Pensieve Header: A failed attempt to solve the two F equations with analytic hair using Tail Scattering, where F is written as an exponential. The equations are derived but not solved.

Ouch! f1 and f2 were treated as constants rather than functions!

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\ScatterAndGlow"];
<< ScatterAndGlow.m

F=TS[Exp[
-1/4 Ar[2,1]+1/4 Ar[1,2]+Y[1,2,1,AH[f1]]+Y[1,2,2,AH[f2]]
]]

Ar[1,0] + Ar[2,0] // F

Simplify[(2 (E^(-((x[2])/4)-f2 x[1] x[2])) (-(E^(((x[1])/4)+f1 x[1] x[2]))+(E^(((x[2])/4)+f2 x[1] x[2]))) (-1+2 f1 x[1]+2 f2 x[2]))/(x[1]-x[2]+4 f1 x[1] x[2]-4 f2 x[1] x[2])]

F21 = TS[Exp[
-1/4 Ar[1,2]+1/4 Ar[2,1]+Y[2,1,2,AH[f1]]+Y[2,1,1,AH[f2]]
]]

lhs1 = Ar[3,0] // S[Exp[Ar[1,3]+Ar[2,3]]] // F

rhs1 = Ar[3,0] // F // S[sigma[1,3],sigma[2,3]]

lhs2 = Ar[1,0] // F21 // S[sigma[1,2]]// S[Exp[1/2 Ar[1,1]]] // S[Exp[1/2 Ar[2,2]]]

rhs2=Ar[1,0] // S[Exp[Expand[1/2(Ar[1,1]+Ar[1,2]+Ar[2,1]+Ar[2,2])]]] // F

(eqns1 = Simplify[{
Coefficient[lhs2, Y[1,2,0]] == Coefficient[rhs2, Y[1,2,0]],
Coefficient[lhs1, Y[1,2,0]] == Coefficient[rhs1, Y[1,2,0]]
}]) // ColumnForm

(eqns2 = Simplify[{
(E^(f2 x[1] x[2]))(Coefficient[lhs2, Y[1,2,0]] - Coefficient[rhs2, Y[1,2,0]]),
(E^(+(1/4) (1+4 f2 x[1]) x[2]))x[1] x[2] (x[1]+x[2]) (x[2]+x[1] (-1-4 f1 x[2]+4 f2 x[2]))/x[3](Coefficient[lhs1, Y[1,2,0]] - Coefficient[rhs1, Y[1,2,0]])
}] /. {x[1] -> x1, x[2] -> x2}) // ColumnForm

(eqns3=Simplify[eqns2 /. {x1-> Random[], x2->Random[]}]) // ColumnForm

The following crashed after a few hours

Solve[(#==0)& /@ eqns3, {f1,f2}]

F

Reduce[
{
g1==(((E^(-((x[2])/4)-f2 x[1] x[2])) (-(E^(((x[1])/4)+f1 x[1] x[2]))+(E^(((x[2])/4)+f2 x[1] x[2]))) (-1+4 f1 x[1]))/(x[1]-x[2]+4 f1 x[1] x[2]-4 f2 x[1] x[2])),
g2==-(((E^(-((x[2])/4)-f2 x[1] x[2])) (-(E^(((x[1])/4)+f1 x[1] x[2]))+(E^(((x[2])/4)+f2 x[1] x[2]))) (-1+4 f2 x[2]))/(-x[1]+x[2]-4 f1 x[1] x[2]+4 f2 x[1] x[2]))
}, {f1,f2}
]