *b1
b29=----------------------------
a1*a51
2 2
- 3*a1*a10 *b1 + 3*a10*a51 *b1
b28=---------------------------------
2
a1 *a51
b27=0
b26=0
b25=0
3*a10*b1
b24=----------
a1
2
- 3*a1*a10*b1 + 3*a51 *b1
b23=----------------------------
2
a1
b22=0
b21=0
b20=0
b19=0
b18=0
b17=0
b16=0
b15=0
b14=0
b13=0
b12=0
b11=0
b10=0
b9=0
3*a51*b1
b8=----------
a1
b6=0
b5=0
b4=0
b3=0
a56=0
a55=0
a54=0
a53=0
a52=0
1
a50= - ---*a10
2
a49=0
a48=0
a47= - a10
2 2
a1*a10 + a10*a51
a46=--------------------
2
a51
a45=0
- 2*a1*a10
a44=-------------
a51
a43=0
2
- 2*a1*a10 + 2*a51
a42=----------------------
a51
a41=0
a40=0
a39=0
a38=0
a37=0
a36=0
a35=0
a34=0
a33=0
a32=0
a31=0
a28=0
a27=0
a26=0
a25=0
a24=0
a23=0
a22=0
1
a21=---*a10
2
a20=0
a19=0
a18=0
2 2
- a1*a10 - a10*a51
a17=-----------------------
2
a51
a16=0
a15=0
a14=0
a13=0
- 2*a1*a10
a12=-------------
a51
2
- 2*a1*a10 + 2*a51
a11=----------------------
a51
a10=a10
a9=0
a8=0
a7=0
a6=0
a5=a51
a4=0
a3=0
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:
b1,a51,a10,a1
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), v=v(t,x) are vector
functions of arbitrary dimension and f(..,..) is the scalar
product between two such vectors:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2 3
df(u,t)=(u *a1*a51 - 2*u *f(v,u)*a1*a10*a51 + 2*u *f(v,u)*a51
2x x x
3 2
+ v *f(u,u)*a51 - 2*f(u,v )*a1*a10*a51*u + f(u,u)*f(v,u)*a10*a51 *v
x x
1 2 2 2
+ ---*f(u,u)*f(v,v)*a10*a51 *u - f(v,u) *a1*a10 *u
2
2 2 2
- f(v,u) *a10*a51 *u)/a51
2 3 2 2
df(u,s)=(u *a1 *a51 *b1 - 3*u *f(v,u)*a1 *a10*a51 *b1
3x 2x
4 2 2
+ 3*u *f(v,u)*a1*a51 *b1 - 3*u *f(u,v )*a1 *a10*a51 *b1
2x x x
3 5 2 2
+ ---*u *f(u,u)*f(v,v)*a51 *b1 - 3*u *f(v,u )*a1 *a10*a51 *b1
2 x x x
4 2 2 2
+ 3*u *f(v,u )*a1*a51 *b1 + 3*u *f(v,u) *a1 *a10 *a51*b1
x x x
2 3 2 5
- 9*u *f(v,u) *a1*a10*a51 *b1 + 3*u *f(v,u) *a51 *b1
x x
4 3
+ 3*v *f(u,u )*a1*a51 *b1 - 3*v *f(u,u)*f(v,u)*a1*a10*a51 *b1
x x x
5 2 2
+ 3*v *f(u,u)*f(v,u)*a51 *b1 - 3*f(v ,u )*a1 *a10*a51 *b1*u
x x x
3
+ 3*f(u,u )*f(v,u)*a1*a10*a51 *b1*v
x
2 2
+ 6*f(u,v )*f(v,u)*a1 *a10 *a51*b1*u
x
3
- 6*f(u,v )*f(v,u)*a1*a10*a51 *b1*u
x
2 2 2 2 4
- 3*f(u,u)*f(v,u) *a1*a10 *a51 *b1*v + 3*f(u,u)*f(v,u) *a10*a51 *b1*v
3 3 2 3
- 3*f(v,u )*f(v,u)*a1*a10*a51 *b1*u + 2*f(v,u) *a1 *a10 *b1*u
x
3 4 2 3
- 2*f(v,u) *a10*a51 *b1*u)/(a1 *a51 )
3 2
df(v,t)=(u *f(v,v)*a51 - v *a1*a51 - 2*v *f(v,u)*a1*a10*a51
x 2x x
3 1 2
+ 2*v *f(v,u)*a51 - ---*f(u,u)*f(v,v)*a10*a51 *v
x 2
2 2 2 2
- 2*f(v,u )*a1*a10*a51*v + f(v,u) *a1*a10 *v + f(v,u) *a10*a51 *v
x
2 2
- f(v,u)*f(v,v)*a10*a51 *u)/a51
4 3
df(v,s)=( - 3*u *f(v,v )*a1*a51 *b1 - 3*u *f(v,u)*f(v,v)*a1*a10*a51 *b1
x x x
5 2 3
+ 3*u *f(v,u)*f(v,v)*a51 *b1 + v *a1 *a51 *b1
x 3x
2 2 4
+ 3*v *f(v,u)*a1 *a10*a51 *b1 - 3*v *f(v,u)*a1*a51 *b1
2x 2x
2 2 4
+ 3*v *f(u,v )*a1 *a10*a51 *b1 - 3*v *f(u,v )*a1*a51 *b1
x x x x
3 5 2 2
+ ---*v *f(u,u)*f(v,v)*a51 *b1 + 3*v *f(v,u )*a1 *a10*a51 *b1
2 x x x
2 2 2 2 3
+ 3*v *f(v,u) *a1 *a10 *a51*b1 - 9*v *f(v,u) *a1*a10*a51 *b1
x x
2 5 2 2
+ 3*v *f(v,u) *a51 *b1 + 3*f(v ,u )*a1 *a10*a51 *b1*v
x x x
3
- 3*f(u,v )*f(v,u)*a1*a10*a51 *b1*v
x
2 2
+ 6*f(v,u )*f(v,u)*a1 *a10 *a51*b1*v
x
3
- 6*f(v,u )*f(v,u)*a1*a10*a51 *b1*v
x
3 3 2 3
+ 3*f(v,v )*f(v,u)*a1*a10*a51 *b1*u - 2*f(v,u) *a1 *a10 *b1*v
x
3 4 2 2 2
+ 2*f(v,u) *a10*a51 *b1*v + 3*f(v,u) *f(v,v)*a1*a10 *a51 *b1*u
2 4 2 3
- 3*f(v,u) *f(v,v)*a10*a51 *b1*u)/(a1 *a51 )