Timeline for One dimensional periodic travelling waves to some pde
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 19, 2019 at 0:41 | comment | added | Michael Engelhardt | I would just run this numerically with a representative grid of initial conditions and see how it behaves. Then you know better what you should or shouldn't be trying to prove. | |
| Nov 18, 2019 at 22:16 | comment | added | Willie Wong | ... explicit solutions even in this case, never mind all periodic solutions. | |
| Nov 18, 2019 at 22:15 | comment | added | Willie Wong | I don't see why one can expect that all periodic solutions have the form you gave. In general by taking Fourier transform you get that a periodic solution will look like $\phi = \sum A_k e^{2\pi ikx / T}$ where $k\in \mathbb{Z}$, and the coefficients $A_k$ satisfy an infinite system of equations that look like (the coefficients may be a bit off) $$ (- 4\pi^2 k^2 / T^2 - 2\pi ck / T + 1)A_k + \sum_{k_1 + k_2 - k_3 = k} A_{k_1} A_{k_2} \overline{A_{k_3}} = 0 $$ In the small amplitude limit one can probably do some sort of perturbative analysis. But I don't have high hopes for... | |
| Nov 18, 2019 at 20:51 | comment | added | Michael Engelhardt | I don't know off-hand. Certainly, there must be more solutions, since there is only one free integration constant in $Ae^{ikx} $, but I don't know whether they're periodic. If there are soliton solutions, maybe one can build soliton-antisoliton chains. I don't know the literature on this, I expect all this is known. | |
| Nov 18, 2019 at 19:32 | history | edited | R. N. Marley | CC BY-SA 4.0 |
deleted 130 characters in body
|
| Nov 18, 2019 at 19:32 | comment | added | R. N. Marley | Thank you, you are right! By the way, if there a way to prove or disprove that all the periodic solutions to that equation have the form $Ae^{ikx}$ ?? | |
| Nov 18, 2019 at 17:39 | comment | added | Michael Engelhardt | If you insert $\phi = Ae^{ikx} $, you get $|A|^2 = 1-k^2 -ck$, so there's a range of solutions for $k$ small enough. Actually, your $Ae^{icx} $ is not a solution for $|A|^2 =1$. | |
| Nov 18, 2019 at 9:03 | comment | added | R. N. Marley | What does if mean adjusting the amplitude? Can You explain that please? | |
| Nov 18, 2019 at 3:20 | comment | added | Michael Engelhardt | You can have other frequencies if you adjust the amplitude accordingly. This equation should have solitonic solutions as well. | |
| Nov 17, 2019 at 19:41 | history | edited | R. N. Marley | CC BY-SA 4.0 |
edited title
|
| Nov 17, 2019 at 19:35 | history | edited | R. N. Marley | CC BY-SA 4.0 |
added 135 characters in body
|
| Nov 17, 2019 at 19:20 | history | asked | R. N. Marley | CC BY-SA 4.0 |