Timeline for How much can one say about this differential equation?
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12 events
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| Sep 5, 2014 at 18:23 | comment | added | Christian Remling | @RobertBryant: One more small remark: periodic (and anti-periodic) solutions are of course very special. You get these precisely at the band edges. | |
| Sep 5, 2014 at 18:21 | comment | added | Christian Remling | @RobertBryant: Of course you can't work with real valued solutions (and you don't) if you are inside a band (equivalently, if $|\mu+\mu^{-1}|<2$). By the way, I've read your interesting discussion with Igor below. However, for me, Floquet "theory" has always had the complete summary: bring $T(p)$ into Jordan normal form ($T$ = transfer matrix over one period). | |
| Sep 5, 2014 at 10:19 | comment | added | Robert Bryant | Actually, if you want real solutions in the 'exponential-periodic' form, you have to go to period $4\pi$. There are two independent solutions $y_1(x) = u_1(x)\mu^x$ and $y_2(x) = u_2(x) \mu^{-x}$, where $u_1$ and $u_2$ satisfy $u_i(x+2\pi) = -u_i(x)$ and where $\mu\approx 1.5557$. (Additionally, you can take $u_2(x)=u_1(-x)$.) No nonzero linear combination of these two solutions is actually periodic for any period. | |
| S Sep 4, 2014 at 19:25 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
replaced tag 'de'; editing for readability
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| Sep 4, 2014 at 19:24 | review | Suggested edits | |||
| S Sep 4, 2014 at 19:25 | |||||
| Sep 4, 2014 at 18:33 | comment | added | Alex R. | @ChristianRemling: You're right. I guess I'm taking this question literally as whether the whole solution is periodic or not (clearly from the picture it's not). The solution is of the form: $$c_1 S(0,2,x/2)+c_2C(0,2,x/2)$$ where the two functions are the Mathieu functions. As far as I know, the "characteristic values of the Mathieu equation" determine for which coefficients $a,b$ in $S(a,b,x)$ and $C(a,b,x)$ you have bona-fide periodic solutions. | |
| Sep 4, 2014 at 18:24 | comment | added | Christian Remling | @AlexR.: It does not imply the solution is oscillating, this follows precisely if you are in a band. However, it completely clarifies the general asymptotics (plane wave or real exponential, modulated by a periodic amplitude). | |
| Sep 4, 2014 at 18:23 | answer | added | Robert Bryant | timeline score: 16 | |
| Sep 4, 2014 at 18:23 | answer | added | Michael Renardy | timeline score: 11 | |
| Sep 4, 2014 at 18:21 | comment | added | Alex R. | @ChristianRemling: This implies the solution is oscillating with varying amplitude, not periodic. Characteristic values of the Mathieu equation tell when the solution is bona-fide periodic, and it's a tricky business. | |
| Sep 4, 2014 at 18:13 | comment | added | Christian Remling | By basic Floquet theory, any solution is a linear combination of solutions of the form $u_j(x)e^{\alpha_j x}$, with $u_j$ $2\pi$ periodic and $e^{\alpha_1+\alpha_2}=1$. Doesn't that tell the whole story? | |
| Sep 4, 2014 at 17:26 | history | asked | Igor Rivin | CC BY-SA 3.0 |