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I am reading through the paper "Dynamics of Membranes Driven by Actin Polymerization" by Nir S. Gov and Ajay Gopinathan. In it a set of coupled differential equations for a mathematical model is given (I have slightly modified the notation for clarity):

$$\frac{\partial n}{\partial t}=D\nabla^2n-\Lambda\kappa H\nabla^4h+\nabla\cdot f_n$$ $$\frac{\partial h}{\partial t}=-\int\text dr'\lambda(r-r')\kappa\nabla^4h(r')+An$$

where $n$ and $h$ are both functions of space and time, $D$, $\Lambda$, $\kappa$, $H$, and $A$ are all constant parameters, $f_n$ is a thermal noise term, and $\lambda$ is an "interaction kernel" whose Fourier transform in q-space is $1/4\eta q$ such that the Fourier transform of the integral ends up being $\kappa q^3/4\eta\cdot\bar h\equiv\omega_q\bar h$

The paper then says they solve this by "first Fourier transforming both equations, and using solutions of the form: $e^{-i(\omega t+q\cdot r)}$. This gives the following system of equations in matrix form

$$ \begin{pmatrix}-i\omega+\omega_D & -Bq^4 \\ -A & -i\omega+\omega_q\end{pmatrix}\begin{pmatrix}n \\ h\end{pmatrix}=\begin{pmatrix}-if_n q \\ 0\end{pmatrix} $$

where $B\equiv-\Lambda\kappa H$ and $\omega_D=Dq^2$

(as an aside, I am confused by what they mean by "using solutions of the form: $e^{-i(\omega t+q\cdot r)}$". Are they just saying the form of the Fourier transform they used?).

Then they go on to say

The dispersion equation of $n$ and $h$, respectively, is given by equating the determinant of the matrix to zero. The two solutions to the eigenvalue equation for $\omega$ are given by...

and then they give the solutions $\omega_n$ and $\omega_h$ for $\omega$ that make the determinant zero and claim that the time-dependent solutions $(n,h)(t)$ are proportional to $\exp(-i\omega_{n,\,h}t)$

I see how the matrix equation resembles an eigenvalue problem with eigenvalues of $i\omega$, but I am not familiar with how the values of $\omega$ that make the determinant $0$ determines the time evolution of the solutions in the way given. Additionally, I have only ever seen solutions being obtained by finding the eigenvalues of the matrix that describes the system, not interpreting the matrix equation as an eigenvalue problem itself.

So, how does solving for when the determinant is equal to zero determine the dispersion equations, and how does that lead to the proposed behavior of the time-dependence?

BioPhysicist
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  • If you ignore the inhomogeneous $f_n$ term and look for a homogenous solution per Fourier component, you see that you have to find a relation between $q$ and $\omega$. This relationship is the dispersion relation. It is basically the Green's function of the equations. – user619894 Dec 06 '22 at 08:48
  • You're looking for solutions with a wave form. Solving the algebraic equation in $\omega$ and $q$ you get the relationship between these two parameters, for the wave to be a solution. Two little things: 1. assuming that form of the solution, they're doing a sort of Fourier transform in time and space; 2. with the forced problem, I think you should perform the Fourier transform of the known $f$ as well, to determine the actual expansion of $n$ and $h$, solving the determinate linear system, without setting the determinant to zero. – basics Dec 08 '22 at 19:26
  • Setting the determinant to zero, you find all the sort of natural undamped resonant ways the free system may evolve – basics Dec 08 '22 at 19:28
  • @basics $f_n$ is a thermal noise term, so all that is given is $\langle f_n(r,t)f_n(r',t')\rangle=2n_0\lambda k_BT\delta(r-r'),\delta(t-t')$ – BioPhysicist Dec 08 '22 at 19:31
  • in order to get an idea about the method, try to approach with the same method the simple hyperbolic problem, $\partial_t u + U \partial_x u = f$, with constant convective velocity $U$ – basics Dec 08 '22 at 19:34
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    I'm from the smartphone tonight. I'll come back to your problem tomorrow from a pc. A very interesting one form me, especially the one about the fluid-membrane interaction on Physics SE. – basics Dec 08 '22 at 19:36

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