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PreprintCC BY 4.0arXiv

Lattice KP type equations arising from eigenfunctions and Dbar problem

  • Leilei Shi
  • Peter van der Kamp
  • Cheng Zhang
  • Da-jun Zhang
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Abstract

In this paper, we construct the lattice Kadomtsev-Petviashvili (KP) type eigenfunction equations. A homogeneous nonlocal ∂̅ problem is considered, from which we are able to define the eigenfunction of the Lax pair of the lattice KP equation. The eigenfunction together with its expansions at infinity and at a finite analytic point provide formulations of the lattice modified KP equation, the lattice Schwarzian KP equation and the Nijhoff-Quispel-Capel KP (NQC-KP) equation. We also consider an inhomogeneous nonlocal ∂̅ problem. It defines the eigenfunction of the Lax pair of the lattice modified KP equation. This eigenfunction generates a direct formulation for the NQC-KP equation, which is different from the previous ones. Explicit solutions of these equations are obtained, from which we can see the difference of the different formulations for same equations.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026lattice,
  title = {{Lattice KP type equations arising from eigenfunctions and Dbar problem}},
  author = {Leilei Shi and Peter van der Kamp and Cheng Zhang and Da-jun Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30190},
  eprint = {2608.30190},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30190},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Lattice KP type equations arising from eigenfunctions and Dbar problem
AU  - Leilei Shi
AU  - Peter van der Kamp
AU  - Cheng Zhang
AU  - Da-jun Zhang
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30190
UR  - https://arxiv.org/abs/2608.30190
AB  - In this paper, we construct the lattice Kadomtsev-Petviashvili (KP) type eigenfunction equations. A homogeneous nonlocal ∂̅ problem is considered, from which we are able to define the eigenfunction of the Lax pair of the lattice KP equation. The eigenfunction together with its expansions at infinity and at a finite analytic point provide formulations of the lattice modified KP equation, the lattice Schwarzian KP equation and the Nijhoff-Quispel-Capel KP (NQC-KP) equation. We also consider an inhomogeneous nonlocal ∂̅ problem. It defines the eigenfunction of the Lax pair of the lattice modified KP equation. This eigenfunction generates a direct formulation for the NQC-KP equation, which is different from the previous ones. Explicit solutions of these equations are obtained, from which we can see the difference of the different formulations for same equations.
ER  -