Mathieu equation for case of negative instability

+ 3 like - 0 dislike
312 views

Let's assume Mathieu equation:
$$\tag 1 \frac{d^{2}y(t)}{dt^{2}} + \omega^{2}(t)y(t) = 0, \quad \omega^{2}(t) = A(t) - 2q(t)cos(2t)$$

Here $A(t), q(t)$ are slowly decreased with time functions (i.e., $|\dot{A}(t)| <A(t), |\dot{q}(t)| < q(t)$), and in initial moment of time $A(t_{0}) < 2q(t_{0})$. Eq. $(1)$ describes, for example, magnetic field evolution in presence of axion field and preheating in an expanding Universe.

I look for exponentially growing solutions of Eq. $(1)$; more precisely, I need to determine the Floquet exponent $\mu$, $y(t) \sim e^{\int \limits_{}^{t} \mu (t)dt}$. If I temporary assume that $q(t), A(t)$ are constans, then $(1)$ is reduced to Mathieu equation for case $A - 2q < 0$, called Mathieu equation with negative instability. Unfortunately, I haven't found an expression for Floquet exponent for this case.

Can someone give the reference in which this case is treated? I note that it is different from case which is described in famous article by Kofman, Linde and Starobinsky, where always $A - 2q > 0$.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.