<p>A pair of coupled nonlinear Schrödinger equations governs optical wave propagation in birefringent fibers, the most important model for nonlinear fiber optics that accounts for complex polarization-dependent pulse dynamics in combination with cross-phase modulation effects that single-component models do not consider. Despite the extensive work on this system previously, the systematic graphical approach in the literature remains a significant absence to approach a comprehensive analytical strategy combining various integration architectures on the conformable derivative scheme and presenting a continuous graphical picture that shows wave morphologies. This study further analytically analyzes the coupled system using a variety of complementary exact integration schemes derived from which we can derive the rich spectrum of the precise traveling wave solutions such as bright, dark, periodic or mixed-type wave structures. The physical behaviors of each solution family are explored, shown using two-dimensional and three-dimensional imaging juxtaposed with polar models to demonstrate how amplitude, phase velocity and localization are sensitively dependent on system parameters. The key characteristic of this work is the access to the conformable fractional derivative, whose order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tau \in (0,1]\)</EquationSource> </InlineEquation> gives a physically relevant nonlocality: when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tau \)</EquationSource> </InlineEquation> departs from unity, the wave profiles demonstrate visible broadening, modulation of amplitude (differences between the positive and negative waveforms), and propagation behavior change toward diffraction as compared to the classical integer-order version a reflection of the memory-like effects of the conformable operator on the wave dynamics. Although much of the work is carried out analytically, it still provides a novel insight into how to treat the model with the same approach based on a conformable derivative; the variety of recovered exact soliton structures; and how fractional-order effects can affect wave behaviours in a materially meaningful way. The current results have direct and immediate relevance for the design and analysis of optical communication systems, including mitigation of Internet bottleneck effects through improved pulse stability and spectral efficiency, suppression of four-wave mixing (FWM) and six-wave mixing (SWM) interactions in wavelength-division multiplexed (WDM) links, reduction of soliton radiation and dispersive energy leakage, and stabilization against soliton turbulence. The conformable fractional parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tau \)</EquationSource> </InlineEquation> provides a physically meaningful, continuously tunable degree of nonlocality that governs memory-like effects absent from integer-order models, rendering these results a novel set of analytic benchmarks for theoretical investigations and numerical solvers of nonlinear wave physics in birefringent media.</p>

Exact traveling wave and soliton solutions of optical solitons in birefringent fibers for two coupled nonlinear Schrödinger equations

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发表日期：

2026年6月18日

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Salim S. Mahmood,
Salisu Ibrahim,
Mohammed R. Ahmed,
N. Hemnath,
Sandip Saha

摘要

A pair of coupled nonlinear Schrödinger equations governs optical wave propagation in birefringent fibers, the most important model for nonlinear fiber optics that accounts for complex polarization-dependent pulse dynamics in combination with cross-phase modulation effects that single-component models do not consider. Despite the extensive work on this system previously, the systematic graphical approach in the literature remains a significant absence to approach a comprehensive analytical strategy combining various integration architectures on the conformable derivative scheme and presenting a continuous graphical picture that shows wave morphologies. This study further analytically analyzes the coupled system using a variety of complementary exact integration schemes derived from which we can derive the rich spectrum of the precise traveling wave solutions such as bright, dark, periodic or mixed-type wave structures. The physical behaviors of each solution family are explored, shown using two-dimensional and three-dimensional imaging juxtaposed with polar models to demonstrate how amplitude, phase velocity and localization are sensitively dependent on system parameters. The key characteristic of this work is the access to the conformable fractional derivative, whose order \(\tau \in (0,1]\) gives a physically relevant nonlocality: when \(\tau \) departs from unity, the wave profiles demonstrate visible broadening, modulation of amplitude (differences between the positive and negative waveforms), and propagation behavior change toward diffraction as compared to the classical integer-order version a reflection of the memory-like effects of the conformable operator on the wave dynamics. Although much of the work is carried out analytically, it still provides a novel insight into how to treat the model with the same approach based on a conformable derivative; the variety of recovered exact soliton structures; and how fractional-order effects can affect wave behaviours in a materially meaningful way. The current results have direct and immediate relevance for the design and analysis of optical communication systems, including mitigation of Internet bottleneck effects through improved pulse stability and spectral efficiency, suppression of four-wave mixing (FWM) and six-wave mixing (SWM) interactions in wavelength-division multiplexed (WDM) links, reduction of soliton radiation and dispersive energy leakage, and stabilization against soliton turbulence. The conformable fractional parameter \(\tau \) provides a physically meaningful, continuously tunable degree of nonlocality that governs memory-like effects absent from integer-order models, rendering these results a novel set of analytic benchmarks for theoretical investigations and numerical solvers of nonlinear wave physics in birefringent media.