I have a system of ten complex equations, with four complex variables u1,u2,v1,v2
and two real variables θ1, θ2
. In these equations there are four real and positive parameters a1, a2, m, k
.
I am trying to find an analytical solution to these equations, if it exists, using Mathematica. Sometimes it keeps running until my computer crashes or it says that the equations are not a quantified system of equations.
This is my Mathematica code:
eq1 = Abs[u1]^2 - Abs[v1]^2 == 0;
eq2 = Abs[u2]^2 - Abs[v2]^2 == 0;
eq3 = (u1 Cosh[θ1] - Conjugate[v1] Sinh[θ1]) + (Conjugate[u2] Cosh[θ2] - v2 Sinh[θ2]) == 0;
eq4 = (v1 Cosh[θ1] - Conjugate[u1] Sinh[θ1]) + (-v2 Cosh[θ2] + Conjugate[u2] Sinh[θ2]) == 0;
eq5 = ((k^2)/(4 m) + a1) Cosh[2 θ1] + a1 Sinh[2 θ1] + ((k^2)/(4 m) + a2) (Abs[u2]^2 + Abs[v2]^2) + a2 (u2 v2 + Conjugate[u2] Conjugate[v2]) - Sqrt[((k^2)/(4 m) + a1)^2 - a1^2] == 0;
eq6 = ((k^2)/(4 m) + a2) Cosh[2 θ2] + a2 Sinh[2 θ2] + ((k^2)/(4 m) + a1) (Abs[u1]^2 + Abs[v1]^2) + a1 (u1 v1 + Conjugate[u1] Conjugate[v1]) - Sqrt[((k^2)/(4 m) + a2)^2 - a2^2] == 0;
eq7 = ((k^2)/(4 m) + a1) (u1 Cosh[θ1] + Conjugate[v1] Sinh[θ1]) + a1 (Conjugate[v1] Cosh[θ1] + u1 Sinh[θ1]) + ((k^2)/(4 m) + a2) (Conjugate[u2] Cosh[θ2] + v2 Sinh[θ2]) + a2 (v2 Cosh[θ2] + Conjugate[u2] Sinh[θ2]) == 0;
eq8 = ((k^2)/(4 m) + a1) Sinh[2 θ1] + a1 Cosh[2 θ1] + ((k^2)/(4 m) + a2) 2 Conjugate[u2] v2 + a2 (Conjugate[u2]^2 + v2^2) == 0;
eq9 = ((k^2)/(4 m) + a1) (v1 Cosh[θ1] + Conjugate[u1] Sinh[θ1]) + a1 (Conjugate[u1] Cosh[θ1] + v1 Sinh[θ1]) + ((k^2)/(4 m) + a2) (v2 Cosh[θ2] + Conjugate[u2] Sinh[θ2]) + a2 (Conjugate[u2] Cosh[θ2] + v2 Sinh[θ2]) == 0;
eq10 = ((k^2)/(4 m) + a2) Sinh[2 θ2] + a1 Cosh[2 θ2] + ((k^2)/(4 m) + a1) 2 Conjugate[u1] v1 + a1 (Conjugate[u1]^2 + v1^2) == 0;
Reduce[eq1&&eq2&&eq3&&eq4&&eq5&&eq6&&eq7&&eq8&&eq9&&eq10,{θ1, θ2, u1, u2, v1, v2}]
This doesn't work. I also tried to simplify the system by using the first two equations and expressing the complex variables in polar representation, and solving the resulting "simpler" system of eight equations.
u1 = r1 Exp[I λ1];
u2 = r2 Exp[I λ2];
v1 = r1 Exp[I μ1];
v2 = r2 Exp[I μ2];
Again, it doesn't work (takes forever or gives me error).
Is there a better or just correct way to solve this system? Should I use Solve
instead?
Is it possible to include the conditions on the parameters (i.e., real and positive) in Reduce
/Solve
? Is it a problem that two of the unknown variables are real and the other four are complex?
Or is it too computationally difficult for Mathematica? What if I try to use some numerical values for the parameters, could that make things easier?
Thank you for your help.