Problem l12o13
Unknowns | Inequalities |
Equations |
Solution 1 |
Relevance |
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Unknowns
All solutions for the following 42 unknowns have to be determined:
a0, a2, a3, a4, a5, a6, b1, ..., b36
(where a4 does not occur in the equations as it can be gauged
to zero by a Galilei transformation).
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.
{a0}, % 1st order system after
% Galilei transformation
{b1,b17}, % 3rd order symmetry
{a3,b6,b7,b13,b10,b8,b9,b14,b11,b16,b15,b12}, % u_t or u_s to involve v
{a5,b18,b19,b20,b29,b22,b21,
b30,b23,b31,b35,b32,b24} % v_t or v_s to involve u
Equations
All comma separated 66 expressions involving 221 terms have to vanish.
All terms are products of one a- and one b-unknown.
a6*b36,
a2*b21,
a2*b23,
a2*b1,
a2*b2,
a2*b4,
a2*b5,
a6*b17,
a6*b25,
a6*b28,
a6*b6,
a6*b33,
a6*b34,
a6*b13,
3*a2*b2 + 2*a2*b3,
a6*b25 + a6*b26,
a6*b25 + a6*b28,
a6*b27 + a6*b28,
a0*b7 - 6*a3*b1,
a0*b6 - 3*a3*b1,
a0*b20 + 6*a5*b17,
a0*b21 + 3*a5*b17,
a3*b36 - 3*a6*b16,
3*a2*b24 - a5*b5,
a3*b21 + a5*b25,
2*a0*b8 + a0*b9 - 6*a3*b1,
a0*b18 + 2*a0*b19 + 6*a5*b17,
a0*b13 - a3*b6 - a6*b7,
a0*b9 - 2*a3*b1 + 2*a3*b17,
a0*b18 - 2*a5*b1 + 2*a5*b17,
a0*b23 + a2*b20 + a5*b21,
a3*b20 + a5*b26 + 2*a6*b21,
a2*b30 + 2*a3*b23 + a5*b30,
a3*b18 + a5*b28 - a5*b9,
a3*b30 + 2*a5*b33 + a6*b30,
a0*b12 + a3*b23 - a3*b4 - a5*b11,
a0*b11 + 2*a2*b8 - 2*a3*b2 - 2*a5*b8,
a0*b30 + a3*b20 + 2*a5*b25 + a5*b26,
3*a0*b24 + 3*a2*b22 + a5*b23 - a5*b4,
a0*b22 + 4*a2*b18 - 2*a5*b2 + 2*a5*b21,
a3*b18 + a5*b27 - a5*b8 + a6*b21,
a2*b31 + a3*b22 - a5*b11 + a5*b31,
a3*b18 - a3*b2 - a5*b6 - a5*b9,
a2*b12 - a3*b24 + 3*a3*b5 + a5*b12,
a2*b35 + 2*a3*b32 - a5*b15 + 2*a5*b35,
2*a3*b12 - a3*b32 + 2*a5*b15 + a6*b12,
a3*b35 - a5*b16 + 3*a5*b36 + a6*b35,
3*a0*b16 - a3*b13 + a3*b33 + a3*b34 - 3*a6*b14,
a0*b14 + a3*b26 - a3*b7 - 2*a6*b8 - 2*a6*b9,
a0*b14 + 2*a3*b25 + 2*a3*b28 - 2*a3*b6 - 4*a6*b9,
a0*b14 + 2*a3*b27 - 2*a3*b6 - 2*a6*b8 - 2*a6*b9,
a0*b35 + a3*b29 - a5*b13 + a5*b33 + a5*b34,
a0*b11 + 2*a2*b9 - 2*a3*b2 + 2*a3*b21 - 2*a5*b9,
a0*b29 + 2*a3*b19 + 2*a5*b25 + 2*a5*b28 - 2*a6*b19,
2*a2*b32 + 3*a3*b24 - a5*b12 + a5*b32 - a6*b24,
3*a2*b6 - a3*b19 + a3*b3 + a5*b7 + a5*b9,
a2*b10 - a3*b22 + 2*a3*b4 + a5*b10 + a5*b11,
a3*b29 + a3*b31 - a5*b14 + 2*a5*b34 + a6*b31,
a3*b10 - a3*b29 + 2*a5*b13 + a5*b14 + a6*b10,
a2*b16 - a3*b15 + a3*b35 - 3*a5*b16 - 2*a6*b15,
a0*b29 + 2*a3*b18 + 2*a5*b25 + 2*a5*b28 - 2*a5*b6 - 2*a6*b18,
3*a0*b22 + 6*a2*b18 + 6*a2*b19 + 2*a5*b20 + 2*a5*b21 - 2*a5*b3,
a0*b10 - 2*a3*b2 + a3*b20 - 2*a3*b3 - a5*b7 - 2*a5*b8 - 2*a5*b9,
a0*b31 + 2*a3*b18 + 2*a3*b19 + a5*b26 + 2*a5*b27 + 2*a5*b28 - a5*b7,
2*a0*b15 + a2*b14 - a3*b10 - a3*b11 + a3*b30 + a3*b31 - 2*a5*b14 - 2*a6*b11,
2*a0*b32 + 2*a2*b29 + 2*a3*b22 - a5*b10 + a5*b29 + a5*b30 + a5*b31 - a6*b22
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
u = u *(a0+a4) + f(v,v)*a3*u + a2*u ,
t x
2
v = v *a4 + f(v,v)*a6*v + a5*u *v
t x
For any solution of the above algebraic conditions for the
a's and b's this system has the following symmetry:
2 2
u = u *b1 + u *f(v,v)*b6 + u *b2*u + u *b3*u + u *f(v,v )*b7
s 3x 2x 2x x x x
2 2 4
+ u *f(v,v) *b13 + u *f(v,v)*b10*u + u *b4*u + f(v ,v )*b8*u
x x x x x
3 3
+ f(v,v )*b9*u + f(v,v )*f(v,v)*b14*u + f(v,v )*b11*u + f(v,v) *b16*u
2x x x
2 3 5 7
+ f(v,v) *b15*u + f(v,v)*b12*u + b5*u ,
2 3
v = u *b18*u*v + u *b19*v + u *v *b20*u + u *f(v,v)*b29*u*v + u *b22*u *v
s 2x x x x x x
2
+ v *b17 + v *f(v,v)*b25 + v *b21*u + v *f(v,v )*b26
3x 2x 2x x x
2 2 4
+ v *f(v,v) *b33 + v *f(v,v)*b30*u + v *b23*u + f(v ,v )*b27*v
x x x x x
2
+ f(v,v )*b28*v + f(v,v )*f(v,v)*b34*v + f(v,v )*b31*u *v
2x x x
3 2 2 4 6
+ f(v,v) *b36*v + f(v,v) *b35*u *v + f(v,v)*b32*u *v + b24*u *v