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SUMMARY:Maria Ntekoume (Rice)
DTSTART:20201026T210000Z
DTEND:20201026T220000Z
DTSTAMP:20260423T005733Z
UID:OARS/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OARS/4/">Hom
 ogenization for the cubic nonlinear Schrödinger equation on ℝ²</a>\nby
  Maria Ntekoume (Rice) as part of OARS Online Analysis Research Seminar\n\
 n\nAbstract\nThe cubic nonlinear Schr\\"odinger equation on $\\mathbb R^2$
  is\ngiven by\n$$i \\partial_t  u +\\Delta u=\\bar g  |u|^2 u\, \\quad u(0
 )=u_0 \\in\nL^2(\\mathbb R^2).$$\nThis equation comes in two flavors\, dep
 ending on the sign of $\\bar g$:\nWhen $\\bar g<0$\, the self-interaction 
 described by the nonlinearity is\nattractive. Heuristiaclly\, the nonlinea
 r part is working to counteract\nthe dispersive effects of the linear part
 \; indeed\, finite time blow-up\nis possible. On the other hand\, the case
  $\\bar g>0$ indicates a\nrepulsive self-interaction. In this regime the q
 uestion of\nwell-posedness for general initial data in $L^2$ was a long-st
 anding\nproblem in the field until its recent resolution by Dodson.\n\nIn 
 this talk we will look at the corresponding inhomogeneous problem\n$$i \\p
 artial_t  u +\\Delta u=g(nx)  |u|^2 u$$\nwith initial data in $L^2$\, wher
 e $g$ does not necessarily have a fixed\nsign. We will discuss how it rela
 tes to the homogeneous NLS above and\nderive sufficient conditions on $g$ 
 to ensure existence and uniqueness\nof global solutions for $n$ large\, as
  well as homogenization.\n
LOCATION:https://researchseminars.org/talk/OARS/4/
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