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<p>This research is focused upon a complex Ginzburg–Landau equation with additional nonlocal nonlinearities of higher order and a Kudryashov-type nonlinear refractive index law. This fully nonlocal model accounts for nonlinear chromatic dispersion and refractive index modulation in a consistent framework for describing stationary optical solitons in inhomogeneous nonlinear optical fibers and waveguides. Using a traveling-wave transformation, the original partial differential equation is converted into a singular planar system that is later regularized for a comprehensive qualitative investigation. The singular planar system is shown to possess a Hamiltonian properties, under certain parameter constraints, which allows the equilibrium points to be defined with bifurcation theory and the Jacobian determinant. The distinct phase portraits for the equilibrium behaviors centers, saddles, and cusps are described for the different discriminant regimes. <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Delta &gt; 0\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Delta &lt; 0\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Delta = 0\)</EquationSource> </InlineEquation>, and are constructed in the (Ω,&#xa0;<i>y</i>) coordinate system. The unified Riccati equation expansion method is employed to derive exact analytical solutions of hyperbolic, trigonometric, and rational types. This formally treats the solution of nonlinear wave propagation in complex dispersive media and substantially enhances the solution approaches of nonlocal nonlinear Schrödinger-type systems.</p>

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Bifurcation analysis and optical solitons of the complex Ginzburg–Landau equation

  • 期刊论文

  • 发表日期：

    2026年5月12日

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  • Hezha Hussni Abdulkareem,
  • Hajar Farhan Ismael

摘要

This research is focused upon a complex Ginzburg–Landau equation with additional nonlocal nonlinearities of higher order and a Kudryashov-type nonlinear refractive index law. This fully nonlocal model accounts for nonlinear chromatic dispersion and refractive index modulation in a consistent framework for describing stationary optical solitons in inhomogeneous nonlinear optical fibers and waveguides. Using a traveling-wave transformation, the original partial differential equation is converted into a singular planar system that is later regularized for a comprehensive qualitative investigation. The singular planar system is shown to possess a Hamiltonian properties, under certain parameter constraints, which allows the equilibrium points to be defined with bifurcation theory and the Jacobian determinant. The distinct phase portraits for the equilibrium behaviors centers, saddles, and cusps are described for the different discriminant regimes. \(\Delta > 0\) , \(\Delta < 0\) , and \(\Delta = 0\) , and are constructed in the (Ω, y) coordinate system. The unified Riccati equation expansion method is employed to derive exact analytical solutions of hyperbolic, trigonometric, and rational types. This formally treats the solution of nonlinear wave propagation in complex dispersive media and substantially enhances the solution approaches of nonlocal nonlinear Schrödinger-type systems.