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Consider the cubic focusing non linear Schrodinger equation in dimension $n\geq 2$:

$$(iu_t+\Delta)=-|u|^2u\qquad u(0,x)=u_0(x)\in L^2(\mathbb{R}^n)$$

Can we find an initial data $u_0\neq 0$ such that $u(t,x)$ is globally defined and converge in $L^2(\mathbb{R}^n)$ to a limit $u$ as $t\rightarrow +\infty$?

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  • $\begingroup$ 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. $\endgroup$
    – Terry Tao
    Commented Oct 22, 2013 at 1:10
  • $\begingroup$ 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. $\endgroup$
    – Terry Tao
    Commented Oct 22, 2013 at 1:15
  • $\begingroup$ 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? $\endgroup$
    – uapu
    Commented Oct 22, 2013 at 17:47
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    $\begingroup$ 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. $\endgroup$
    – Terry Tao
    Commented Oct 30, 2013 at 17:11

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