System of nonlinear complex-valued equations using Solve

Question

I'm trying to figure out the proper syntax to use in Mathematica to solve the following system of $6$ nonlinear complex-valued equations in $6$ unknowns $a,b,c,d,e,f$, where $a,d,f \in \mathbb{R}$ and $b,c,e \in \mathbb{C}$: $$\begin{cases} a^2-|b|^2-|c|^2=1, \\ |b|^2-d^2-|e|^2=-1, \\ |c|^2-|e|^2-f^2=-1, \\ ab-bd-c\bar{e}=0, \\ \bar{b}c-de-ef=0, \\ \bar{c}a-\bar{e}\bar{b}-f\bar{c}=0, \end{cases}$$ where $a^2 \geq 1$. I wanted the solution to have a mix of complex and real values. After some brute force I actually found a particular solution: $$a=d= \pm{\sqrt{2}}, b= \pm{i}, c=e= 0,f=\pm{1}.$$ I was wondering if Mathematica using Solve could find other solutions. Here is the code that I used:

 Solve[a^2 - Abs[b]^2 - Abs[c]^2 == 
   1 && Abs[b]^2 - d^2 - Abs[e]^2 ==-1 && Abs[c]^2 - Abs[e]^2 - f^2 == -1 && 
 a b - b d - c e\[Conjugate] == 0 && b\[Conjugate] c - d e - e f == 0 && 
  c\[Conjugate] a - e\[Conjugate] b\[Conjugate] - f c\[Conjugate] == 
   0, {a, b, c, d, e, f}, ?]

My apologies, I still don't have the hang on how to enter Mathematica code here, even after keeping four spaces. My issue is what to put in "?" for the system to be solved.

Answer 1

I would tackle this as follows

eqn = a^2 - Abs[b]^2 - Abs[c]^2 == 1 && 
  Abs[b]^2 - d^2 - Abs[e]^2 == -1 && Abs[c]^2 - Abs[e]^2 - f^2 == -1 &&
   a b - b d - c e\[Conjugate] == 0 && 
  b\[Conjugate] c - d e - e f == 0 && 
  c\[Conjugate] a - e\[Conjugate] b\[Conjugate] - f c\[Conjugate] == 0
(* a^2 - Abs[b]^2 - Abs[c]^2 == 
  1 && -d^2 + Abs[b]^2 - Abs[e]^2 == -1 && -f^2 + Abs[c]^2 - 
   Abs[e]^2 == -1 && 
 a b - b d - c Conjugate[e] == 0 && -d e - e f + c Conjugate[b] == 0 &&
  a Conjugate[c] - f Conjugate[c] - Conjugate[b] Conjugate[e] == 0 *)

Define separate variables for the real and imaginary parts of those that are complex

subst = Thread[
  Flatten[{Re /@ {b, c, e}, Im /@ {b, c, e}}] -> {rb, rc, re, ib, ic, 
    ie}]
(* {Re[b] -> rb, Re[c] -> rc, Re[e] -> re, Im[b] -> ib, Im[c] -> ic, 
 Im[e] -> ie} *)

Rewrite the equations as real expressions using these

eqn2 = ComplexExpand[(Re /@ eqn) && (Im /@ eqn), {b, c, e}, 
   TargetFunctions -> {Re, Im}] /. subst
(* a^2 - ib^2 - ic^2 - rb^2 - rc^2 == 
  1 && -d^2 + ib^2 - ie^2 + rb^2 - re^2 == -1 && -f^2 + ic^2 - ie^2 + 
   rc^2 - re^2 == -1 && -ic ie + a rb - d rb - rc re == 0 && 
 ib ic + rb rc - d re - f re == 0 && 
 ib ie + a rc - f rc - rb re == 0 && 
 a ib - d ib + ie rc - ic re == 0 && -d ie - f ie + ic rb - ib rc == 
  0 && -a ic + f ic + ie rb + ib re == 0 *)

vars = Flatten[{a, d, f, Re /@ {b, c, e}, Im /@ {b, c, e}}] /. subst
(* {a, d, f, rb, rc, re, ib, ic, ie} *)

I now find that Solve and Reduce struggle to solve these equations. The relevant expressions would be

(*Solve[eqn2,vars,Reals]*)
(*Solve[eqn2,vars,Reals]*)

Looking for a numerical solution seems to give a dense set of solutions, suggesting that we don't really have 9 independent equations. To solve the equations 100 times from random starting points, evaluate

Table[FindRoot[eqn2, 
   Transpose[{vars, RandomReal[{-1, 1}, Length[vars]]}], 
   MaxIterations -> 1000], {100}] // Chop

EDIT

There is a trap for the unwary here. If we don't specify the solution domain, Complex is assumed by default and we get incorrect answers. For example

NSolve[eqn2, vars]

returns 512 solutions, but most of them are complex.

NSolve[eqn2, vars, Reals]

finds only two solutions, for some reason. In comparison, FindRoot finds large numbers of apparently valid solutions.