5
$\begingroup$

Hi,

Consider a system of linear differential equations $$ {d f \over dz} = A(z) f, $$ where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, Floquet theorem applies.

I am curious to know if there exists a generalization of Floquet theorem to the case, where $z \in \mathbb{C}$ and $A(z)$ is a doubly-periodic elliptic function of $z$.

Thanks,

Victor

$\endgroup$
  • 1
    $\begingroup$ The Lamé equation dlmf.nist.gov/29 seems to be the canonical example analogous to the Mathieu equation. $\endgroup$ – J. M. isn't a mathematician Oct 12 '10 at 6:31
  • $\begingroup$ I don't know if it is relevant to your purpose, but the Wiki page says that the Floquet theorem generalizes to the Bloch theorem in higher dimensions. $\endgroup$ – timur Nov 1 '11 at 0:17
3
$\begingroup$

After some searching around, it looks to me that it was Picard who thought of generalizing Floquet theory to linear ODEs with doubly-periodic coefficients (though according to this paper, Floquet himself worked a bit on this generalization, deriving his theory for ODEs with singly periodic coefficients from the more general case).

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Hi JM -- Thanks for the paper. Very useful. I definitely should improve my google-searching skills, as I could not find anything sensible. Anyway, I am sort of amazed that such a natural question is not discussed in detail in any but a very few papers. When this question first came up in my (physics) research, I was hoping to quickly find a book or review on it. All the best $\endgroup$ – Victor Galitski Oct 12 '10 at 13:49
2
$\begingroup$

Another, far reaching, aspect of Floquet theory in differential equations $$a_0(z)f^{(n)}(z)+\cdots+a_nf(z)=0$$ with holomorphic coefficients is Fuchs theory of monodromy, where the leading coefficient $a_0$ has a zeros at $z_0$. You cannot solve a Cauchy problem at $z_0$, but you can solve it in a pointed disk $D\setminus z_0$. When you follow a circle around $z_0$, the coefficients look periodic.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.