# Solve a system of complex equations with parameters

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?

This is an extended comment. Use

eqsx = (TrigToExp[Subtract@@@{eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10}]/.{
θ1->Log[t1],θ2->Log[t2],u1->r1+I*i1,u2->r2+I*i2,v1->r3+I*i3,v2->r4+I*i4}
)//Map[ReIm,#]&//Flatten//ComplexExpand//DeleteCases[#,0]&//Map[Denominator[Together[#]]&,#]*#&//Expand;


to rewrite the equations in polynomial form in the new real unknown t1, t2, r1, i1, ..., r4, i4. (Not polynomial in the parameters because of some square root, but we are going to fix the parameters below.)

Example: Let us fix the parameters and see what we get:

Block[{k=1/3,m=2,a1=3,a2=5},
(* {} *)


Similar with Reduce, it returns False. Of course this is just an example, but it came out like this for every other choice of parameters I made.

The system of equations in this case is

Block[{k=1/3,m=2,a1=3,a2=5},
eqsx]
(*
{i1^2-i3^2+r1^2-r3^2,
i2^2-i4^2+r2^2-r4^2,
r2 t1+r4 t1+r1 t2+r3 t2+r1 t1^2 t2-r3 t1^2 t2+r2 t1 t2^2-r4 t1 t2^2,
-i2 t1+i4 t1+i1 t2-i3 t2+i1 t1^2 t2+i3 t1^2 t2-i2 t1 t2^2-i4 t1 t2^2,
-r2 t1-r4 t1+r1 t2+r3 t2-r1 t1^2 t2+r3 t1^2 t2+r2 t1 t2^2-r4 t1 t2^2,
i2 t1-i4 t1-i1 t2+i3 t2+i1 t1^2 t2+i3 t1^2 t2-i2 t1 t2^2-i4 t1 t2^2,
1/9-(2 Sqrt[433] t1^2)/9+(722 i2^2 t1^2)/9-160 i2 i4 t1^2+(722 i4^2 t1^2)/9+(722 r2^2 t1^2)/9+160 r2 r4 t1^2+(722 r4^2 t1^2)/9+(433 t1^4)/9,
0,
1/9-(2 Sqrt[721] t2^2)/9+(434 i1^2 t2^2)/9-96 i1 i3 t2^2+(434 i3^2 t2^2)/9+(434 r1^2 t2^2)/9+96 r1 r3 t2^2+(434 r3^2 t2^2)/9+(721 t2^4)/9,
0,
(r2 t1)/9-(r4 t1)/9+(r1 t2)/9-(r3 t2)/9+433/9 r1 t1^2 t2+433/9 r3 t1^2 t2+721/9 r2 t1 t2^2+721/9 r4 t1 t2^2,
-((i2 t1)/9)-(i4 t1)/9+(i1 t2)/9+(i3 t2)/9+433/9 i1 t1^2 t2-433/9 i3 t1^2 t2-721/9 i2 t1 t2^2+721/9 i4 t1 t2^2,
-(1/9)-80 i2^2 t1^2+1444/9 i2 i4 t1^2-80 i4^2 t1^2+80 r2^2 t1^2+1444/9 r2 r4 t1^2+80 r4^2 t1^2+(433 t1^4)/9,
-40 i2 r2+(361 i4 r2)/9-(361 i2 r4)/9+40 i4 r4,
-((r2 t1)/9)+(r4 t1)/9-(r1 t2)/9+(r3 t2)/9+433/9 r1 t1^2 t2+433/9 r3 t1^2 t2+721/9 r2 t1 t2^2+721/9 r4 t1 t2^2,
(i2 t1)/9+(i4 t1)/9+(i1 t2)/9+(i3 t2)/9-433/9 i1 t1^2 t2+433/9 i3 t1^2 t2-721/9 i2 t1 t2^2+721/9 i4 t1 t2^2,
-(145/9)-48 i1^2 t2^2+868/9 i1 i3 t2^2-48 i3^2 t2^2+48 r1^2 t2^2+868/9 r1 r3 t2^2+48 r3^2 t2^2+(577 t2^4)/9,
-24 i1 r1+(217 i3 r1)/9-(217 i1 r3)/9+24 i3 r3}
*)


One could make the equations polynomial also in the parameters and use elimination to see what conditions the parameters have to satisfy to get a solution, or if in fact there is no solution for any value of the parameters.