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.
Abs[]
, you can just putAbs[]^2
and that should work. $\endgroup$Abs[]
, round brackets are use for grouping such as(a+b)*c
$\endgroup$