Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of multisoliton solutions of the gravitational Hartree equation in three dimensions

Yutong Wu

TL;DR

The work establishes the existence of multisoliton solutions to the 3D gravitational Hartree equation $iu_t+\Delta u-\phi_{|u|^2}u=0$ whose centers track expansive $m$-body dynamics across hyperbolic, parabolic, or hyperbolic–parabolic regimes. By constructing high-order approximate multisolitons and developing a robust modulation framework, the authors reduce the problem to controlling a small error in $H^1$ via a bootstrap, coercivity of linearized operators $L_+$ and $L_-$, and a Lyapunov-type functional $\mathcal G(\varepsilon)$. A key innovation is a detailed dipole-like expansion of the nonlocal Hartree interaction, enabling cancellation of long-range effects up to order $N$ and allowing the error to decay despite the nonlocal tail $\phi_{|u|^2}(x)\sim 1/|x|$. The result generalizes the two-soliton construction of Krieger–Martel–Raphaël to arbitrary $m$, providing a concrete realization of multisolitons consistent with the $m$-body dynamics and advancing the soliton-resolution program for nonlocal dispersive systems.

Abstract

We prove the existence of multisoliton solutions of the three-dimensional gravitational Hartree equation whose trajectories follow many body dynamics of hyperbolic, parabolic or hyperbolic-parabolic types. This work generalizes and improves the result of Krieger-Martel-Raphaël on two-soliton solutions.

Existence of multisoliton solutions of the gravitational Hartree equation in three dimensions

TL;DR

The work establishes the existence of multisoliton solutions to the 3D gravitational Hartree equation whose centers track expansive -body dynamics across hyperbolic, parabolic, or hyperbolic–parabolic regimes. By constructing high-order approximate multisolitons and developing a robust modulation framework, the authors reduce the problem to controlling a small error in via a bootstrap, coercivity of linearized operators and , and a Lyapunov-type functional . A key innovation is a detailed dipole-like expansion of the nonlocal Hartree interaction, enabling cancellation of long-range effects up to order and allowing the error to decay despite the nonlocal tail . The result generalizes the two-soliton construction of Krieger–Martel–Raphaël to arbitrary , providing a concrete realization of multisolitons consistent with the -body dynamics and advancing the soliton-resolution program for nonlocal dispersive systems.

Abstract

We prove the existence of multisoliton solutions of the three-dimensional gravitational Hartree equation whose trajectories follow many body dynamics of hyperbolic, parabolic or hyperbolic-parabolic types. This work generalizes and improves the result of Krieger-Martel-Raphaël on two-soliton solutions.
Paper Structure (16 sections, 21 theorems, 215 equations)

This paper contains 16 sections, 21 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The m-body problem
  4. The main result
  5. Approximate solutions
  6. Definition of approximate solutions
  7. Accuracy of approximate solutions
  8. Reduction of the problem
  9. Uniform estimates
  10. Modulation estimates
  11. Estimates of the modulation
  12. Control of the parameters
  13. Control of the error
  14. The parabolic and the hyperbolic-parabolic case
  15. The approximate trajectory
  16. ...and 1 more sections

Key Result

Theorem 1

For an expansive solution of eq m-body problem centered at the origin, there exists $(a_1, \cdots, a_m) \in \mathcal{X}$ such that Moreover, if $a_j=a_k$ for some $j \neq k$, then $|\alpha_j- \alpha_k| \sim t^\frac{2}{3}$ as $t \to +\infty$.

Theorems & Definitions (40)

  • Theorem : Marchal-Saari StructureOfExpansive
  • Theorem : Maderna-Venturelli nbodyhyperbolic; Polimeni-Terracini ExistenceofNbodyproblem
  • Theorem 1
  • Remark
  • Corollary 2
  • Remark
  • Definition 2.1: Admissible functions
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • ...and 30 more