I recently started studying about reaction-diffusion system and Hopf bifurcation theory. I realize that Fourier transform/series is a quite useful tool here but it's been many years since I got engaged with such stuff. Hence, I apologize in advance if the following questions are trivial.
QUESTIONS:
- How do we know that $\lambda$ denotes the eigenvalue of $(3),(4)$? I understand that the right-hand side of $(5),(6)$ follows by applying Fourier transform in $(3),(4)$ and I can also verify that the functions $u,v$ as defined solve $(5),(6)$. However I do not get how do we know that $\lambda$ stands for the eigenvalue. Is it by definition?
- It seems to me that there is a step I'm missing, in which Fourier series are introduced. What is that step? Why do we use $u(x,t)=e^{ikx+\lambda t}\hat{u}, \; v(x,t)=e^{ikx+\lambda t}\hat{v}$ in order to solve $(3),(4)$? Is this a standard technique?
I would really appreciate if someone could shed some light on this. I can feel I'm missing something really basic but so far, I haven't found any good reference to understand this concept. Many many thanks in advance!
When computing the different parts of the equations with the ansatz $e^{ikx+\lambda t}$, we get a multiple of $\lambda$ in the lefthand side when computing the time derivatives. This appears in equations (5) and (6) and we have something of the form $\lambda w = Aw$, so we have an eigenvalue equation.
Here Fourier series are not really used as such. We simply notice that the functions $e^{ikx}$ are eigenfunctions of $\partial^2_x$ so we can use them to find eigenfunctions of the linearization of the entire righthand side. This is the same thing as trying to solve $\dot{x} = Ax$ by writing $x$ as a linear combination of eigenvectors of $A$, which diagonalizes the system.