Elliptic solutions, gauge normal forms and a degenerate four-wave-mixing projection for coherent couplers
Abstract
We relate a complex Hamiltonian extension of the asymmetric conservative coherent coupler to three-mode degenerate four-wave mixing. A trajectory-dependent reciprocal gauge preserves the modal products and gives an autonomous normal form with meromorphic envelopes; we construct a projection from degenerate four-wave mixing onto this normal form. Building on established elliptic power-transfer and phase descriptions, we reconstruct the complex envelopes explicitly in Weierstrass σ and ζ functions for unequal propagation constants and self-phase coefficients with symmetric cross-phase coupling. This representation exposes the noninteger powers of sigma ratios introduced generically by unequal self-phase coefficients, and their removal by the gauge. The lifted modes are exponentially gauged Kronecker kernels, equivalently genus-one Baker-Akhiezer functions, and the conserved Hamiltonian product relation is represented by a four-point Frobenius-Stickelberger determinant identity. These mode-level relationships place the coupler within a wider family of integrable continuous-wave systems; numerical comparisons verify the formulae and their evaluation.
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BibTeX
@misc{hesketh2026elliptic,
title = {{Elliptic solutions, gauge normal forms and a degenerate four-wave-mixing projection for coherent couplers}},
author = {Graham Hesketh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.19216},
eprint = {2605.19216},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.19216},
note = {CC0},
}
RIS
TY - UNPB
TI - Elliptic solutions, gauge normal forms and a degenerate four-wave-mixing projection for coherent couplers
AU - Graham Hesketh
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2605.19216
UR - https://arxiv.org/abs/2605.19216
AB - We relate a complex Hamiltonian extension of the asymmetric conservative coherent coupler to three-mode degenerate four-wave mixing. A trajectory-dependent reciprocal gauge preserves the modal products and gives an autonomous normal form with meromorphic envelopes; we construct a projection from degenerate four-wave mixing onto this normal form. Building on established elliptic power-transfer and phase descriptions, we reconstruct the complex envelopes explicitly in Weierstrass σ and ζ functions for unequal propagation constants and self-phase coefficients with symmetric cross-phase coupling. This representation exposes the noninteger powers of sigma ratios introduced generically by unequal self-phase coefficients, and their removal by the gauge. The lifted modes are exponentially gauged Kronecker kernels, equivalently genus-one Baker-Akhiezer functions, and the conserved Hamiltonian product relation is represented by a four-point Frobenius-Stickelberger determinant identity. These mode-level relationships place the coupler within a wider family of integrable continuous-wave systems; numerical comparisons verify the formulae and their evaluation.
ER -