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CATEGORIES:Geometric Analysis and Partial Differential Equati
ons seminar
SUMMARY:Two soliton solutions to the gravitational Hartree
equation - Pierre Raphael (Toulouse)
DTSTART;TZID=Europe/London:20090126T160000
DTEND;TZID=Europe/London:20090126T170000
UID:TALK16019AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/16019
DESCRIPTION:I will consider the three dimensional gravitationa
l Hartree system i∂_t u + ∆u − φu = 0 where φ is t
he Poisson gravitational field \n∆φ = |u|^2\n\nThi
s system arises in particular as a mean field limi
t of many body quantum systems in gravitational in
teraction and is a \ncanonical model of Schrödinge
r type equation with nonlocal nonlinearity. \n\nTh
e existence and uniqueness of well localized perio
dic solutions u(t\, x) = Q(x)e^{it} for this syst
em is well known and the Galilean invariance appli
ed to these solutions yields explicit travelling w
aves with straight line trajectory and constant sp
eed. The orbital stability of these travelling wav
es with ground state profille Q is a consequence o
f variational techniques introduced in the 80’s. \
n\nThe question we ask is the existence of multiso
litary waves for this problem. These are known in
other related settings to be the building blocks f
or the description of the long time dynamics of th
e system. In the case of power like local nonlinea
rities\, multisolitary waves have been constructe
d in the recent years by Martel\, Merle and Rodnia
nski\, Soffer\, Schlag where each wave evolves asy
mpotically according to the free Galilean motion.
We shall see that for the Hartree problem\, the l
ong range structure of the gravitational field cre
ates a strong coupling between the solitons and he
nce a non trivial asymptotic dynamic for their cen
ter of mass. Our main result is the existence of n
on dispersive two soliton like solutions which ce
nter of mass repoduce the nontrapped dynamics of t
he two body problem in Newtonian gravity\, that is
a planar trajectory with either hyperbolic or \np
arabolic asymptotic motion.\n \nThis is joint work
with Joachim Krieger (UPenn) and Yvan Martel (Ver
sailles). \n\n
LOCATION:CMS\, MR13
CONTACT:Prof. Mihalis Dafermos
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