Solution 2 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
b160=0
b158=0
b157=0
b156=0
b155=0
b154=0
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
b140=0
b139=0
b138=0
b137=0
b136=0
b135=0
1 3 1 2
---*a11 *b1 - ---*a11*a43 *b1
4 4
b134=-------------------------------
3
a1
b133=0
3 2 3
---*a11 *b1 + ---*a11*a43*b1
2 2
b132=------------------------------
2
a1
9 3 2
---*a11*a43*b1 + ---*a43 *b1
4 4
b131=------------------------------
2
a1
3 2 3
---*a11 *b1 + ---*a11*a43*b1
4 4
b130=------------------------------
2
a1
b129=0
b128=0
b127=0
3
- ---*a43*b1
2
b125=---------------
a1
3
- ---*a11*b1
2
b124=---------------
a1
3
- ---*a11*b1
2
b123=---------------
a1
3 3
- ---*a11*b1 - ---*a43*b1
2 2
b122=----------------------------
a1
3
- ---*a11*b1
2
b121=---------------
a1
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
b65=0
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
1 3 1 2
- ---*a11 *b1 + ---*a11*a43 *b1
4 4
b47=----------------------------------
3
a1
b46=0
b45=0
b44=0
9 3 2
---*a11*a43*b1 + ---*a43 *b1
4 4
b43=------------------------------
2
a1
3 2 3
---*a11 *b1 + ---*a11*a43*b1
2 2
b42=------------------------------
2
a1
3 2 3
---*a11 *b1 + ---*a11*a43*b1
4 4
b41=------------------------------
2
a1
b40=0
b38=0
b37=0
b36=0
3
---*a43*b1
2
b34=------------
a1
3
---*a11*b1
2
b32=------------
a1
3 3
---*a11*b1 + ---*a43*b1
2 2
b31=-------------------------
a1
3
---*a11*b1
2
b30=------------
a1
3
---*a11*b1
2
b29=------------
a1
b28=0
b27=0
b26=0
b25=0
b24=0
b23=0
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
b8=0
b6=0
b5=0
b4=0
b3=0
a56=0
a55=0
a54=0
a53=0
a52=0
a51=0
a50=0
a49=0
a48=0
a47=0
1 2 1
---*a11 - ---*a11*a43
4 4
a46=------------------------
a1
a45=0
a44=a11
a42=a11
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
a21=0
a20=0
a19=0
a18=0
1 2 1
- ---*a11 + ---*a11*a43
4 4
a17=---------------------------
a1
a16=0
a15=0
a14=0
a13=a43
a12=a11
a10=0
a9=0
a8=0
a7=0
a6=0
a5=0
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,a43,a11,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
df(u,t)=(u *a1 + u *f(v,u)*a1*a11 + f(u,v )*a1*a11*u + f(v,u )*a1*a43*u
2x x x x
1 2 2 1 2
- ---*f(v,u) *a11 *u + ---*f(v,u) *a11*a43*u)/a1
4 4
3 3 2 3 2
df(u,s)=(u *a1 *b1 + ---*u *f(v,u)*a1 *a11*b1 + ---*u *f(u,v )*a1 *a11*b1
3x 2 2x 2 x x
3 2 3 2
+ ---*u *f(v,u )*a1 *a11*b1 + ---*u *f(v,u )*a1 *a43*b1
2 x x 2 x x
3 2 2 3 2
+ ---*u *f(v,u) *a1*a11 *b1 + ---*u *f(v,u) *a1*a11*a43*b1
4 x 4 x
3 2 3 2
+ ---*f(v ,u )*a1 *a11*b1*u + ---*f(u,v )*f(v,u)*a1*a11 *b1*u
2 x x 2 x
3 3 2
+ ---*f(u,v )*f(v,u)*a1*a11*a43*b1*u + ---*f(v,u )*a1 *a43*b1*u
2 x 2 2x
9 3 2
+ ---*f(v,u )*f(v,u)*a1*a11*a43*b1*u + ---*f(v,u )*f(v,u)*a1*a43 *b1*u
4 x 4 x
1 3 3 1 3 2 3
- ---*f(v,u) *a11 *b1*u + ---*f(v,u) *a11*a43 *b1*u)/a1
4 4
2
df(v,t)=( - v *a1 + v *f(v,u)*a1*a11 + f(u,v )*a1*a43*v + f(v,u )*a1*a11*v
2x x x x
1 2 2 1 2
+ ---*f(v,u) *a11 *v - ---*f(v,u) *a11*a43*v)/a1
4 4
3 3 2 3 2
df(v,s)=(v *a1 *b1 - ---*v *f(v,u)*a1 *a11*b1 - ---*v *f(u,v )*a1 *a11*b1
3x 2 2x 2 x x
3 2 3 2
- ---*v *f(u,v )*a1 *a43*b1 - ---*v *f(v,u )*a1 *a11*b1
2 x x 2 x x
3 2 2 3 2
+ ---*v *f(v,u) *a1*a11 *b1 + ---*v *f(v,u) *a1*a11*a43*b1
4 x 4 x
3 2 3 2
- ---*f(v ,u )*a1 *a11*b1*v - ---*f(u,v )*a1 *a43*b1*v
2 x x 2 2x
9 3 2
+ ---*f(u,v )*f(v,u)*a1*a11*a43*b1*v + ---*f(u,v )*f(v,u)*a1*a43 *b1*v
4 x 4 x
3 2 3
+ ---*f(v,u )*f(v,u)*a1*a11 *b1*v + ---*f(v,u )*f(v,u)*a1*a11*a43*b1*v
2 x 2 x
1 3 3 1 3 2 3
+ ---*f(v,u) *a11 *b1*v - ---*f(v,u) *a11*a43 *b1*v)/a1
4 4