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PreprintCC BY 4.0arXiv

Lax Pair for the Kirchhoff-Pohozaev equation

  • Dario Bambusi
  • Emanuele Haus
  • Simone Marrocco
  • Michela Procesi
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Abstract

We prove that the Kirchhoff-Pohozaev equation on any compact connected Riemannian manifold without boundary admits a Lax pair (L,B), where L is a self-adjoint finite-rank perturbation of the Laplace-Beltrami operator and B is a skew-adjoint operator of finite rank, both depending analytically on the phase space variables. We study the spectrum of L using the perturbative determinant which is known to determine its spectrum and to admit a product representation similar to that of the generating function in KdV. We use this to show that the dynamics of the integrals of motion constructed in \cite{BoitiManfrin2026,BHMP,campos} admits a Lax pair formulation with the same operator L and to get a new proof of the involutivity of these integrals.

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Cite

BibTeXRIS
BibTeX
@misc{bambusi2026pair,
  title = {{Lax Pair for the Kirchhoff-Pohozaev equation}},
  author = {Dario Bambusi and Emanuele Haus and Simone Marrocco and Michela Procesi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34522},
  eprint = {2610.34522},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34522},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Lax Pair for the Kirchhoff-Pohozaev equation
AU  - Dario Bambusi
AU  - Emanuele Haus
AU  - Simone Marrocco
AU  - Michela Procesi
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34522
UR  - https://arxiv.org/abs/2610.34522
AB  - We prove that the Kirchhoff-Pohozaev equation on any compact connected Riemannian manifold without boundary admits a Lax pair (L,B), where L is a self-adjoint finite-rank perturbation of the Laplace-Beltrami operator and B is a skew-adjoint operator of finite rank, both depending analytically on the phase space variables. We study the spectrum of L using the perturbative determinant which is known to determine its spectrum and to admit a product representation similar to that of the generating function in KdV. We use this to show that the dynamics of the integrals of motion constructed in \cite{BoitiManfrin2026,BHMP,campos} admits a Lax pair formulation with the same operator L and to get a new proof of the involutivity of these integrals.
ER  -