Timeline for Nonlinear PDE and Green functions
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 16, 2013 at 22:06 | comment | added | user36539 | May the theory of Colombeau's generalized functions be usefull for this problem. | |
| Jun 7, 2013 at 7:25 | comment | added | Jon | @timur:The point is that I am interested in an approximation that involves the formal limit $\lambda\rightarrow\infty$. What you are proposing is for case $\lambda\rightarrow 0$. From a physical standpoint, the ability to manage a strongly coupled PDE cannot be underestimated. | |
| Jun 7, 2013 at 7:20 | comment | added | timur | Wouldn't you also get a good approximation by using a fundamental solution of the linear part only? Or perhaps the nonlinear fundamental solution gives a better result? | |
| Jun 7, 2013 at 7:18 | comment | added | Jon | @timur: This is quite interesting. If you consider such kind of "nonlinear" Green functions, you can get a good approximation to the problem $\partial^\phi+V(\phi)=j$ with a leading order $\phi\approx\int d^Dx'G(x-x')j(x')$, similarly to the linear case. This represents the leading term of a strong coupling series and, for the simplest case of an ODE, it is just a small time expansion. So, the rescaling $\sqrt{\lambda}t$ accounts for the range of validity of this approximation. | |
| Jun 7, 2013 at 7:03 | comment | added | timur | Depending on the nonlinearity, it may or may not be straightforward to give a meaning to this. Since there is no superposition principle, I would guess a fundamental solution might not be that useful. Out of curiosity, what use would it have? | |
| Jun 7, 2013 at 6:26 | history | edited | Jon | CC BY-SA 3.0 |
Expanded question to clarify the content
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| Jun 7, 2013 at 6:06 | comment | added | Jon | @DeaneYang: Yes, I will clarify this in the question. | |
| Jun 6, 2013 at 22:17 | comment | added | Deane Yang | In the third equation you have $\partial_t^2$ but in the first two equations you have $\partial^2$. Could you clarify what you mean by the latter? Also, could you explain how you get the third equation from a "gradient expansion"? | |
| Jun 6, 2013 at 21:25 | history | asked | Jon | CC BY-SA 3.0 |