Problem l1323o23
Unknowns | Inequalities |
Equations |
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Unknowns
All solutions for the following 34 unknowns have to be determined:
a1, ..., a10, b1, ..., b24
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,a6},
{a4,a5},
{a7,a8,a10,a9},
{b1,b12},
{b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11},
{b12,b13,b14,b15,b16,b17,b18,
b19,b20,b21,b22,b23,b24}
Equations
All comma separated 102 expressions involving 529 terms have to vanish.
All terms are products of one a- and one b-unknown.
a4*b24,
a3*b16,
a3*b2,
a3*b5,
a5*b16,
a6*b24,
a4*b11 - 3*a4*b24,
a1*b24 - a6*b24,
2*a6*b16 - 3*a8*b12,
2*a3*b18 - a8*b5,
2*a1*b2 - 3*a2*b1,
2*a1*b3 - 9*a2*b1,
a1*b3 - 3*a2*b1,
a2*b5 - 2*a3*b4,
3*a3*b19 - 2*a9*b5,
3*a10*b12 - 2*a6*b21,
a4*b13 - a7*b7,
a4*b21 - 4*a6*b24,
a4*b20 - 8*a6*b24,
2*a10*b7 - a4*b20,
a4*b22 - 4*a6*b24,
4*a10*b11 - a5*b11 - 2*a5*b24,
2*a10*b11 - 2*a10*b24 - a5*b24,
3*a1*b2 + 4*a1*b3 - 18*a2*b1,
2*a1*b4 - a2*b2 - 21*a3*b1,
30*a1*b5 - 14*a3*b2 - 3*a3*b3,
6*a10*b12 + 3*a4*b16 - 4*a6*b21,
3*a10*b12 - a6*b20 - a6*b21,
a1*b20 + 6*a10*b12 - 3*a6*b20,
2*a10*b21 + 2*a5*b21 - 3*a8*b11,
2*a10*b6 - a4*b20 + 4*a6*b24,
2*a1*b11 - 2*a10*b7 + a4*b20,
2*a6*b15 + 3*a6*b16 - 3*a7*b12 - 9*a8*b12,
a2*b16 - 4*a6*b18 + a8*b16 + 6*a9*b12,
a6*b14 + a6*b15 - 3*a7*b12 - 3*a8*b12,
10*a1*b4 - 2*a2*b2 - a2*b3 - 70*a3*b1,
a1*b13 - a6*b13 - a7*b1 + a7*b12,
a1*b13 - a6*b13 - a6*b14 + 3*a7*b12,
a1*b14 - 2*a6*b13 - 5*a6*b14 + 12*a7*b12,
a1*b21 + 3*a10*b12 - a6*b20 - a6*b21,
6*a10*b12 + a4*b15 - 2*a6*b22 - 3*a8*b7,
2*a3*b21 + 6*a5*b18 - 3*a8*b10 + 2*a9*b21,
a1*b20 - 2*a10*b1 + 2*a10*b12 - a6*b20,
12*a10*b12 + a4*b14 - 4*a6*b20 - 2*a6*b22,
6*a10*b12 + a4*b14 - 2*a6*b22 - a7*b7,
a4*b17 + 2*a5*b13 - a7*b9 - 6*a9*b7,
2*a1*b8 + a4*b14 - 6*a5*b1 - 2*a7*b7,
2*a1*b6 + a1*b7 - 3*a4*b1 - a6*b7,
a1*b7 - a4*b1 + a4*b12 - a6*b7,
a3*b10 + 2*a5*b19 - 7*a5*b5 - 2*a9*b10,
3*a1*b16 - 2*a6*b15 - 3*a6*b16 + 3*a7*b12 + 9*a8*b12,
a1*b14 - 2*a6*b13 - a6*b14 - a7*b1 + 4*a7*b12,
12*a10*b12 + a4*b15 - 2*a6*b20 - 4*a6*b21 - 2*a6*b22,
6*a10*b12 + a4*b15 - 2*a6*b21 - 2*a6*b22 - 3*a8*b6,
4*a10*b16 + 6*a4*b18 + 2*a5*b15 - 4*a6*b23 - 3*a8*b9,
a1*b22 + 6*a10*b12 + a4*b13 - 2*a6*b20 - a6*b22,
6*a10*b12 + a4*b14 - a6*b20 - 2*a6*b22 - a7*b6,
6*a10*b12 + 3*a4*b13 - 2*a6*b22 - 2*a7*b6 - a7*b7,
2*a1*b8 + a4*b13 + a4*b14 - 9*a5*b1 - 3*a7*b7,
a4*b17 + 2*a5*b13 - 3*a5*b2 - a7*b9 - 6*a9*b7,
2*a1*b6 + 2*a1*b7 - 4*a4*b1 + a4*b12 - 2*a6*b6,
2*a10*b10 - a3*b24 - 5*a5*b23 + 6*a9*b11 - 4*a9*b24,
2*a10*b10 - 5*a3*b11 + 3*a5*b10 - 2*a5*b23 + 4*a9*b11,
a1*b15 - 2*a6*b13 - a6*b15 + 3*a7*b12 - 3*a8*b1 + 3*a8*b12,
a1*b15 - 2*a6*b13 - 2*a6*b14 - 3*a6*b15 + 9*a7*b12 + 9*a8*b12,
2*a2*b18 + 3*a3*b15 - 6*a6*b19 - a8*b18 - a8*b4 + 4*a9*b16,
3*a2*b19 + 4*a3*b17 - 4*a7*b5 - 3*a8*b19 + 2*a9*b18 - 2*a9*b4,
2*a10*b16 + a4*b17 + 2*a5*b13 - 2*a6*b23 - a7*b9 - 6*a9*b6,
2*a3*b22 + 9*a4*b19 + 2*a5*b17 - 2*a7*b10 + 2*a9*b22 - 6*a9*b9,
2*a10*b19 + 2*a10*b5 - 5*a3*b23 - 9*a5*b19 + 6*a9*b10 - 2*a9*b23,
2*a10*b22 - 2*a10*b9 + 5*a4*b23 + 2*a5*b20 + 2*a5*b22 - 4*a7*b11,
8*a1*b11 - 4*a10*b6 - 4*a10*b7 + 2*a4*b21 + 2*a4*b22 - 3*a4*b9,
4*a10*b16 + 2*a2*b21 + 5*a5*b15 - 10*a6*b23 + 2*a7*b21 + a8*b21 - 3*a8*b8,
8*a1*b11 - 2*a10*b6 - 6*a10*b7 + 2*a4*b20 + a4*b21 + a4*b22 - a4*b8,
2*a10*b8 + 2*a10*b9 - 3*a2*b11 - 5*a4*b23 - 2*a5*b20 + 2*a5*b8 + 4*a7*b11,
4*a1*b11 - 2*a10*b6 - 2*a10*b7 + a4*b21 + a4*b22 - a4*b8 + 3*a5*b6,
4*a1*b11 - 4*a10*b7 + a4*b21 + a4*b22 - a4*b8 + 3*a5*b7 - 4*a6*b11,
5*a2*b15 - 10*a6*b17 - 30*a6*b18 + 4*a7*b16 + a8*b15 + 6*a8*b16 - 3*a8*b3 + 90*
a9*b12,
a1*b17 + a2*b13 - a6*b17 + a7*b16 - a7*b2 - a8*b13 - 6*a9*b1 + 6*a9*b12,
6*a2*b13 + 7*a2*b14 - 20*a6*b17 + 2*a7*b15 + 2*a7*b16 - 4*a7*b3 - a8*b14 + 120*
a9*b12,
4*a1*b8 + 2*a4*b13 + a4*b15 - 3*a4*b2 - 18*a5*b1 - 2*a7*b6 - 2*a7*b7 - 3*a8*b7,
30*a1*b10 + 5*a4*b17 + 2*a5*b14 - 6*a5*b2 - 5*a5*b3 - 2*a7*b8 - 2*a7*b9 - 30*a9
*b7,
2*a1*b8 + 6*a1*b9 + 3*a2*b6 + a4*b15 - 2*a4*b3 - 18*a5*b1 - 2*a7*b6 - 6*a8*b6,
3*a2*b10 - 2*a3*b8 + 9*a4*b19 + 2*a5*b17 - 6*a5*b4 - 2*a7*b10 - 2*a9*b8 - 6*a9*
b9,
2*a1*b10 + a2*b9 + 7*a3*b6 + a4*b18 - a4*b4 - 2*a5*b2 - a8*b9 - 2*a9*b6,
a1*b9 + a2*b7 + a4*b16 - a4*b2 - 2*a5*b1 + 2*a5*b12 - a6*b9 - a8*b7,
a2*b10 + 2*a3*b9 + a4*b19 - 5*a4*b5 + a5*b18 - a5*b4 - a8*b10 - a9*b9,
2*a10*b21 + 2*a10*b22 - 2*a10*b8 + a2*b24 + 4*a5*b20 - 4*a7*b11 + 4*a7*b24 - a8
*b24,
2*a10*b9 - 2*a2*b11 + 3*a4*b10 - 2*a4*b23 - a5*b21 - a5*b22 + a5*b8 + 2*a8*b11,
6*a1*b18 + a2*b15 - 2*a6*b17 - 6*a6*b18 + 2*a7*b16 - a8*b15 + 3*a8*b16 - 3*a8*
b2 + 18*a9*b12,
9*a1*b19 + a2*b17 + 9*a3*b13 - 9*a6*b19 + a7*b18 - a7*b4 - a8*b17 + 6*a9*b16 -
6*a9*b2,
4*a1*b8 + 3*a1*b9 + a4*b14 + a4*b15 - a4*b3 - 18*a5*b1 - 2*a7*b6 - 2*a7*b7 - 3*
a8*b7,
3*a1*b9 + 2*a2*b6 + a2*b7 + a4*b16 - 3*a4*b2 - 6*a5*b1 - a6*b9 - 2*a8*b6 - a8*
b7,
8*a2*b17 + 42*a3*b13 + 15*a3*b14 - 72*a6*b19 + 2*a7*b18 - 8*a7*b4 - 5*a8*b17 +
6*a9*b15 + 30*a9*b16 - 6*a9*b3,
8*a10*b16 + 2*a2*b22 + 5*a4*b17 + 12*a5*b13 + 4*a5*b14 - 20*a6*b23 + 2*a7*b22 -
2*a7*b8 - 5*a7*b9 + a8*b22,
2*a1*b8 + 6*a1*b9 + 3*a2*b7 + a4*b15 + 3*a4*b16 - 2*a4*b3 - 18*a5*b1 - 2*a6*b8
- 2*a7*b7 - 6*a8*b7,
2*a1*b10 + a2*b9 + 7*a3*b7 + a4*b18 - a4*b4 + 2*a5*b16 - 2*a5*b2 - 2*a6*b10 -
a8*b9 - 2*a9*b7,
5*a1*b17 + 11*a2*b13 + 2*a2*b14 - 15*a6*b17 + a7*b15 + 6*a7*b16 - 5*a7*b2 - 2*
a7*b3 - 2*a8*b13 - 2*a8*b14 + 90*a9*b12,
5*a1*b23 - 2*a10*b13 + 2*a10*b16 - 2*a10*b2 + a2*b20 + 5*a5*b13 - 5*a6*b23 + a7
*b21 + a7*b22 - a7*b8 - a8*b20,
2*a10*b14 - 2*a10*b15 - 2*a10*b16 + 2*a10*b3 - 4*a2*b20 - 6*a5*b13 - 7*a5*b14 +
20*a6*b23 - 2*a7*b20 - 2*a7*b21 - 2*a7*b22 + 4*a7*b8 + a8*b20,
2*a10*b17 - 2*a10*b18 + 2*a10*b4 - 5*a2*b23 - 8*a3*b20 - 8*a5*b17 + 8*a7*b10 -
2*a7*b23 + 5*a8*b23 - 2*a9*b20 - 6*a9*b21 - 6*a9*b22 + 6*a9*b8,
24*a1*b10 + 2*a2*b8 + 3*a2*b9 + 2*a4*b17 + 6*a4*b18 - 6*a4*b4 + 2*a5*b15 - 12*
a5*b2 - 4*a5*b3 - 2*a7*b9 - 2*a8*b8 - 3*a8*b9 - 12*a9*b6 - 12*a9*b7
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.
3 3 7
u = u *a1 + u *a2*u + f(v,v )*a4 + f(v,v)*a5*u + a3*u
t 2x x x
2 3 2 6
v = u *a7*u *v + v *a6 + v *a8*u + f(v,v)*a10*u *v + a9*u *v
t x 2x x
For any solution of the above algebraic conditions for the
a's and b's this system has the following symmetry:
3 2 2 2 6
u = u *b1 + u *b2*u + u *b3*u + u *f(v,v)*b8*u + u *b4*u
t 3x 2x x x x
3 2 2
+ f(v ,v )*b6 + f(v,v )*b7 + f(v,v )*b9*u + f(v,v) *b11*u
x x 2x x
6 10
+ f(v,v)*b10*u + b5*u
2 2 2
v = u *b13*u *v + u *b14*u*v + u *v *b15*u + u *f(v,v)*b20*u*v
t 2x x x x x
5 3 2 6
+ u *b17*u *v + v *b12 + v *b16*u + v *f(v,v)*b21*u + v *b18*u
x 3x 2x x x
2 2 5 9
+ f(v,v )*b22*u *v + f(v,v) *b24*u*v + f(v,v)*b23*u *v + b19*u *v
x