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PreprintCC BY 4.0arXiv

Vector rogue wave patterns associated with generalized Hermite and Okamoto polynomials

  • Hejiaqi Chen
  • Dongwei Wu
  • Chengfa Wu
  • Guangxiong Zhang
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Abstract

We establish rogue wave patterns associated with the fourth Painlevé equation (PIV) in the multi-component nonlinear Schrödinger and Hirota equations. The generalized Hermite and generalized Okamoto polynomials arise in representations of rational solutions of PIV, and we show that their roots determine two classes of rogue wave patterns when one of the internal parameters of rogue wave solutions is large. Specifically, the generalized Hermite polynomials arise from rogue wave solutions represented by Schur-polynomial determinants with consecutive indices, whereas the generalized Okamoto polynomials arise from analogous determinants with index jumps of three. Numerical examples for both equations agree with the predictions.

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Cite

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BibTeX
@misc{chen2026vector,
  title = {{Vector rogue wave patterns associated with generalized Hermite and Okamoto polynomials}},
  author = {Hejiaqi Chen and Dongwei Wu and Chengfa Wu and Guangxiong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14457},
  eprint = {2609.14457},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14457},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Vector rogue wave patterns associated with generalized Hermite and Okamoto polynomials
AU  - Hejiaqi Chen
AU  - Dongwei Wu
AU  - Chengfa Wu
AU  - Guangxiong Zhang
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14457
UR  - https://arxiv.org/abs/2609.14457
AB  - We establish rogue wave patterns associated with the fourth Painlevé equation (P_IV) in the multi-component nonlinear Schrödinger and Hirota equations. The generalized Hermite and generalized Okamoto polynomials arise in representations of rational solutions of P_IV, and we show that their roots determine two classes of rogue wave patterns when one of the internal parameters of rogue wave solutions is large. Specifically, the generalized Hermite polynomials arise from rogue wave solutions represented by Schur-polynomial determinants with consecutive indices, whereas the generalized Okamoto polynomials arise from analogous determinants with index jumps of three. Numerical examples for both equations agree with the predictions.
ER  -