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PreprintCC BY 4.0arXiv

Smooth stationary water waves with multiple vortex spikes

  • Zhiqiang Fan
  • Yuchen Wang
  • Youquan Zheng
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Abstract

For any prescribed positive integer n, we establish the existence of a family of stationary solutions to the two dimensional capillary-gravity water wave equations in finite depth, whose stream functions exhibit n well-separated vertically aligned vortex spikes with alternating signs. The construction is based on a nonlinear gluing scheme for a singularly perturbed semilinear elliptic equation coupled to the Bernoulli equation on the free surface. Our results extend the single-spike construction to arbitrary finite multi-spike configurations and reveal a mechanism through which exponentially localized vortex spikes can coexist with an almost flat free surface.

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Cite

BibTeXRIS
BibTeX
@misc{fan2026smooth,
  title = {{Smooth stationary water waves with multiple vortex spikes}},
  author = {Zhiqiang Fan and Yuchen Wang and Youquan Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35129},
  eprint = {2610.35129},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35129},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Smooth stationary water waves with multiple vortex spikes
AU  - Zhiqiang Fan
AU  - Yuchen Wang
AU  - Youquan Zheng
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35129
UR  - https://arxiv.org/abs/2610.35129
AB  - For any prescribed positive integer n, we establish the existence of a family of stationary solutions to the two dimensional capillary-gravity water wave equations in finite depth, whose stream functions exhibit n well-separated vertically aligned vortex spikes with alternating signs. The construction is based on a nonlinear gluing scheme for a singularly perturbed semilinear elliptic equation coupled to the Bernoulli equation on the free surface. Our results extend the single-spike construction to arbitrary finite multi-spike configurations and reveal a mechanism through which exponentially localized vortex spikes can coexist with an almost flat free surface.
ER  -