Problem l21o24
Unknowns | Inequalities |
Equations |
Solution 1 |
Solution 2 |
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Unknowns
All solutions for the following 37 unknowns have to be determined:
a1, ..., a9, b1, ..., b28
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,a7},
{a3,a4,a6,a5},
{a8},
{b1,b7,b18},
{b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b12,b13,b14,b15,b16,b17},
{b18,b19,b20,b21,b22,b23,b24,b25,b26,b27,b28}
Equations
All comma separated 74 expressions involving 452 terms have to vanish.
All terms are products of one a- and one b-unknown.
a5*b17 - a6*b16 + 4*a6*b28 - 6*a9*b17,
a6*b27 - a8*b17 + 2*a9*b28,
a2*b17 + a5*b28 - a6*b12 + 2*a6*b27 - 3*a8*b17 - 2*a9*b16,
24*a1*b17 + 8*a3*b28 + 16*a4*b28 + a5*b15 + 4*a6*b24 - 8*a6*b8 - 4*a6*b9 - 4*a9
*b13 - 4*a9*b14 - 8*a9*b15,
6*a1*b17 - a4*b16 + 6*a4*b28 + a5*b14 - a6*b11 + 4*a6*b23 + 4*a6*b26 - 4*a6*b8
- 6*a7*b17 - 6*a9*b14,
6*a1*b17 - a3*b16 + 2*a3*b28 + 4*a4*b28 + a5*b13 - a6*b10 + 4*a6*b25 - 4*a6*b8
- 4*a9*b13 - 2*a9*b14,
a5*b27 + 2*a6*b22 - a8*b16 + 4*a8*b28,
a4*b27 + 4*a6*b19 - a8*b14 + 2*a9*b26,
a4*b27 + 4*a6*b19 - a6*b2 - a8*b14 - 2*a9*b8,
8*a6*b19 - 8*a7*b28 - a8*b15 + 4*a9*b25 + 4*a9*b26,
a3*b27 + 4*a6*b19 - 4*a7*b28 - a8*b13 + 2*a9*b23 + 2*a9*b26,
a2*b16 - a5*b12 + 2*a5*b27 + 4*a6*b22 - 3*a6*b4 - 4*a8*b16 - 2*a9*b12,
2*a1*b13 + 2*a1*b15 - a3*b10 + 2*a3*b24 + 2*a3*b25 - 24*a6*b1 - 6*a9*b6,
a1*b14 + a4*b23 + a4*b26 - a4*b8 + a5*b7 - 4*a6*b1 + 4*a6*b18 - a7*b14 - 4*a9*
b7,
4*a1*b16 + a2*b15 + 2*a3*b27 + 2*a4*b27 + a5*b24 - a5*b9 - 4*a6*b2 - 2*a8*b15 -
2*a9*b10 - 2*a9*b11,
2*a1*b14 + 2*a1*b15 - a4*b11 + 2*a4*b23 + 2*a4*b24 + 2*a4*b26 - 24*a6*b1 - 2*a7
*b14 - 4*a9*b5 - 12*a9*b7,
4*a1*b16 + a3*b27 + 3*a4*b27 - 2*a5*b8 + 2*a6*b20 - 4*a6*b3 - a8*b13 - a8*b14 -
a8*b15 - 2*a9*b9,
4*a1*b14 + 2*a1*b15 + 2*a3*b26 + 4*a4*b23 + 2*a4*b24 + 2*a4*b26 - a4*b9 - 32*a6
*b1 - 2*a7*b15 - 4*a9*b6 - 16*a9*b7,
2*a1*b16 + a2*b13 - a3*b12 + a3*b27 + a4*b27 + a5*b25 - a5*b8 - 2*a6*b2 - 2*a8*
b13 - a9*b10 - a9*b11,
2*a1*b16 + a2*b14 - a4*b12 + 2*a4*b27 + a5*b23 + a5*b26 - a5*b8 - 2*a6*b2 + 2*
a6*b21 - 2*a7*b16 - 2*a8*b14 - 2*a9*b11,
2*a1*b13 + 2*a1*b14 + 2*a3*b23 - 2*a3*b8 + 2*a4*b25 + 2*a4*b26 - 2*a4*b8 + a5*
b6 - 16*a6*b1 - 2*a7*b13 - 2*a9*b6 - 8*a9*b7,
2*a1*b13 + a1*b14 - 2*a3*b8 + a4*b23 + 2*a4*b25 + a4*b26 - a4*b8 + a5*b5 - 12*
a6*b1 - a7*b14 - 2*a9*b5 - 6*a9*b7,
2*a1*b13 + 2*a1*b14 + 4*a1*b15 - a3*b11 + 2*a3*b24 + 2*a3*b26 - a4*b10 + 2*a4*
b23 + 2*a4*b24 + 2*a4*b25 - 48*a6*b1 - 2*a7*b13 - 4*a9*b5 - 6*a9*b6 - 12*a9*b7,
8*a1*b13 + 4*a1*b14 + 6*a1*b15 + 4*a3*b23 + 2*a3*b24 + 4*a3*b25 + 2*a3*b26 - 2*
a3*b9 + 4*a4*b24 + 4*a4*b25 + 2*a4*b26 - a4*b9 - 96*a6*b1 - 2*a7*b15 - 8*a9*b5
- 12*a9*b6 - 24*a9*b7,
a6*b20 - 2*a7*b28 + a9*b23 + a9*b24,
a6*b21 + 2*a9*b23,
a4*b19 - a8*b7,
a5*b22 - a5*b4 - a8*b12,
a1*b7 - a4*b1 + a4*b18 - a7*b7,
2*a4*b22 + 2*a5*b19 - a8*b11 + 2*a8*b26,
2*a1*b5 + a1*b6 - 6*a3*b1 - 4*a4*b1 - a7*b6,
2*a1*b12 - 2*a2*b8 + 2*a4*b22 - a5*b3 - a8*b9,
2*a4*b22 + 2*a5*b19 - a5*b2 - a8*b11 - 2*a8*b8,
a2*b27 + 2*a5*b22 - a8*b12 + 2*a8*b27 - 2*a9*b22,
2*a3*b19 + 2*a4*b19 - 2*a7*b26 - a8*b6 + 8*a9*b18,
2*a3*b19 + a4*b19 - 2*a7*b25 - a8*b5 + 6*a9*b18,
2*a1*b12 + a2*b10 + 2*a3*b22 - 3*a3*b4 - 2*a5*b2 - 2*a8*b10,
a1*b6 + 2*a1*b7 - 2*a3*b1 + 2*a3*b18 - 4*a4*b1 - a7*b6,
2*a3*b22 + 2*a5*b19 - 2*a7*b27 - a8*b10 + 2*a8*b25 + 2*a9*b21,
4*a5*b19 - 4*a7*b27 + a8*b24 + 2*a8*b25 + 2*a8*b26 - a8*b9,
2*a1*b8 + 2*a1*b9 + 2*a3*b19 - a3*b2 + 2*a3*b20 - 12*a5*b1 - 3*a8*b6,
4*a1*b8 + 2*a3*b19 + 2*a4*b19 + a4*b20 - 8*a5*b1 - a8*b6 - 4*a8*b7,
a1*b27 + a5*b19 - a7*b27 + a8*b23 + a8*b26 - a8*b8 - 2*a9*b19,
2*a1*b12 + a2*b11 + 2*a4*b22 - 3*a4*b4 - 2*a5*b2 + 2*a5*b21 - 2*a7*b12 - 2*a8*
b11,
a1*b11 + 2*a2*b7 - a4*b2 + a4*b21 - 2*a5*b1 + 2*a5*b18 - a7*b11 - 2*a8*b7,
2*a1*b5 + 2*a1*b6 + a1*b7 - 8*a3*b1 - 7*a4*b1 + a4*b18 - 2*a7*b5 - a7*b7,
a1*b10 + a1*b11 + a2*b6 - a3*b2 + a3*b21 - a4*b2 - 4*a5*b1 - a7*b10 - a8*b6,
2*a1*b10 + a1*b11 + 2*a2*b5 - 2*a3*b2 - a4*b2 + a4*b21 - 6*a5*b1 - a7*b11 - 2*
a8*b5,
4*a1*b12 + 2*a3*b22 + 2*a4*b22 - 2*a5*b2 + a5*b20 - 2*a5*b3 - a8*b10 - a8*b11 -
a8*b9,
2*a1*b11 + a1*b9 + a4*b20 + 2*a4*b21 - 2*a4*b3 - 8*a5*b1 - a7*b9 - a8*b6 - 4*a8
*b7,
2*a1*b8 + 2*a1*b9 + 2*a4*b19 - a4*b2 + 2*a4*b20 + a4*b21 - 12*a5*b1 - 2*a7*b8 -
2*a8*b5 - 6*a8*b7,
4*a1*b10 + 2*a1*b11 + 3*a1*b9 + 2*a3*b20 + 2*a3*b21 - 4*a3*b3 + a4*b20 - 2*a4*
b3 - 24*a5*b1 - a7*b9 - 4*a8*b5 - 3*a8*b6,
a4*b20 - 2*a7*b26 + 8*a9*b18,
a5*b20 - 2*a7*b27 + a8*b24 + 2*a9*b21,
a5*b20 - 2*a7*b27 + 2*a8*b23 + a8*b24,
2*a3*b20 + a4*b20 - 2*a7*b24 - 4*a7*b25 - 2*a7*b26 + 24*a9*b18,
a5*b21 + 2*a8*b23,
a4*b21 - 2*a7*b24 + 12*a9*b18,
a3*b21 - 2*a7*b23 - 2*a7*b24 + 12*a9*b18,
a7*b23 - 2*a9*b18,
a2*b4,
a2*b2,
2*a2*b22 - a8*b4,
a1*b3 - 3*a2*b1,
a1*b2 - 4*a2*b1,
3*a1*b4 - a2*b2 - a2*b3,
2*a2*b19 - 2*a7*b22 + a8*b20 - a8*b3,
a1*b19 - a7*b19 - a8*b1 + a8*b18,
2*a1*b22 + 2*a2*b19 - 2*a7*b22 - a8*b2 + a8*b21,
a2*b20 - 2*a7*b22 + a8*b21,
a1*b20 - 2*a7*b19 - a7*b20 + 4*a8*b18,
a2*b21,
a1*b21 - 2*a7*b20 - a7*b21 + 6*a8*b18,
a7*b21 - 2*a8*b18
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.
2 2
u = u *a1 + f(v ,v )*a3 + f(v,v )*a4 + f(v,v) *a6 + f(v,v)*a5*u + a2*u ,
t 2x x x 2x
v = v *a7 + f(v,v)*a9*v + a8*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:
2
uu = u *b1 + u *f(v,v)*b8 + u *b2*u + u *b3 + u *f(v,v )*b9
t 4x 2x 2x x x x
+ f(v ,v )*b5 + f(v ,v )*b6 + f(v ,v )*f(v,v)*b13 + f(v ,v )*b10*u
2x 2x x 3x x x x x
2
+ f(v,v )*b7 + f(v,v )*f(v,v)*b14 + f(v,v )*b11*u + f(v,v ) *b15
4x 2x 2x x
3 2 2 3
+ f(v,v) *b17 + f(v,v) *b16*u + f(v,v)*b12*u + b4*u ,
v = u *b19*v + u *v *b20 + v *b18 + v *f(v,v)*b23 + v *b21*u
t 2x x x 4x 2x 2x
2
+ v *f(v,v )*b24 + f(v ,v )*b25*v + f(v,v )*b26*v + f(v,v) *b28*v
x x x x 2x
2
+ f(v,v)*b27*u*v + b22*u *v