Complex nonlinear equation solution

Question

I am trying to find an analytic/numeric solution for a set of equations involving functions, summation, and complex quantities.

Definitions:

step = 1;
num = 15;
rho = Range[step , num, step];
theta = Most[Range[0, 2 Pi, 2 Pi/10]];
area[l_] := \[Pi] (2 step l - step^2) 1/Length[theta]

(I choose functions below arbitarily)

K11[a_, b_] := I a + a b + a Cos[b]
K12[a_, b_] := Conjugate[K11[a, b]]
K21[a_, b_] := (I a + a Sin[b])^2 + a Cos[(2 b)/3]
K22[a_, b_] := Conjugate[K21[a, b]]
C1[a_, b_] := K11[a, b] - K12[a, b]

Equations:

eq1 = -a1 + I Sum[  Conjugate[ (r1[\[Rho], \[Theta]] K11[\[Rho], \[Theta]] + r2[\[Rho], \[Theta]] K12[\[Rho], \[Theta]])] area[\[Rho]], {\[Rho], rho}, {\[Theta], theta}];
eq2 = -a2 +  I Sum[  Conjugate[ (r1[\[Rho], \[Theta]] K21[\[Rho], \[Theta]] + r2[\[Rho], \[Theta]] K22[\[Rho], \[Theta]])] area[\[Rho]], {\[Rho], rho}, {\[Theta], theta}];
eq3 = -4 Im[r1[\[Rho], \[Theta]] (K11[\[Rho], \[Theta]] a1 + K21[\[Rho], \[Theta]] a2)] -  C1[\[Rho], \[Theta]] (n1[\[Rho], \[Theta]] + 1) + C2 (1 - n1[\[Rho], \[Theta]]) (*C2 is a parameter*);
eq4 = -4 Im[ r2[\[Rho], \[Theta]] (K12[\[Rho], \[Theta]] a1 + K22[\[Rho], \[Theta]] a2)] - C1[\[Rho], \[Theta]] (n2[\[Rho], \[Theta]] + 1) + 0 (1 - n2[\[Rho], \[Theta]]) ;
eq5 = -1 r1[\[Rho], \[Theta]] + I n1[\[Rho], \[Theta]] Conjugate [(K11[\[Rho], \[Theta]] a1 +  K21[\[Rho], \[Theta]] a2)] ;
eq6 = -1 r2[\[Rho], \[Theta]] + I n2[\[Rho], \[Theta]] Conjugate [(K12[\[Rho], \[Theta]] a1 + K22[\[Rho], \[Theta]] a2)] ;

Condition and variables:

eqn={eq1==0,eq2==0,eq3==0,eq4==0,eq5==0,eq6==0}

Var= {a1,a2,r1,r2,n1,n2}

Problem:

Here, I want to plot a graph between (a1^2 vs C2). For that, I need to get analytic or numeric solution of the variables most of which are the functions of position and angle as given above. I couldn't deal using elimination method and solve feature.

Any help is appreciated.Thank you

Answer 1

I didn't save the working version I had, but perhaps this will help.

step=1;
num=15;
rho=Range[step , num, step];
theta=Most[Range[0,2 Pi,2 Pi/10]];
area[l_]:=Pi(2 step l-step^2)/Length[theta];
K11[a_,b_]:=I a+a b+a Cos[b];
K12[a_,b_]:=Conjugate[K11[a,b]];
K21[a_,b_]:=(I a+a Sin[b])^2+a Cos[2 b/3];
K22[a_,b_]:=Conjugate[K21[a,b]];
C1[a_,b_]:=K11[a, b]-K12[a,b];
eq1=Simplify[-a1+I Sum[Conjugate[r1 K11[ρ,θ]+r2 K12[ρ,θ]]area[ρ],{ρ,rho},{θ,theta}]];
eq2=Simplify[-a2+I Sum[Conjugate[r1 K21[ρ,θ]+r2 K22[ρ,θ]]area[ρ],{ρ,rho},{θ,theta}]];
eq3= -4 Im[r1(K11[ρ,θ]a1+K21[ρ,θ]a2)]-C1[ρ,θ](n1+1)+C2(1-n1);
eq4= -4 Im[r2(K12[ρ,θ]a1+K22[ρ,θ]a2)]-C1[ρ,θ](n2+1)+0 (1-n2);
eq5= -1 r1+I n1 Conjugate[K11[ρ,θ]a1+K21[ρ,θ]a2];
eq6= -1 r2+I n2 Conjugate[K12[ρ,θ]a1+K22[ρ,θ]a2];
Var= {a1,a2,r1,r2,n1,n2};
Reduce[eq1==0&&eq2==0,Var]

You can then use Simplify and N on the output from Reduce to make the result much smaller and possibly see what you are looking for in that. The Simplify on eq1 and eq2 make those dramatically simpler because of your summation over regular angles.

I'm not certain if your complete system will solve or not, I didn't wait for that because as you mentioned, you were just making up arbitrary equations for each Knn.

I'm not certain what it is in the way you wrote the code and was provoking Reduce to say that it could not solve the system. I did see that warning once, but I didn't save the code at that point to try to track down the reason for that.

I'm not certain whether your r1,r2 are real radius or not. If they are and you tell Reduce that Element[r1|r2,Reals] then the code speeds up dramatically. Anything that you assume, but you don't tell Mathematica, then it will assume the worst case with everything as unknown complex, possibly zero, possibly infinite, possibly etc. Are ρ and θ real? If so then telling Mathematica that might make your problem much easier to solve.

So I'm not sure where to go, if anywhere, with this now. I can't even seem to get it to solve eq3 for C2. Can you at least reproduce what I've shown here? Can you try to track down what the source of the "Reduce can't solve" warning?