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Could anybody please make a suggestion how can I implement this program which attempts to solve $2$ dimensional complex system of equations? My computer is not able to solve... After 3 hours the program is still running and no results...

G3[x1_, x2_] = {2 ((0. + 0. I) + (316.127 + 912.071 I) x1^2 Conjugate[
      x1] + (0.0660957 + 0.175946 I) x1 x2 Conjugate[
      x1] - (316.128 + 912.074 I) x1 x2 Conjugate[
      x2] + (0.00475099 + 0.0576605 I) x2^2 Conjugate[x2]), 
 2 ((0. + 0. I) + (0.00475099 + 0.0576605 I) x1^2 Conjugate[
      x1] - (316.128 + 912.074 I) x1 x2 Conjugate[
      x1] + (0.0660957 + 0.175946 I) x1 x2 Conjugate[
      x2] + (316.127 + 912.071 I) x2^2 Conjugate[x2])}

a=-0.0000324659 - 0.0406768 I;

Solve[{(r*a-I)*x1, (r*a-I)*x2} == 
   G3[x1, x2], {x1, x2, r}]
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  • $\begingroup$ You have two equations and three unknowns. Which unknown will be allowed to be a parameter (in terms of which the others get solved)? $\endgroup$ – Daniel Lichtblau Jan 27 '15 at 22:02
  • $\begingroup$ $r$ is the parameter. I need a solution with $r \in R$ and $Im(x2) \ \text{or} \ Im(x1)=0$ $\endgroup$ – user149901 Jan 27 '15 at 22:04
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I seem to get only the trivial (all zero) solution. I had to separate into explicit real and imaginary parts (of both variables and equations), and equate those latter separately. Also I rationalized since Solve will do that anyway, and be grumpy about it.

G3[x1_, x2_] = 
  Rationalize[
   Rationalize[{2 ((0. + 0. I) + (316.127 + 912.071 I) x1^2 Conjugate[
           x1] + (0.0660957 + 0.175946 I) x1 x2 Conjugate[
           x1] - (316.128 + 912.074 I) x1 x2 Conjugate[
           x2] + (0.00475099 + 0.0576605 I) x2^2 Conjugate[x2]), 
      2 ((0. + 0. I) + (0.00475099 + 0.0576605 I) x1^2 Conjugate[
           x1] - (316.128 + 912.074 I) x1 x2 Conjugate[
           x1] + (0.0660957 + 0.175946 I) x1 x2 Conjugate[
           x2] + (316.127 + 912.071 I) x2^2 Conjugate[x2])} /. {x1 -> 
       x1 + I*y1, x2 -> x2 + I*y2}], 0];

a = Rationalize[-0.0000324659 - 0.0406768 I, 0];
exprs = {(r*a - I)*(x1 + I*y1), (r*a - I)*(x2 + I*y2)} - G3[x1, x2];
e2 = Flatten[ComplexExpand[{Re[#], Im[#]}]] & /@ exprs;

Solve[e2 == 0, {x1, x2, y1, y2}]

(* Out[342]= {{x1 -> 0, x2 -> 0, y1 -> 0, y2 -> 0}} *)

Here is e2, if it matters.

In[353]:= e2

(* Out[353]= {-((324659 r x1)/10000000000) - (316127 x1^3)/500 - (
  660957 x1^2 x2)/5000000 + (79032 x1 x2^2)/125 - (475099 x2^3)/
  50000000 + y1 + (25423 r y1)/625000 + (912071 x1^2 y1)/500 - (
  456037 x2^2 y1)/250 - (316127 x1 y1^2)/500 - (660957 x2 y1^2)/
  5000000 + (912071 y1^3)/500 + (87973 x1^2 y2)/250000 + (
  115321 x2^2 y2)/1000000 + (87973 y1^2 y2)/250000 + (79032 x1 y2^2)/
  125 - (475099 x2 y2^2)/50000000 - (456037 y1 y2^2)/250 + (
  115321 y2^3)/1000000, -x1 - (25423 r x1)/625000 - (912071 x1^3)/
  500 - (87973 x1^2 x2)/250000 + (456037 x1 x2^2)/250 - (115321 x2^3)/
  1000000 - (324659 r y1)/10000000000 - (316127 x1^2 y1)/500 + (
  79032 x2^2 y1)/125 - (912071 x1 y1^2)/500 - (87973 x2 y1^2)/
  250000 - (316127 y1^3)/500 - (660957 x1^2 y2)/5000000 - (
  475099 x2^2 y2)/50000000 - (660957 y1^2 y2)/5000000 + (
  456037 x1 y2^2)/250 - (115321 x2 y2^2)/1000000 + (79032 y1 y2^2)/
  125 - (475099 y2^3)/50000000, -((475099 x1^3)/50000000) - (
  324659 r x2)/10000000000 + (79032 x1^2 x2)/125 - (660957 x1 x2^2)/
  5000000 - (316127 x2^3)/500 + (115321 x1^2 y1)/1000000 + (
  87973 x2^2 y1)/250000 - (475099 x1 y1^2)/50000000 + (79032 x2 y1^2)/
  125 + (115321 y1^3)/1000000 + y2 + (25423 r y2)/625000 - (
  456037 x1^2 y2)/250 + (912071 x2^2 y2)/500 - (456037 y1^2 y2)/
  250 - (660957 x1 y2^2)/5000000 - (316127 x2 y2^2)/500 + (
  87973 y1 y2^2)/250000 + (912071 y2^3)/
  500, -((115321 x1^3)/1000000) - x2 - (25423 r x2)/625000 + (
  456037 x1^2 x2)/250 - (87973 x1 x2^2)/250000 - (912071 x2^3)/500 - (
  475099 x1^2 y1)/50000000 - (660957 x2^2 y1)/5000000 - (
  115321 x1 y1^2)/1000000 + (456037 x2 y1^2)/250 - (475099 y1^3)/
  50000000 - (324659 r y2)/10000000000 + (79032 x1^2 y2)/125 - (
  316127 x2^2 y2)/500 + (79032 y1^2 y2)/125 - (87973 x1 y2^2)/
  250000 - (912071 x2 y2^2)/500 - (660957 y1 y2^2)/5000000 - (
  316127 y2^3)/500} *)
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  • $\begingroup$ Thank you for the answer. So for any $r$ the only possible solution is the trivial one? This seems to be strange to be honest... $\endgroup$ – user149901 Jan 27 '15 at 22:21
  • $\begingroup$ I agree it is unexpected. Actually the reason I showed the modified system, e2, is in case I made a mistake that someone might catch. $\endgroup$ – Daniel Lichtblau Jan 27 '15 at 22:29
  • $\begingroup$ It is interesting when I add the parameter $r$ in the list $\{x1,x2,y1,y2\}$ of the function $Solve$ the program stucks again. $\endgroup$ – user149901 Jan 27 '15 at 22:57

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