I have these non-linear coupled differential equations: $$ \frac{\mathrm{d}A_1}{\mathrm{d}z} = C_1 A_1^* A_2 \\ \frac{\mathrm{d}A_2}{\mathrm{d}z} = C_2 A_1^2 $$ Where $C_1$ and $C_2$ are constants and $A_1$ and $A_2$ vary in $z$.
Basically, what I'd like to do is, write those two equations in the form (or something similar): $$ \frac{\mathrm{d}}{\mathrm{d}z} \begin{bmatrix} A_1 \\ A_2 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} A_1 \\ A_2 \end{bmatrix} $$
I'm after finding the eigenvectors/eigenvalues of the system. Any advice on how to proceed is appreciated.
My current idea is to some how linearise the system using a Taylor Series approach; I believe these types of systems are normally linearised about a steady-state condition; however, I believe the solution to this is sinusoidal, and so I'm not sure about what point to linearise. I believe I might be able to somehow linearise about a sinusoid using Fourier theory, though not sure.
Thank you.
If you want to determine eigenmodes you must employ some kind of linearisation, that is the most basic idea is to choose some point $(A_{10},A_{20})$ and then determine the linear system of equations that enables you to determine eigenmodes of your system, but limited only to the short interval around the chosen point.
$$ \frac{\mathrm{d}A_1}{\mathrm{d}t} = C_1 A_1^* A_2 \\ \frac{\mathrm{d}A_2}{\mathrm{d}t} = C_2 A_1^2 $$
So we choose some point $(A_{10},A_{20})$ and linearise:
$$ \frac{dA_1}{dt}=f_1(A_1,A_2)$$ $$ \frac{d\Delta A_1}{dt}=\frac{df_1}{dA_1}|_{A_{10},A_{20}}\Delta A_1+\frac{df_1}{dA_2}|_{A_{10},A_{20}}\Delta A_2$$
$$ \frac{dA_2}{dt}=f_2(A_1,A_2)$$ $$ \frac{d\Delta A_2}{dt}=\frac{df_2}{dA_1}|_{A_{10},A_{20}}\Delta A_1+\frac{df_2}{dA_2}|_{A_{10},A_{20}}\Delta A_2$$
This enables us to determine:
$$ \frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} A_1 \\ A_2 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} A_1 \\ A_2 \end{bmatrix} $$
with $a,b,c,d$ being the linearisation coefficients calculated from the upper partial derivations.
$$ \frac{d\Delta A_1}{dt}=C_1A_{20}\Delta A_1+C_1 A_{10}\Delta A_2$$
$$ \frac{d\Delta A_2}{dt}=2C_2 A_{10}\Delta A_1+0\Delta A_2$$
$$ \frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} \Delta A_1 \\ \Delta A_2 \end{bmatrix} = \begin{bmatrix} C_1 A_{20} & C_1A_{10} \\ 2C_2 A_{10} & 0 \end{bmatrix} \begin{bmatrix} \Delta A_1 \\ \Delta A_2 \end{bmatrix} $$
From this point on you can determine eigenvectors and eigenmodes of the system, although it would be helpful we had more info on the system, is it some kind of physical system or control system, or something like that...