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I have the following two equations with complex $\alpha, \beta$: $$0 = -i \omega_m \beta - ig|\alpha|^2 - \frac{\gamma}{2}\beta$$ $$0 = \alpha(i\Delta - \frac{\kappa}{2}) - i\epsilon - ig\alpha(\beta + \beta^*)$$ The other parameters $g, \omega_m, \Delta, \kappa, \gamma, \epsilon$ are real.

To solve this system for $\alpha$, I solved the first equation for $\beta$ and got $$\beta = \frac{g|\alpha|^2}{\frac{i\gamma}{2} - \omega_m}$$

I then inserted this into the second line and obtained $$\alpha\left(i\Delta - \frac{\kappa}{2} + \frac{2ig^2\omega_m|\alpha|^2}{\frac{\gamma^2}{4} + \omega_m}\right) = i\epsilon$$However, in this equation, both $\alpha$ and $|\alpha|^2$ appear, and I don't know how to solve this.

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The last equation is of the form $\,\alpha\left(u + i\left(v + w\, |\alpha|^2\right)\right) = i \epsilon\,$ with $\,u,v,w,\epsilon \in \mathbb R\,$.

Taking squared magnitudes on both sides:

$$ |\alpha|^2\left(u^2 + \left(v+ w\,|\alpha|^2\right)^2\right) = \epsilon^2 $$

This is a cubic equation with real coefficients in $\,|\alpha|^2\,$ which can be solved either algebraically or numerically. Substituting the positive real solution(s) $\,|\alpha|^2\,$ in the first equation of the system gives $\,\beta\,$, then substituting the respective $\,\beta\,$ in the second equation gives $\,\alpha\,$.

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  • $\begingroup$ Thanks. Solving this with Mathematica yields some messy and lengthy expressions. Does there even exist an analytic expression? $\endgroup$ Commented May 8, 2022 at 9:19
  • $\begingroup$ @LionCereals The cubic is solvable by radicals, see e.g. here for example. There are also trigonometric and hyperbolic forms, depending on the nature of the roots. That said, the calculations can be laborious. $\endgroup$
    – dxiv
    Commented May 8, 2022 at 16:50

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