Timeline for Mathematical difference between solitons and traveling waves for a non-linear dispersive PDE
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31, 2023 at 10:57 | comment | added | Carlo Beenakker | yes, any square integrable solution $f(x-ct)$ in $\mathbb{R}$ is a localised wave, but you will not find it in a linear PDE. | |
| Jan 31, 2023 at 9:46 | comment | added | Niser | Thank you ! I have one additional question. In general, many PDEs are known to be locally well-posed in Sobolev spaces $H^s(\mathbb{R})$ with $s$ not so much negative. Thus, for example, if we exhibit a traveling wave that belongs to $L^2(\mathbb{R}),$ then of course, we have $\lim_{s\to\pm}f(s)=0$. This means that every traveling wave in $L^2(\mathbb{R})$ is a homoclinic soliton ? (The spatial variable here is $x\in\mathbb{R}$). | |
| Jan 31, 2023 at 9:06 | comment | added | Carlo Beenakker | yes, that is what we understand as a "localized" wave. | |
| Jan 31, 2023 at 7:54 | comment | added | Niser | So, for the two examples in Q2, since we have $\lim_{x\to2\pi}\mathrm{e}^{ix}=1\neq 0,$ then it is not localized ? And the same goes for the second example. In other words, we need to have the support of $f$ strictly included in a box of the periodic domain? | |
| Jan 31, 2023 at 7:17 | comment | added | Carlo Beenakker | if you have periodic boundary conditions you would want the spatial extent of the soliton to be smaller than the size of the domain, so that you can see if it is spatially localized or not; if it extends over the entire domain the notion of a "solitary wave" is no longer applicable. | |
| Jan 31, 2023 at 6:30 | comment | added | Niser | I think it is what happen with long time behavior, but I want to be sure ;-) | |
| Jan 31, 2023 at 6:16 | comment | added | Niser | Thank you ! Even if we are in a periodic domain, we should look at what happens when $s\to\pm \infty$ and not the limit when $s$ tends to the boundary of the domain? Because I wonder if this limit means that we are looking at the behavior of the initial data in long time behavior, or rather what happens when we approach the edge of our space. | |
| Jan 31, 2023 at 6:15 | vote | accept | Niser | ||
| Jan 30, 2023 at 13:14 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 395 characters in body
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| Jan 30, 2023 at 12:47 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |