Problem l2313o23
Unknowns | Inequalities |
Equations |
Solution 1 |
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Unknowns
All solutions for the following 35 unknowns have to be determined:
a1, ..., a13, b1, ..., b22
Inequalities
Each of the following lists represents one inequality which states
that not all unknowns in this list may vanish. These inequalities
filter out solutions which are trivial for the application.
{a1,a4,a9},
{a3,a4,a8,a7,a6,a5},
{a12,a11,a10},
{b1,b4,b12},
{b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11},
{b12,b13,b14,b15,b16,b17,b18,
b19,b20,b21,b22}
Equations
All comma separated 114 expressions involving 694 terms have to vanish.
All terms are products of one a- and one b-unknown.
a2*b14,
a2*b2,
3*a10*b12 - 2*a9*b14,
a1*b2 - 6*a2*b1,
a1*b2 - 4*a2*b1,
3*a10*b12 - a9*b13,
3*a11*b12 - 2*a9*b16,
3*a10*b4 - a4*b13,
3*a12*b12 - 2*a9*b19,
2*a11*b4 - a4*b15,
3*a13*b12 - 2*a9*b21,
a12*b4 - a4*b18,
6*a13*b12 - a9*b22,
6*a13*b21 + a8*b19,
9*a10*b12 - 2*a9*b13 - 3*a9*b14,
a1*b13 + 9*a10*b12 - 3*a9*b13,
a10*b14 - a10*b2 + 2*a2*b13,
3*a11*b12 - a9*b15 - a9*b16,
6*a11*b12 + 3*a4*b14 - 2*a9*b17,
2*a10*b16 + 2*a2*b16 + 3*a5*b14,
a1*b15 + 6*a11*b12 - 3*a9*b15,
6*a11*b12 - 2*a9*b15 - a9*b17,
6*a11*b12 + a4*b13 - 2*a9*b17,
6*a12*b12 + a4*b16 - a9*b20,
6*a12*b12 - 2*a9*b19 - a9*b20,
12*a12*b12 + a4*b15 - 2*a9*b20,
6*a12*b12 - 2*a9*b18 - a9*b20,
12*a12*b12 + a3*b17 - 3*a9*b20,
12*a12*b12 + a4*b17 - 2*a9*b20,
18*a13*b12 + a4*b19 - 2*a9*b22,
9*a13*b12 - 3*a9*b21 - a9*b22,
72*a13*b12 + a3*b20 - 12*a9*b22,
72*a13*b12 + a4*b20 - 8*a9*b22,
3*a1*b14 + 9*a10*b12 - 2*a9*b13 - 3*a9*b14,
a1*b13 - 3*a10*b1 + 3*a10*b12 - a9*b13,
a1*b16 + 3*a11*b12 - a9*b15 - a9*b16,
6*a11*b12 - a9*b15 - 2*a9*b16 - a9*b17,
6*a11*b12 + 3*a3*b14 - 2*a9*b16 - 2*a9*b17,
a1*b15 - 2*a11*b1 + 2*a11*b12 - a9*b15,
a1*b17 + 6*a11*b12 - 2*a9*b15 - a9*b17,
6*a11*b12 + a3*b13 - a9*b15 - 2*a9*b17,
2*a10*b17 - 3*a10*b6 + 2*a2*b17 + 2*a5*b13,
2*a1*b5 - 9*a10*b4 + 3*a4*b13 - 9*a5*b1,
a1*b5 - 3*a10*b4 + a4*b13 - 3*a5*b1,
a1*b4 - a4*b1 + a4*b12 - a9*b4,
2*a10*b6 - a2*b6 - 2*a5*b14 + 2*a5*b2,
a1*b19 + 3*a12*b12 - 2*a9*b18 - a9*b19,
12*a12*b12 - 4*a9*b18 - 4*a9*b19 - a9*b20,
6*a12*b12 + a3*b16 - 2*a9*b19 - a9*b20,
a1*b18 - a12*b1 + a12*b12 - a9*b18,
a1*b20 + 12*a12*b12 - 8*a9*b18 - a9*b20,
12*a12*b12 + a3*b15 - 4*a9*b18 - 2*a9*b20,
2*a1*b7 - 6*a11*b4 + 3*a4*b15 - 6*a6*b1,
18*a13*b12 + a3*b19 - 6*a9*b21 - 2*a9*b22,
6*a12*b21 + 4*a13*b19 + a7*b19 + 2*a8*b16,
a12*b11 - 6*a13*b22 - 8*a8*b18 - a8*b20,
a12*b11 + 6*a13*b9 - 8*a8*b18 + a8*b7,
3*a10*b3 - 6*a11*b12 - 2*a3*b13 - a4*b13 + 2*a9*b17,
2*a10*b5 + 3*a10*b6 + 2*a2*b5 - 2*a5*b13 + 3*a5*b2,
4*a10*b19 + 2*a11*b16 + a2*b19 + 2*a5*b16 + 3*a6*b14,
2*a11*b3 - 12*a12*b12 - 2*a3*b15 - a4*b15 + 2*a9*b20,
a12*b3 - 18*a13*b12 - 2*a3*b18 - a4*b18 + 2*a9*b22,
a1*b3 + 2*a1*b4 - 2*a3*b1 + 2*a3*b12 - 3*a4*b1 - a9*b3,
6*a10*b21 + 4*a11*b19 + 2*a12*b16 + a5*b19 + 2*a6*b16 + 3*a7*b14,
6*a11*b21 + 4*a12*b19 + 2*a13*b16 + a6*b19 + 2*a7*b16 + 3*a8*b14,
a12*b21 + a12*b22 - a12*b9 - 2*a13*b18 + a7*b18 + a8*b15,
14*a13*b11 - a7*b11 + a8*b10 - 8*a8*b21 - 8*a8*b22 + 8*a8*b9,
4*a1*b5 - 3*a10*b3 - 9*a10*b4 + 2*a3*b13 + 2*a4*b13 + 3*a4*b14 - 18*a5*b1,
2*a1*b5 + 6*a1*b6 - 6*a10*b3 + 2*a3*b13 + 6*a3*b14 - 3*a3*b2 - 18*a5*b1,
3*a1*b3 + a1*b4 - 6*a3*b1 - 4*a4*b1 + a4*b12 - a9*b3 - a9*b4,
2*a1*b10 - 2*a12*b3 - 6*a12*b4 + a3*b20 + 2*a4*b19 + a4*b20 - 12*a7*b1,
3*a1*b11 - 3*a13*b3 - 15*a13*b4 + a3*b22 + 3*a4*b21 + 2*a4*b22 - 24*a8*b1,
4*a1*b5 + 3*a1*b6 - 3*a10*b3 - 9*a10*b4 + 2*a3*b13 + 2*a4*b13 + 3*a4*b14 - 18*
a5*b1,
a1*b6 - 2*a10*b4 + 4*a2*b4 + a4*b14 - a4*b2 - 2*a5*b1 + 2*a5*b12 - a9*b6,
4*a10*b20 - 3*a10*b8 + 2*a11*b17 - 2*a11*b6 + a2*b20 + 2*a5*b15 + 2*a5*b17 + 4*
a6*b13,
4*a1*b7 - a11*b3 - 9*a11*b4 + a3*b15 + 3*a4*b15 + a4*b16 + a4*b17 - 12*a6*b1,
2*a10*b8 + 2*a11*b6 - a2*b8 - a5*b16 - a5*b17 + a5*b5 - 2*a6*b14 + 2*a6*b2,
12*a1*b9 - a12*b3 - 15*a12*b4 + 2*a3*b18 + 10*a4*b18 + a4*b19 + a4*b20 - 18*a7*
b1,
2*a11*b11 + a12*b10 - 6*a12*b22 - 4*a13*b20 - 6*a7*b18 - a7*b20 - 8*a8*b15 - 2*
a8*b17,
2*a11*b11 + a12*b10 + 6*a12*b9 + 4*a13*b7 - 6*a7*b18 + a7*b7 - 8*a8*b15 + 2*a8*
b5,
2*a10*b15 + 3*a10*b16 + 3*a10*b17 - 3*a10*b5 - 2*a11*b13 + 2*a11*b14 - 2*a11*b2
+ 5*a2*b15 + 5*a5*b13,
2*a1*b5 + 6*a1*b6 - 3*a10*b3 - 9*a10*b4 + 2*a4*b13 + 6*a4*b14 - 3*a4*b2 - 18*a5
*b1 - 2*a9*b5,
3*a1*b6 - 2*a10*b3 + 4*a2*b3 + 2*a3*b14 - 2*a3*b2 + a4*b14 - a4*b2 - 6*a5*b1 -
a9*b6,
4*a1*b7 + a1*b8 - a11*b3 - 9*a11*b4 + a3*b15 + 3*a4*b15 + a4*b16 + a4*b17 - 12*
a6*b1,
4*a10*b7 + 3*a10*b8 + 2*a11*b5 + 2*a11*b6 + a2*b7 - 2*a5*b15 + 2*a5*b5 - 4*a6*
b13 + 3*a6*b2,
6*a1*b8 - 6*a11*b3 - 6*a11*b4 + 4*a3*b16 + 4*a3*b17 - 3*a3*b6 + 2*a4*b16 + 2*a4
*b17 - 24*a6*b1,
a1*b8 - 4*a11*b4 + a4*b16 + a4*b17 - a4*b5 + 3*a5*b4 - 4*a6*b1 + 4*a6*b12 - a9*
b8,
6*a1*b8 - 4*a11*b3 - 12*a11*b4 + 2*a3*b17 + 6*a4*b16 + 4*a4*b17 - 3*a4*b6 - 24*
a6*b1 - 2*a9*b8,
6*a1*b10 - 6*a12*b3 - 18*a12*b4 + 4*a3*b19 + 2*a3*b20 - a3*b8 + 2*a4*b19 + 4*a4
*b20 - 36*a7*b1,
a1*b10 - 6*a12*b4 + a4*b19 + a4*b20 - a4*b7 + 2*a6*b4 - 6*a7*b1 + 6*a7*b12 - a9
*b10,
6*a1*b10 - 4*a12*b3 - 24*a12*b4 + a3*b20 + 6*a4*b19 + 5*a4*b20 - a4*b8 - 36*a7*
b1 - 2*a9*b10,
18*a1*b11 - 18*a13*b3 - 90*a13*b4 - a3*b10 + 12*a3*b21 + 4*a3*b22 + 6*a4*b21 +
14*a4*b22 - 144*a8*b1,
a1*b11 - 8*a13*b4 + a4*b21 + a4*b22 - a4*b9 + a7*b4 - 8*a8*b1 + 8*a8*b12 - a9*
b11,
18*a1*b11 - 12*a13*b3 - 108*a13*b4 + 2*a3*b22 - a4*b10 + 18*a4*b21 + 16*a4*b22
- 144*a8*b1 - 6*a9*b11,
2*a1*b7 + 2*a1*b8 - 2*a11*b3 - 6*a11*b4 + a3*b15 + 2*a3*b16 - a3*b5 + a4*b15 +
2*a4*b17 - 12*a6*b1,
2*a1*b7 + 2*a1*b8 - a11*b3 - 9*a11*b4 + 2*a4*b15 + 2*a4*b16 + 2*a4*b17 - a4*b5
- 12*a6*b1 - 2*a9*b7,
2*a1*b7 + 2*a1*b8 - 2*a11*b3 - 6*a11*b4 + a3*b15 + a3*b17 + a4*b15 + 2*a4*b16 +
a4*b17 - 12*a6*b1,
2*a1*b10 + 6*a1*b9 - 2*a12*b3 - 12*a12*b4 + 2*a3*b18 + 2*a3*b19 - a3*b7 + 4*a4*
b18 + 2*a4*b20 - 18*a7*b1,
2*a1*b10 + 6*a1*b9 - a12*b3 - 15*a12*b4 + 6*a4*b18 + 2*a4*b19 + 2*a4*b20 - a4*
b7 - 18*a7*b1 - 6*a9*b9,
4*a1*b10 + 12*a1*b9 - 4*a12*b3 - 24*a12*b4 + 4*a3*b18 + a3*b20 + 8*a4*b18 + 4*
a4*b19 + 3*a4*b20 - 36*a7*b1,
2*a11*b21 + 2*a11*b22 - 2*a11*b9 + a12*b19 + a12*b20 - a12*b7 - 4*a13*b15 + 2*
a6*b18 + 2*a7*b15 + 2*a8*b13,
6*a12*b11 + 6*a13*b10 - a6*b11 - 3*a7*b21 - 3*a7*b22 + 3*a7*b9 - 4*a8*b19 - 4*
a8*b20 + 4*a8*b7 + a8*b8,
3*a1*b8 - 2*a11*b3 - 6*a11*b4 + 2*a3*b16 - 2*a3*b5 + a4*b16 + 3*a4*b17 - a4*b5
+ 3*a5*b3 - 12*a6*b1 - a9*b8,
3*a10*b10 - 6*a10*b22 - 4*a11*b20 + 2*a11*b8 - 2*a12*b17 + a12*b6 - 2*a5*b18 -
a5*b20 - 4*a6*b15 - 2*a6*b17 - 6*a7*b13,
3*a1*b10 - 2*a12*b3 - 12*a12*b4 + 2*a3*b19 - 2*a3*b7 + a4*b19 + 3*a4*b20 - a4*
b7 + 2*a6*b3 - 18*a7*b1 - a9*b10,
3*a10*b11 + 2*a11*b10 - 6*a11*b22 - 4*a12*b20 + a12*b8 - 2*a13*b17 - 4*a6*b18 -
a6*b20 - 6*a7*b15 - 2*a7*b17 - 8*a8*b13,
3*a1*b11 - 2*a13*b3 - 18*a13*b4 + 2*a3*b21 - 2*a3*b9 + a4*b21 + 3*a4*b22 - a4*
b9 + a7*b3 - 24*a8*b1 - a9*b11,
3*a10*b10 + 6*a10*b9 + 4*a11*b7 + 2*a11*b8 + 2*a12*b5 + a12*b6 - 2*a5*b18 + a5*
b7 - 4*a6*b15 + 2*a6*b5 - 6*a7*b13 + 3*a7*b2,
3*a10*b11 + 2*a11*b10 + 6*a11*b9 + 4*a12*b7 + a12*b8 + 2*a13*b5 - 4*a6*b18 + a6
*b7 - 6*a7*b15 + 2*a7*b5 - 8*a8*b13 + 3*a8*b2,
4*a10*b18 + 3*a10*b19 + 3*a10*b20 - 3*a10*b7 + 2*a11*b16 + 2*a11*b17 - 2*a11*b5
- 4*a12*b13 + a12*b14 - a12*b2 + 4*a2*b18 + 4*a5*b15 + 4*a6*b13,
6*a10*b10 + 6*a11*b8 + 6*a12*b6 - 3*a2*b10 - 2*a5*b19 - 2*a5*b20 + 2*a5*b7 - a5
*b8 - 4*a6*b16 - 4*a6*b17 + 4*a6*b5 + a6*b6 - 6*a7*b14 + 6*a7*b2,
3*a10*b21 + 3*a10*b22 - 3*a10*b9 + 2*a11*b18 + 2*a11*b19 + 2*a11*b20 - 2*a11*b7
- 2*a12*b15 + a12*b16 + a12*b17 - a12*b5 - 6*a13*b13 + 3*a5*b18 + 3*a6*b15 + 3
*a7*b13,
10*a11*b11 + 10*a12*b10 + 10*a13*b8 - 3*a5*b11 - a6*b10 - 4*a6*b21 - 4*a6*b22 +
4*a6*b9 - 6*a7*b19 - 6*a7*b20 + 6*a7*b7 + a7*b8 - 8*a8*b16 - 8*a8*b17 + 8*a8*
b5 + 3*a8*b6,
4*a10*b11 + 4*a11*b10 + 4*a12*b8 + 4*a13*b6 - 2*a2*b11 - a5*b10 - a5*b21 - a5*
b22 + a5*b9 - 2*a6*b19 - 2*a6*b20 + 2*a6*b7 - 3*a7*b16 - 3*a7*b17 + 3*a7*b5 +
a7*b6 - 4*a8*b14 + 4*a8*b2
Relevance
In the following evolutionary system u=u(t,x) is a scalar function,
v=v(t,x) is a vector function and f(..,..) stands for the scalar
product of both vector arguments of f.
4 3
u = u *a1 + f(v ,v )*a3 + f(v,v )*a4 + f(v,v) *a8 + f(v,v) *a7*u
t 2x x x 2x
2 2 3 4
+ f(v,v) *a6*u + f(v,v)*a5*u + a2*u
3 2 2 3
v = v *a9 + f(v,v) *a13*v + f(v,v) *a12*u*v + f(v,v)*a11*u *v + a10*u *v
t 2x
For any solution of the above algebraic conditions for the
a's and b's this system has the following symmetry:
3 2 2 3
u = u *b1 + u *f(v,v) *b9 + u *f(v,v) *b7*u + u *f(v,v)*b5*u + u *b2*u
t 3x x x x x
3
+ f(v ,v )*b3 + f(v,v )*b4 + f(v,v )*f(v,v) *b11
x 2x 3x x
2 2 3
+ f(v,v )*f(v,v) *b10*u + f(v,v )*f(v,v)*b8*u + f(v,v )*b6*u
x x x
2 2
v = u *f(v,v) *b18*v + u *f(v,v)*b15*u*v + u *b13*u *v + v *b12
t x x x 3x
3 2 2 3
+ v *f(v,v) *b21 + v *f(v,v) *b19*u + v *f(v,v)*b16*u + v *b14*u
x x x x
2 2
+ f(v,v )*f(v,v) *b22*v + f(v,v )*f(v,v)*b20*u*v + f(v,v )*b17*u *v
x x x