Complete Weierstrass elliptic function solutions and canonical coordinates for four-wave mixing in nonlinear optical fibres

TL;DR

This paper derives complete analytic solutions for four-wave mixing in nonlinear optical fibers using Weierstrass elliptic functions, revealing a universal canonical form and invariance properties.

Contribution

It introduces a full set of solutions in terms of elliptic functions and uncovers a new invariance property under coordinate transformations.

Findings

Solutions expressed in Weierstrass elliptic functions for all initial conditions.

Canonical form with a universal, parameter-free structure.

Numerical validation confirms the analytical solutions.

Abstract

Complete analytic solutions to quasi-continuous-wave four-wave mixing in nonlinear optical fibres are presented in terms of Weierstrass elliptic $\wp$, $\zeta$, and $\sigma$ functions, providing the full complex envelopes for all four waves under arbitrary initial conditions. A sequence of coordinate transformations reveals a canonical form with universal parameter-free structure. Remarkably, these transformations depend explicitly on the propagation variable yet preserve the structural form of the differential equations, an invariance property not previously reported for four-wave mixing. In the canonical coordinates, solutions become single-valued meromorphic Kronecker theta functions, establishing connections with other integrable nonlinear optical systems. The Hamiltonian conservation is shown to arise from the Frobenius-Stickelberger determinant. Numerical validation confirms the solutions using open-source Python libraries.