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Asymptotic Stability and Classification of Multi-solitons for Klein–Gordon Equations

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Abstract

Focusing on multi-solitons for the Klein–Gordon equations, in the first part of this paper, we establish their conditional asymptotic stability. In the second part of this paper, we classify pure multi-solitons which are solutions converging to multi-solitons in the energy space as \(t\rightarrow \infty \). Using Strichartz estimates developed in our earlier work (Chen and Jendrej in Strichartz estimates for Klein–Gordon equations with moving potentials, 2022) and the modulation techniques, we show that if a solution stays close to the multi-soliton family, then it scatters to the multi-soliton family in the sense that the solution will converge in large time to a superposition of Lorentz-transformed solitons (with slightly modified velocities), and a radiation term which is in main order a free wave. Moreover, we construct a finite-codimension centre-stable manifold around the well-separated multi-soliton family. Finally, given different Lorentz parameters and arbitrary centers, we show that all the corresponding pure multi-solitons form a finite-dimension manifold.

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Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. In particular, note that all pure multi-soliton solutions constructed in [18] meet this condition.. Moreover, Corollary 1.6 below will give a full characterization of the set of solutions satisfying this condition.

  2. Using more classical notations, \({\mathcal {D}}^{\frac{\nu }{2}}L_{x}^{2}\) is nothing but \(H^{-\nu /2}\).

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Acknowledgements

We would like to thank Marius Beceanu and Chongchun Zeng for enlightening discussions.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Gong Chen

  2. CNRS & LAGA, Université Sorbonne Paris Nord, UMR 7539, 99 av J.-B. Clément, 93430, Villetaneuse, France

    Jacek Jendrej

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  1. Gong Chen
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Correspondence to Gong Chen.

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Chen, G., Jendrej, J. Asymptotic Stability and Classification of Multi-solitons for Klein–Gordon Equations. Commun. Math. Phys. 405, 7 (2024). https://doi.org/10.1007/s00220-023-04904-5

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