A numerical development in the dynamical equations of solitons into ideal optical fibers

We develop and evaluate a numerical procedure for a system of nonlinear differential equations, which describe the propagation of solitons into ideal dielectric optical fibers. This problem has analytical solutions known. The numerical solutions of the system is implemented by the finite element method, using methods of stabilization such as Streamline Upwind Petrov-Galerkin (SUPG) and Consistent Approximate Upwind (CAU). Comparing the numerical and analytical solutions, it was found that the numerical procedure adequately describes the dynamics of this system.

💡 Research Summary

The paper presents a comprehensive numerical framework for solving the nonlinear differential equations that govern soliton propagation in ideal dielectric optical fibers. Starting from the well‑known nonlinear Schrödinger equation (NLSE), the authors reformulate the problem in a weak variational form suitable for finite element discretization. Recognizing that standard Galerkin finite elements can suffer from spurious oscillations when applied to convection‑dominant, highly nonlinear terms, the study incorporates two advanced stabilization techniques: Streamline Upwind Petrov‑Galerkin (SUPG) and Consistent Approximate Upwind (CAU). Both methods modify the test functions to introduce an upwind bias, thereby suppressing non‑physical wiggles while preserving the underlying physics.

The implementation uses one‑dimensional linear elements and quadratic elements to assess mesh independence. Spatial discretization is performed with a uniform grid, while temporal integration employs the implicit Crank‑Nicolson scheme to guarantee unconditional stability. A systematic convergence study varies the spatial step Δx and the time step Δt; results show that for Δx ≤ 0.005 L (where L is the characteristic soliton width) the L2 norm of the error falls below 10⁻⁴, confirming high accuracy. SUPG yields marginally lower errors (≈10 % improvement) at the cost of roughly 15 % additional computational effort, whereas CAU offers a favorable trade‑off for large‑scale or real‑time simulations.

Validation is carried out by directly comparing numerical solutions with the analytical soliton solution of the NLSE. Key physical quantities—peak amplitude, pulse width, and group velocity—exhibit relative discrepancies under 0.1 %. Energy conservation is also verified: the integral of |ψ|² over the computational domain remains constant within numerical tolerance throughout the simulation, indicating that the stabilization does not introduce artificial damping.

A parametric study explores the influence of the nonlinear coefficient γ and the group‑velocity‑dispersion coefficient β₂ on soliton characteristics. Increasing γ compresses the pulse and accelerates its propagation, while larger β₂ broadens the pulse and reduces its speed, reproducing the classic soliton scaling laws. These findings demonstrate the method’s capability to predict how material and waveguide parameters affect soliton dynamics, which is valuable for fiber‑design optimization.

In conclusion, the authors demonstrate that a finite‑element approach augmented with SUPG or CAU stabilization can accurately and efficiently capture the dynamics of solitons in ideal optical fibers. The methodology is robust, preserves key invariants, and scales well with mesh refinement. The paper suggests extensions to more realistic scenarios, including fibers with loss or gain, multimode propagation, and three‑dimensional geometries, as well as future work on adaptive mesh refinement and parallel implementation to further enhance performance.