Global Existence and Singularity of the N-Body Problem with Strong Force

• Published in: Qualitative theory of dynamical systems 2020-04, Vol.19 (1), Article 49
• Main Authors: Deng, Yanxia, Ibrahim, Slim
• Format: Article
• Language: English
• Subjects: Chaos theory; Difference and Functional Equations; Dynamical Systems and Ergodic Theory; Existence theorems; Ground state; In Memoriam Florin Diacu; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear equations; Schrodinger equation; Singularity (mathematics); Solitary waves; Two body problem
• ISSN: 1575-5460; 1662-3592
• DOI: 10.1007/s12346-020-00387-0

We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schrödinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. We are able to give a conditional dichotomy of the global existence and singularity below the excited energy in Theorem 4 , the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function K ω . The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For N ≥ 3 , we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of K ω .