I am studying an exercise paper and I am encountering the following problem. With $u_i(t)=u_i$ ($i=1,2$) we have the system:
\begin{equation} \begin{split} i \dot{u_1} & = -(\omega_0 + i \beta) u_1 + \chi |u_1|^2 u_1 + \gamma u_2 \\ i \dot{u_2} & = -(\omega_0 - i \beta) u_2 + \chi |u_2|^2 u_2 + \gamma u_1. \end{split} \end{equation}
He wants to identify the two conserved quantities and also write the system in a form of $S_i$'s. The Stokes parameters are the following:
\begin{equation} \begin{split} & S_0 = |u_1|^2 + |u_2|^2,\quad S_3 = |u_1|^2 - |u_2|^2,\\ & S_1 = u_1 u_2^* + u_1^* u_2,\quad S_2 = i(u_1 u_2^* - u_1^* u_2). \end{split} \end{equation}
So how do I proceed in solving the system or finding the conserved quantities? Any ideas would be helpful, because I can't find much stuff about Stokes parameters online also.
*Hint: for $\beta=0$, power $N$ and energy $E$ are conserved: \begin{equation} \begin{split} N = |u_1|^2 + |u_2|^2 \\ E = N + \frac{\chi}{2} \left( |u_1|^4 + |u_2|^4 \right). \end{split} \end{equation}
Thanks in advance
**Bonus question: is there any particular way to derive the equations for the Stokes parameters?