Timeline for Focusing NLS: $L^2$ convergence of a solution as $t\rightarrow +\infty$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2013 at 17:11 | comment | added | Terry Tao | The soliton resolution conjecture suggests that the answer should be "no", but to prove this rigorously is beyond current methods (except for sufficiently small data). The enemy here is a "breather" solution which is time periodic but not of the solitary wave form $u(t,x) = e^{iEt} Q(x)$, in which case the time average is likely to be non-zero. | |
| Oct 22, 2013 at 17:47 | comment | added | uapu | Thank you Terence! The original question is whatever exists a solution $u(t,x)$ such that $1/T\int_0^Tu(t,x)dt$ converge in $L^2$ to a limit different than 0. Do you have any suggestions? | |
| Oct 22, 2013 at 1:15 | comment | added | Terry Tao | Note for sufficiently small and regular data, the solution scatters to a solution of the linear Schrodinger equation (as can be shown by Strichartz estimates, see e.g. Cazenave's book ams.org/mathscinet-getitem?mr=2002047 ). For larger data, one also sees soliton type behaviour, as well as various blowup solutions... but none of these scenarios corresponds to convergence to a time-independent profile. | |
| Oct 22, 2013 at 1:10 | comment | added | Terry Tao | It depends to some extent on what you mean by a solution to the equation, but under reasonable interpretations the answer will be no, because the limit $u$ will have to be a steady-state solution to NLS, which is necessarily zero (from a Pohozaev identity argument), which then makes $u_0=0$ by conservation of mass. | |
| Oct 21, 2013 at 20:00 | review | First posts | |||
| Oct 21, 2013 at 20:01 | |||||
| Oct 21, 2013 at 19:42 | history | asked | uapu | CC BY-SA 3.0 |