<p>In this paper, we study a three-coupled variable-coefficient nonlinear Schrödinger system, crucial for picosecond pulse propagation in inhomogeneous optical fibers and modern communication advancements. Using the Riemann-Hilbert method, we analyze this system based on a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$4 \times 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> matrix spectral problem from its Lax pair. After constructing relevant analytic functions and solving the Riemann-Hilbert problem, we obtain the N-soliton solutions under non-reflective boundary conditions. Subsequently, we analyze the dynamical behavior of multi-soliton solutions, revealing a series of physically meaningful phenomena like elastic collisions, soliton reflection, and picosecond pulse propagation patterns. Additionally, we investigate how the nonlinear coefficient <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the group velocity dispersion coefficient <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> affect soliton solutions, clarifying their role in governing soliton trajectories and amplitude. This research provides essential theoretical support for precisely controlling picosecond pulses in heterogeneous optical fibers. This work enriches the soliton dynamics theory of the aforementioned system, provides critical guidance for multi-component optical fiber systems, and offers further theoretical support for advancing high-performance optical fiber communication technologies.</p>

N-Soliton Solutions of the Three-Coupled Nonlinear Schrödinger Equations with Variable Coefficients in Optical Fibers

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发表日期：

2026年3月13日

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Lingling Zhang,
Meng Zhang

摘要

In this paper, we study a three-coupled variable-coefficient nonlinear Schrödinger system, crucial for picosecond pulse propagation in inhomogeneous optical fibers and modern communication advancements. Using the Riemann-Hilbert method, we analyze this system based on a $4 \times 4$ $4 \times 4$ matrix spectral problem from its Lax pair. After constructing relevant analytic functions and solving the Riemann-Hilbert problem, we obtain the N-soliton solutions under non-reflective boundary conditions. Subsequently, we analyze the dynamical behavior of multi-soliton solutions, revealing a series of physically meaningful phenomena like elastic collisions, soliton reflection, and picosecond pulse propagation patterns. Additionally, we investigate how the nonlinear coefficient $\gamma (z)$ $γ (z)$ and the group velocity dispersion coefficient $\beta (z)$ $β (z)$ affect soliton solutions, clarifying their role in governing soliton trajectories and amplitude. This research provides essential theoretical support for precisely controlling picosecond pulses in heterogeneous optical fibers. This work enriches the soliton dynamics theory of the aforementioned system, provides critical guidance for multi-component optical fiber systems, and offers further theoretical support for advancing high-performance optical fiber communication technologies.