Csdrhrt

I want to perform a linear stability analysis of the following system of PDEs to determine if oscillations arise and - in case they arise - the conditions for them to arise:

begin{align}
frac{partial c_1}{partial t} & = D_1 frac{partial^2 c_1}{partial x^2} + a_1 frac{partial c_1}{partial x} + f_1(c_1, c_2) \ frac{partial c_2}{partial t} & = D_2 frac{partial^2 c_1}{partial x^2} + a_2 frac{partial c_1}{partial x} + f_2(c_1, c_2) \ frac{partial c_1}{partial x} Big |_{x = 0} & = A + a_1 frac{partial c_1}{partial x} Big |_{x = 0} + f_1(c_1, c_2) Big |_{x = 0}
end{align}

Hereby, $D_1$, $D_2$, $a_1$, $a_2$ are constants and $f_1(c_1, c_2)$ and $f_2(c_1, c_2)$ both are nonlinear functions.

My idea:

We make the following ansatz for a stationary state solution: $c_{1,2}^* = A_{1,2} e^{-kx}$

It holds

begin{align}
0 & = -k^2 D_1 c_1^* - k a_1 c_1^* + f_1(c_1^*, c_2^*) \ 0 & = - k^2 D_2 c_2^* - k a_2 c_2^* + f_2(c_1^*, c_2^*) \ - k c_1^*(0) & = A - k a_1 c_1^*(0) + f_1(c_1^*(0), c_2^*(0)).
end{align}

This must hold in particular for $x = 0$ which gives

begin{align}
0 & = -k^2 D_1 A_1 - k a_1 A_1 + f_1(A_1, A_2) \ 0 & = - k^2 D_2 A_2 - k a_2 A_2 + f_2(A_1, A_2) \ - k A_1 & = A - k a_1 A_1 + f_1(A_1, A_2).
end{align}

Introducing $y_{1,2} = c_{1,2} - c_{1,2}^*$ it holds

begin{align}
frac{partial y_1}{partial t} & = D_1 frac{partial^2 y_1}{partial x^2} - k^2 D_1 c_1^* + a_1 frac{partial y_1}{partial x} - k a_1 c_1^* + f_1(y_1 + c_1^*, y_2 + c_2^*) \ frac{partial y_2}{partial t} & = D_2 frac{partial^2 y_2}{partial x^2} - k^2 D_2 c_2^* + a_2 frac{partial y_2}{partial x} - k a_2 c_2^* + f_2(y_1 + c_1^*, y_2 + c_2^*) \ frac{partial y_1}{partial t} Big |_{x = 0} & = A + a_1 frac{partial y_1}{partial x} Big |_{x = 0} - k a_1 A_1 + f_1(y_1 + c_1^*, y_2 + c_2^*) Big |_{x = 0}.
end{align}

Using both a Taylor expansion of $f_1(y_1 + c_1^*, y_2 + c_2^*)$ and $f_2(y_1 + c_1^*, y_2 + c_2^*)$ around $(c_1^*, c_2^*)$ and the equations above we get

begin{align}
frac{partial y_1}{partial t} & = D_1 frac{partial^2 y_1}{partial x^2} + a_1 frac{partial y_1}{partial x} + frac{partial f_1}{partial c_1} (c_1^*, c_2^*) y_1 + frac{partial f_1}{partial c_2} (c_1^*, c_2^*) y_2 \ frac{partial y_2}{partial t} & = D_2 frac{partial^2 y_2}{partial x^2} + a_2 frac{partial y_2}{partial x} + frac{partial f_2}{partial c_2} (c_1^*, c_2^*) y_1 + frac{partial f_2}{partial c_2} (c_1^*, c_2^*) y_2 \ frac{partial y_1}{partial t} Big |_{x = 0} & = a_1 frac{partial y_1}{partial x} Big |_{x = 0} - k A_1 + frac{partial f_2}{partial c_1} (c_1^*, c_2^*) y_1 Big |_{x = 0} + frac{partial f_2}{partial c_2} (c_1^*, c_2^*) y_2 Big |_{x = 0}.
end{align}

For $y_{1,2}$ we make the ansatz $y_{1,2} = B_{1,2} e^{-kx} e^{i omega t}$ which gives

begin{align}
i omega y_1 & = k^2 D_1 y_1 - k a_1 y_1 + frac{partial f_1}{partial c_1} (c_1^*, c_2^*) y_1 + frac{partial f_1}{partial c_2} (c_1^*, c_2^*) y_2 \ i omega y_2 & = k^2 D_2 y_2 - k a_2 y_2 + frac{partial f_2}{partial c_1} (c_1^*, c_2^*) y_1 + frac{partial f_2}{partial c_2} (c_1^*, c_2^*) y_2 \ i omega y_1 Big |_{x = 0} & = - k a_1 y_1 Big |_{x = 0} - k A_1 + frac{partial f_2}{partial c_1} (c_1^*, c_2^*) y_1 Big |_{x = 0} + frac{partial f_2}{partial c_2} (c_1^*, c_2^*) y_2 Big |_{x = 0}.
end{align}

We get

begin{align}
begin{pmatrix}
i omega - k^2 D_1 + k a_1 - frac{partial f_1}{partial c_1} (c_1^*, c_2^*) && - frac{partial f_1}{partial c_2} (c_1^*, c_2^*)
\
- frac{partial f_2}{partial c_1} (c_1^*, c_2^*) && i omega - k^2 D_2 + k a_2 - frac{partial f_2}{partial c_2} (c_1^*, c_2^*)
end{pmatrix}
begin{pmatrix}
y_1
\
y_2
end{pmatrix}
=
begin{pmatrix}
0
\
0
end{pmatrix}.
end{align}

This system of linear equations has non trivial solutions if and only if the determinant of the matrix vanishes:

$Big (i omega - k^2 D_1 + k a_1 - frac{partial f_1}{partial c_1} (c_1^*, c_2^*) Big ) Big (i omega - k^2 D_2 + k a_2 - frac{partial f_2}{partial c_2} (c_1^*, c_2^*) Big ) - frac{partial f_1}{partial c_2} (c_1^*, c_2^*) frac{partial f_2}{partial c_1} (c_1^*, c_2^*) = 0$

This equation must hold in particular for $x = 0$ which gives

$(i omega - k^2 D_1 + k a_1 - frac{partial f_1}{partial c_1} (A_1, A_2)) (i omega - k^2 D_2 + k a_2 - frac{partial f_2}{partial c_2} (A_1, A_2)) - frac{partial f_1}{partial c_2} (A_1, A_2) frac{partial f_2}{partial c_1} (A_1, A_2) = 0.$

In order to determine the frequency $omega$, we need to know $k$, $A_1$ and $A_2$ which are given by the equations for $c_{1,2}^*$ above, evaluated at $x = 0$.

Does this approach work or are there conceptual errors? Is there perhaps another way to determine if oscillations occur and - in case they occur - to determine the conditions for $omega$?