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 $\sigma$ and $\zeta$ 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.

arXiv commentRevised title, abstract and introduction to clarify contributions and prior work. Corrected elliptic identities and conventions. Expanded discussion of gauge normal forms, branching and the projection from degenerate four-wave mixing. Added an asymmetric physical example and updated references