Solving a dynamical system - separation

I wish to solve the following dynamical system

$$\frac{dA}{dt}=\left(1+i\alpha\right)A-\left(1+i\beta\right)|A|^{2}A$$

What I did was substitute $A=R(t)e^{i\theta(t)}$, thus getting to

$$\frac{dR\left(t\right)}{dt}+iR\left(t\right)\frac{d\theta\left(t\right)}{dt}=\left(1+i\alpha\right)R\left(t\right)-\left(1+i\beta\right)R^{3}\left(t\right)$$

I was wondering how and if I can make it into two different independent equations, in order to find $\frac{dR}{dt} = 0 $ solutions. Meaning, is there anyway to transform this equation into the following form? $$\frac{d}{dt}\begin{pmatrix}R\left(t\right)\\ \theta(t) \end{pmatrix}=\begin{pmatrix}...\\ ... \end{pmatrix}$$ Any advice will be greatly appreciated.

Both $R$ and $θ$ are supposed to be real. So the real part of the equation and the imaginary part have to hold independently, so for $R>0$ $$ R'=R-R^3,\\ θ'=α-βR^2. $$