Timeline for Does $i\partial_tu = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta u$?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 27 at 16:00 | vote | accept | Dispersion | ||
| May 27 at 8:46 | answer | added | Willie Wong | timeline score: 2 | |
| May 19 at 16:42 | comment | added | Dispersion | Thank you for your comments. How do you show that the decay rate is still $t^{-1/2}$ near $k\neq 0$? Do you have any references that expand on the concept of group velocity dispersion as it relates to my question? | |
| May 18 at 1:58 | comment | added | Willie Wong | Also, I am not sure about your association of dispersion with the ratio between phase and group velocities. For studying wave packets and other localized initial data, what you are looking at really should be the group velocity dispersion which is the rate of change of the group velocity as you change the localized frequency, which is something like the second derivative of $\omega$ in your notation. | |
| May 18 at 1:46 | comment | added | Willie Wong | So in fact your claim that localized initial data for the first equation will "spread" like $\Delta x \approx t^{1/2}$ is I believe incorrect. | |
| May 18 at 1:44 | comment | added | Willie Wong | The short answer is that the Schroedinger equation (which is Galilean invariant) has the same dispersion at all frequencies, so that the worst case estimate given by $L^1$--$L^\infty$ is essentially homogeneous across frequencies, and translates to the spreading that you see when you look at the Gaussian initial data. But for the first equation the dispersion is not the same at all frequencies; at $k = 0$ you have a highly degenerate critical point which makes the worst case $L^1$--$L^\infty$ estimate. But localized near $k \neq 0$ the decay rate is still $t^{-1/2}$. | |
| May 18 at 1:31 | comment | added | Dispersion | Right, sorry about that. It's now fixed. | |
| May 18 at 1:31 | history | edited | Dispersion | CC BY-SA 4.0 |
edited title and typo in post.
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| May 17 at 23:51 | comment | added | LSpice | Aren't the two equations in your title the same? | |
| May 17 at 23:41 | history | asked | Dispersion | CC BY-SA 4.0 |