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

Graham Hesketh

TL;DR

This work provides complete analytic solutions for quasi-continuous-wave four-wave mixing in nonlinear optical fibres by formulating the coupled envelopes with Weierstrass elliptic functions ($\wp$, $\zeta$, $\sigma$). A sequence of coordinate transformations exposes a canonical, parameter-free form in which the solutions are Kronecker theta functions, and the Hamiltonian conservation is tied to the Frobenius–Stickelberger determinant. The approach supports arbitrary initial conditions, including pump depletion and phase mismatch, and reveals an invariance of the interaction structure under a $z$-dependent transformation. Numerical validation using open-source libraries confirms accuracy and demonstrates practical computability for applications in wavelength conversion, parametric amplification, and quantum light sources.

Abstract

Complete analytic solutions to quasi-continuous-wave four-wave mixing in nonlinear optical fibres are presented in terms of Weierstrass elliptic $\wp$, $ζ$, and $σ$ 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.

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

TL;DR

This work provides complete analytic solutions for quasi-continuous-wave four-wave mixing in nonlinear optical fibres by formulating the coupled envelopes with Weierstrass elliptic functions (, , ). A sequence of coordinate transformations exposes a canonical, parameter-free form in which the solutions are Kronecker theta functions, and the Hamiltonian conservation is tied to the Frobenius–Stickelberger determinant. The approach supports arbitrary initial conditions, including pump depletion and phase mismatch, and reveals an invariance of the interaction structure under a -dependent transformation. Numerical validation using open-source libraries confirms accuracy and demonstrates practical computability for applications in wavelength conversion, parametric amplification, and quantum light sources.

Abstract

Complete analytic solutions to quasi-continuous-wave four-wave mixing in nonlinear optical fibres are presented in terms of Weierstrass elliptic , , and 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.
Paper Structure (20 sections, 58 equations, 4 figures)

This paper contains 20 sections, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The quasi-continuous-wave four-wave mixing system
  3. The normalised system
  4. Conserved quantities
  5. Solutions for modal powers in terms of Weierstrass wp elliptic functions
  6. Solutions for modes in terms of Weierstrass sigma, zeta functions
  7. When quartic terms cancel through special parameter values
  8. Transforming to canonical coordinates
  9. Removing cross-phase modulation with a gauge transform
  10. Reducing the quartic to a cubic via local normalisation
  11. A parameterless system via rescaling
  12. Numerical validation and implementation
  13. Conclusion
  14. Parameter definitions in the original coordinates
  15. Initial condition relations
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Real part of abstract modal power $u_jv_j$: analytic solution \ref{['eq:uv-wp']} (dashed lines) compared with numerical integration (symbols).
  • Figure 2: Imaginary part of abstract modal power $u_jv_j$: analytic solution \ref{['eq:uv-wp']} (dashed lines) compared with numerical integration (symbols).
  • Figure 3: Intensity $|A_j|^2$ of physical field amplitudes: analytic solution \ref{['eq:u-v-quartic']} converted via \ref{['eq:u-v-A']} (dashed lines) compared with numerical integration (symbols).
  • Figure 4: Phase $\phi_j$ of physical field amplitudes: analytic solution \ref{['eq:u-v-quartic']} converted via \ref{['eq:u-v-A']} (dashed lines) compared with numerical integration (symbols).