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<p>This paper investigates a key class of generalized third-order partial differential equations, which are essential for modeling ultrashort pulse propagation in optical fibers. By employing the generalized trial equation method in conjunction with the complete discrimination system for polynomial, we derive eleven exact solutions. These include solitary wave solutions, discontinuous periodic wave solutions, rational function solutions, and double-periodic solutions based on Jacobi elliptic functions. These solutions provide a comprehensive framework for understanding nonlinear wave dynamics, particularly in optical fiber applications. The recent findings demonstrate that these methods can handle more complex nonlinear dynamics than traditional models. The new solutions not only aid in modeling pulse distortion issues in optical communication systems but also extend their applicability to fields such as plasma physics, fluid dynamics, and quantum communications. These solutions exhibit significant flexibility, capable of modeling both periodic and non-periodic wave phenomena, making them valuable tools for experimental physicists and engineers in the field of advanced nonlinear optics. Furthermore, advanced visualization techniques, including three-dimensional representations, have been employed to further validate these theoretical results, clearly illustrating the wave behavior under various physical conditions. This research contributes significantly to the theoretical understanding of nonlinear wave dynamics and provides valuable support for the practical application of these solutions.</p>

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Exact Solutions of the Generalized Third-Order Nonlinear Schrödinger Equation Using the Trial Equation Method and the Complete Discriminant System for Polynomial Method with Applications in Optical Fibers

  • 期刊论文

  • 发表日期：

    2025年3月15日

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  • Na Sun,
  • Yuxue Wang

摘要

This paper investigates a key class of generalized third-order partial differential equations, which are essential for modeling ultrashort pulse propagation in optical fibers. By employing the generalized trial equation method in conjunction with the complete discrimination system for polynomial, we derive eleven exact solutions. These include solitary wave solutions, discontinuous periodic wave solutions, rational function solutions, and double-periodic solutions based on Jacobi elliptic functions. These solutions provide a comprehensive framework for understanding nonlinear wave dynamics, particularly in optical fiber applications. The recent findings demonstrate that these methods can handle more complex nonlinear dynamics than traditional models. The new solutions not only aid in modeling pulse distortion issues in optical communication systems but also extend their applicability to fields such as plasma physics, fluid dynamics, and quantum communications. These solutions exhibit significant flexibility, capable of modeling both periodic and non-periodic wave phenomena, making them valuable tools for experimental physicists and engineers in the field of advanced nonlinear optics. Furthermore, advanced visualization techniques, including three-dimensional representations, have been employed to further validate these theoretical results, clearly illustrating the wave behavior under various physical conditions. This research contributes significantly to the theoretical understanding of nonlinear wave dynamics and provides valuable support for the practical application of these solutions.