Multisoliton solutions and blow up for the $L^2$-critical Hartree equation

TL;DR

This paper constructs multisoliton solutions for the $L^2$-critical Hartree equation, revealing finite-time blow-up behaviors and complex dynamics influenced by many-body interactions and inverse square potentials.

Contribution

It introduces a novel construction of multisoliton solutions for the $L^2$-critical Hartree equation with detailed analysis of blow-up phenomena and dynamics under inverse square potentials.

Findings

Finite-time collision blow-up in parabolic dynamics

Multi-point blow-up in hyperbolic dynamics

Extension of previous methods to handle degeneracy

Abstract

We construct multisoliton solutions for the $L^2$-critical Hartree equation with trajectories asymptotically obeying a many-body law for an inverse square potential. Precisely, we consider the $m$-body hyperbolic and parabolic non-trapped dynamics. The pseudo-conformal symmetry then implies finite-time collision blow up in the latter case and a solution blowing up at $m$ distinct points in the former case. The approach we take is based on the ideas of [Krieger-Martel-Rapha\"el, 2009] and the third author's recent extension. The approximation scheme requires new aspects in order to deal with a certain degeneracy for generalized root space elements.