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New time-periodic solutions to the cubic conformal wave equation on ๐•Š3: I Spherical symmetry

  • Filip Ficek
  • Maciej Maliborski
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Abstract

We present a rigorous construction of novel time-periodic solutions to the radially symmetric cubic conformal wave equation on the Einstein cylinder โ„ร—๐•Š3. Their existence, hinted by the earlier numerical studies (arXiv:2504.10513), complements prior results on Cantor sets of infinitesimally small solutions. Our proof, based on the contraction principle, requires extensive computations that are performed on a computer by the means of rational arithmetic. We are able to significantly reduce their amount by a thorough analysis of the inverse of the associated linear operator, leading to sharper analytical bounds, as well as introduction of simplified versions of certain relevant operators. We provide examples of three solutions with specific frequency; however, our techniques and tools can be directly adapted to prove the existence of other solutions conjectured in (arXiv:2504.10513).

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Cite

BibTeXRIS
BibTeX
@misc{ficek2026timeperiodic,
  title = {{New time-periodic solutions to the cubic conformal wave equation on $\mathbbS^3$: I Spherical symmetry}},
  author = {Filip Ficek and Maciej Maliborski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32600},
  eprint = {2610.32600},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32600},
}
RIS
TY  - UNPB
TI  - New time-periodic solutions to the cubic conformal wave equation on ๐•Š^3: I Spherical symmetry
AU  - Filip Ficek
AU  - Maciej Maliborski
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32600
UR  - https://arxiv.org/abs/2610.32600
AB  - We present a rigorous construction of novel time-periodic solutions to the radially symmetric cubic conformal wave equation on the Einstein cylinder โ„ร—๐•Š^3. Their existence, hinted by the earlier numerical studies (arXiv:2504.10513), complements prior results on Cantor sets of infinitesimally small solutions. Our proof, based on the contraction principle, requires extensive computations that are performed on a computer by the means of rational arithmetic. We are able to significantly reduce their amount by a thorough analysis of the inverse of the associated linear operator, leading to sharper analytical bounds, as well as introduction of simplified versions of certain relevant operators. We provide examples of three solutions with specific frequency; however, our techniques and tools can be directly adapted to prove the existence of other solutions conjectured in (arXiv:2504.10513).
ER  -