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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.19216 |
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| _version_ | 1866916025442762752 |
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| author | Hesketh, Graham |
| author_facet | Hesketh, Graham |
| contents | Complete analytic solutions for the coherent coupler with arbitrary propagation constants and self- and cross-phase modulation coefficients are presented in terms of Weierstrass elliptic $\wp$, $ζ$, and $σ$ functions, giving the full complex envelopes for both modes under generic initial conditions. Jensen's coupler emerges as a special case of the general system. The mode solutions contain factors of the form $\exp(r\log R(z))$, where $R(z)$ is a ratio of Weierstrass $σ$ functions, giving a multi-valued branch structure that is removable by a gauge transformation. A projection from the three-mode degenerate four-wave mixing system onto the two-mode coupler is identified, and the corresponding degenerate-system solutions are single-valued meromorphic Kronecker theta functions. This connection establishes the coherent coupler as a reduction of a broader class of integrable parametric processes and opens a pathway to leveraging known expansions of Kronecker theta functions for further analysis of nonlinear coupler dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19216 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete Weierstrass elliptic function solutions for coherent couplers and their relation to degenerate four-wave mixing Hesketh, Graham Exactly Solvable and Integrable Systems Complete analytic solutions for the coherent coupler with arbitrary propagation constants and self- and cross-phase modulation coefficients are presented in terms of Weierstrass elliptic $\wp$, $ζ$, and $σ$ functions, giving the full complex envelopes for both modes under generic initial conditions. Jensen's coupler emerges as a special case of the general system. The mode solutions contain factors of the form $\exp(r\log R(z))$, where $R(z)$ is a ratio of Weierstrass $σ$ functions, giving a multi-valued branch structure that is removable by a gauge transformation. A projection from the three-mode degenerate four-wave mixing system onto the two-mode coupler is identified, and the corresponding degenerate-system solutions are single-valued meromorphic Kronecker theta functions. This connection establishes the coherent coupler as a reduction of a broader class of integrable parametric processes and opens a pathway to leveraging known expansions of Kronecker theta functions for further analysis of nonlinear coupler dynamics. |
| title | Complete Weierstrass elliptic function solutions for coherent couplers and their relation to degenerate four-wave mixing |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2605.19216 |