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Advanced soliton structures and elliptic wave patterns in a sixth-order nonlinear Schrödinger equation using improved
Mina M Fahim1,2, Hamdy M Ahmed3, K A Dib4
1Basic Science Department, Faculty of Engineering, The British University in Egypt, El shorouk, Cairo, Egypt. Mina.fahim@bue.edu.eg.
Scientific Reports
|December 17, 2025
Summary
This study extends the nonlinear Schrödinger equation (NLSE) to sixth-order, revealing new soliton and wave structures. These findings enhance understanding of nonlinear dynamics in optical fiber and waveguide systems.
Area of Science:
- Nonlinear physics
- Optics
- Wave propagation
Background:
- The nonlinear Schrödinger equation (NLSE) is crucial for modeling wave phenomena.
- Higher-order nonlinear and dispersive effects are vital in advanced optical systems.
- Existing NLSE models may not capture complex dynamics fully.
Purpose of the Study:
- To investigate a sixth-order integrable extension of the NLSE.
- To model higher-order nonlinear and dispersive effects in optical systems.
- To uncover novel analytical solutions and understand nonlinear wave behavior.
Main Methods:
- Utilized the Improved Modified Extended Tanh Function Method.
- Derived exact analytical solutions for the sixth-order NLSE.
- Employed two- and three-dimensional graphical simulations for analysis.
Main Results:
- Obtained a comprehensive family of exact analytical solutions.
- Discovered bright solitons, dark solitons, singular solitons, and singular periodic solutions.
- Identified new soliton and elliptic wave structures, including Jacobi and Weierstrass functions.
Conclusions:
- The sixth-order NLSE extension reveals rich nonlinear dynamics.
- Clarified the transition between periodic and localized wave behaviors.
- Findings have potential implications for ultrafast optical and nonlinear waveguide technologies.
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