2
$\begingroup$

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 $\omega$?

$\endgroup$

0

You must log in to answer this question.