Abstract

I will consider the three dimensional gravitational Hartree system i∂_t u + ∆u − φu = 0 where φ is the Poisson gravitational field ∆φ = |u|²

This system arises in particular as a mean field limit of many body quantum systems in gravitational interaction and is a canonical model of Schrödinger type equation with nonlocal nonlinearity.

The 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 travelling waves with ground state profille Q is a consequence of variational techniques introduced in the 80’s.

The question we ask is the existence of multisolitary waves for this problem. These are known in other related 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, multisolitary waves have been constructed in the recent years by Martel, Merle and Rodnianski, Soffer, Schlag where each wave evolves asympotically according to the free Galilean motion. We shall see that for the Hartree problem, the long range structure of the gravitational field creates a strong coupling between the solitons and hence a non trivial asymptotic dynamic for their center of mass. Our main result is the existence of non dispersive two soliton like solutions which center of mass repoduce the nontrapped dynamics of the two body problem in Newtonian gravity, that is a planar trajectory with either hyperbolic or parabolic asymptotic motion.

This is joint work with Joachim Krieger (UPenn) and Yvan Martel (Versailles).