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PreprintCC BY 4.0arXiv

The \textsc{ABC}-equation revisited: missed integrable cases. Part I: classification and (quasi) integrable structures

  • I. S. Krasil'shchik
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Abstract

We consider the \textsc{ABC}-equation Auxuyt + Buyuxt + Cutuxy = 0, A, B, C∈ℝ, which was proved to be integrable in the case A + B + C = 0 (the Veronese web equation). Our study reveals ten more cases that admit integrable structures that, to the best of our knowledge, where not known before. The techniques is based essentially on the theory of the tangent and cotangent coverings.

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Cite

BibTeXRIS
BibTeX
@misc{krasilshchik2026textscabcequation,
  title = {{The \textscABC-equation revisited: missed integrable cases. Part I: classification and (quasi) integrable structures}},
  author = {I. S. Krasil'shchik},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20672},
  eprint = {2610.20672},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20672},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The \textsc{ABC}-equation revisited: missed integrable cases. Part I: classification and (quasi) integrable structures
AU  - I. S. Krasil'shchik
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20672
UR  - https://arxiv.org/abs/2610.20672
AB  - We consider the \textsc{ABC}-equation Au_xu_yt + Bu_yu_xt + Cu_tu_xy = 0, A, B, C∈ℝ, which was proved to be integrable in the case A + B + C = 0 (the Veronese web equation). Our study reveals ten more cases that admit integrable structures that, to the best of our knowledge, where not known before. The techniques is based essentially on the theory of the tangent and cotangent coverings.
ER  -