Bifurcation Analysis of Cubic Type Nonlinear Schrödinger Equation with Dispersion and Attenuation in Optical Fiber

Saputri, Rika (Mathematics Graduated Program, Universitas Syiah Kuala, Banda Aceh 23111, Indonesia)
Ramli, Marwan (Department of Mathematics, Universitas Syiah Kuala, Banda Aceh 23111, Indonesia)
Ikhwan, Muhammad (Department of Mathematics, Universitas Syiah Kuala, Banda Aceh 23111, Indonesia)
Mardi, Harish Abdillah (Department of Mathematics, Universitas Syiah Kuala, Banda Aceh 23111, Indonesia)

Zero : Jurnal Sains, Matematika, dan Terapan
Vol 9, No 2 (2025): Zero: Jurnal Sains Matematika dan Terapan

This study investigates the bifurcation of the cubic Nonlinear Schrödinger Equation (NLSE) in optical fiber media by considering dispersion and attenuation effects. The NLSE models light pulse propagation, with γ representing the strength of nonlinearity. Analytical derivations yield stationary solutions, while numerical simulations using the Newton–Raphson method and eigenvalue analysis verify stability. Results show a critical bifurcation at γ=0: for γ ≤ 0, the system exhibits one unstable fixed point, whereas for γ > 0, three fixed points appear, with simulations confirming that only the two nontrivial branches are stable. This corresponds to a pitchfork bifurcation and a stability transition governed primarily by nonlinearity. Although attenuation is included in the model, its contribution is negligible, indicating that bifurcation behavior is dominated by γ. Compared with previous studies focusing on dispersion-only NLSE or fractional/extended models, this work highlights the decisive role of nonlinearity in determining fixed points and stability in optical fibers.