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SUMMARY:Two soliton solutions to the gravitational Hartree equation - Pier
 re Raphael (Toulouse)
DTSTART:20090126T160000Z
DTEND:20090126T170000Z
UID:TALK16019@talks.cam.ac.uk
CONTACT:Prof. Mihalis Dafermos
DESCRIPTION:I will consider the three dimensional gravitational Hartree sy
 stem i∂_t u + ∆u − φu = 0 where φ is the Poisson gravitational fie
 ld \n∆φ = |u|^2\n\nThis system arises in particular as a mean field lim
 it of many body quantum systems in gravitational interaction and is a \nca
 nonical model of Schrödinger type equation with nonlocal nonlinearity. \n
 \nThe existence and uniqueness of well localized periodic solutions u(t\, 
 x) = Q(x)e^{it}  for this system is well known and the Galilean invariance
  applied to these solutions yields explicit travelling waves with straight
  line trajectory and constant speed. The orbital stability of these travel
 ling waves with ground state profille Q is a consequence of variational te
 chniques introduced in the 80’s. \n\nThe question we ask is the existenc
 e of multisolitary waves for this problem. These are known in other relate
 d settings to be the building blocks for the description of the long time 
 dynamics of the system. In the case of power like local nonlinearities\, m
 ultisolitary  waves have been constructed in the recent years by Martel\, 
 Merle and Rodnianski\, Soffer\, Schlag where each wave evolves asympotical
 ly according to the free Galilean motion.  We shall see that for the Hartr
 ee problem\, the long range structure of the gravitational field creates a
  strong coupling between the solitons and hence a non trivial asymptotic d
 ynamic for their center of mass. Our main result is the existence of non d
 ispersive two  soliton like solutions which center of mass repoduce the no
 ntrapped dynamics of the two body problem in Newtonian gravity\, that is a
  planar trajectory with either hyperbolic or \nparabolic asymptotic motion
 .\n \nThis is joint work with Joachim Krieger (UPenn) and Yvan Martel (Ver
 sailles). \n\n
LOCATION:CMS\, MR13
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