Complete Weierstrass elliptic function solutions for coherent couplers and their relation to degenerate four-wave mixing

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 $\wp$, $\zeta$, and $\sigma$ 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 $\sigma$ 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.