Hyperelliptic sigma solutions of polynomial Hamiltonian systems
Abstract
We study polynomial Hamiltonian systems whose Hamiltonians consist of two weighted products of canonical modes minus a polynomial in their powers. This class extends the structure of familiar genus-one coupled-mode systems from nonlinear optics. We construct local analytic solutions in terms of sigma functions and Abelian integrals on smooth odd-degree hyperelliptic curves, subject to stated regularity conditions. We associate Onishi's determinant identities with the conservation-law structure of these systems, extending the connection previously identified between Frobenius-Stickelberger determinants and conserved quantities in genus-one nonlinear optics. Explicit genus-two and genus-three examples include bounded oscillatory scalar motion. We also classify the Hamiltonian polynomials that eliminate the additional mode exponentials and show that their elementary modes solve second-order equations with self-generating polynomial potentials. We evaluate the solutions using Generapy, a purpose-built Python package for algebraic curves and Abelian functions, and validate them against independent fourth-order Runge-Kutta integration.
Related
- Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory
- Superintegrability of a resonant ghostly tri-Hamiltonian model: polynomial first integrals and transcendental completion
- Hamiltonian of open string field theory for time-dependent solutions
- Beyond the polynomial ansatz: regular AdS3× M8 solutions of M-theory
- Nonintegrability theorem on perturbed integrable Hamiltonian dynamical systems with two independent resonances
BibTeX
@misc{hesketh2026hyperelliptic,
title = {{Hyperelliptic sigma solutions of polynomial Hamiltonian systems}},
author = {Graham Hesketh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22784},
eprint = {2610.22784},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22784},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperelliptic sigma solutions of polynomial Hamiltonian systems
AU - Graham Hesketh
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.22784
UR - https://arxiv.org/abs/2610.22784
AB - We study polynomial Hamiltonian systems whose Hamiltonians consist of two weighted products of canonical modes minus a polynomial in their powers. This class extends the structure of familiar genus-one coupled-mode systems from nonlinear optics. We construct local analytic solutions in terms of sigma functions and Abelian integrals on smooth odd-degree hyperelliptic curves, subject to stated regularity conditions. We associate Onishi's determinant identities with the conservation-law structure of these systems, extending the connection previously identified between Frobenius-Stickelberger determinants and conserved quantities in genus-one nonlinear optics. Explicit genus-two and genus-three examples include bounded oscillatory scalar motion. We also classify the Hamiltonian polynomials that eliminate the additional mode exponentials and show that their elementary modes solve second-order equations with self-generating polynomial potentials. We evaluate the solutions using Generapy, a purpose-built Python package for algebraic curves and Abelian functions, and validate them against independent fourth-order Runge-Kutta integration.
ER -