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

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Abstract

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 \(K_\omega \). The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For \(N\ge 3\), we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of \(K_\omega \).

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Notes

  1. If we choose a different frequency \(\omega \), the computations seem to be more complicated.

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Acknowledgements

We are grateful to Kenji Nakanishi, Ernesto Perez-Chavela and Cristina Stoica for insightful discussions. We would like to thank Belaid Moa for the numerical simulations on the MacMillan problem. The first author is partially supported by the NSERC Grant. The second author is supported by the NSERC Grant Nos. 371637-2014 and 371637-2019.

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Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, La Jolla, CA, USA

    Yanxia Deng

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

    Slim Ibrahim

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  1. Yanxia Deng
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  2. Slim Ibrahim
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Correspondence to Slim Ibrahim.

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In memory of Florin Diacu.

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Deng, Y., Ibrahim, S. Global Existence and Singularity of the N-Body Problem with Strong Force. Qual. Theory Dyn. Syst. 19, 49 (2020). https://doi.org/10.1007/s12346-020-00387-0

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