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CATEGORIES:Partial Differential Equations seminar
SUMMARY:Construction of infinite-bump solutions for non-li
near-Schrödinger-like equations - Prof. Robert Mag
nus (Reykjavik/DPMMS)
DTSTART;TZID=Europe/London:20121022T150000
DTEND;TZID=Europe/London:20121022T160000
UID:TALK40673AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/40673
DESCRIPTION:In 1986 Floer and Weinstein gave a rigorous proof
that a solution of the stationary\, one-dimensiona
l\, cubic\, non-linear Schrödinger equation\,\n\n_
−ε^2^ u′′ + V(x)u − u^3^ = 0\, −∞ < x < ∞_\n\
nexists that concentrates at a non-degenerate crit
ical point of _V_ as _ε → 0_. This means that the
solution differs substantially from _0_ only in a
small neighbourhood of the critical point. Y-G. Oh
then constructed solutions that concentrate simul
taneously at a finite set of non-degenerate critic
al points\, the so-called multibump solutions. Thi
s work led to much interest in the concentration p
roperties of solutions of this\, and more general
equations\, in *R* ^n^ and with more general non-l
inearities. These have the form\n\n_−ε^2^ ∆u + F (
V (x)\, u) = 0\, x ∈ *R* ^n^_.\n\nIn this talk I
shall discuss the construction of solutions that c
oncentrate simultaneously at an infinite set. I sh
all try to explain the key ideas\, derived from PD
E theory and functional analysis\, used in the pro
ofs.
LOCATION:CMS\, MR11
CONTACT:Prof. Mihalis Dafermos
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