Generalized Trial Equation Scheme and EnhanceModified Extended Tanh Function Method with Nonparaxial Pulse Propagation to the Cubic–Quintic Nonlinear Helmholtz Equation

• Tang Nan ^[1] ; Jalil Manafian ^[2] ; Onur Alp Ilhan ^[3] ; Somaye Malmir ^[4] ; A. A. Fattah ^[5] ; K. H. Mahmoud ^[5] ; A. Majeed Ahmed ^[6] ; Y. Mohammed Nasr ^[7]
  1. [1] Anhui University of Science and Technology

     Anhui University of Science and Technology

     China

     
  2. [2] Lankaran State University

     Lankaran State University

     Sri Lanka

     
  3. [3] Erciyes University

     Erciyes University

     Turquía

     
  4. [4] Payame Noor University

     Payame Noor University

     Irán

     
  5. [5] College of Khurma University College, Taif University
  6. [6] Al-Mustaqbal University
  7. [7] Mazaya University College

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, some optical soliton solutions for the cubic-quintic nonlinear Helmholtz equation are extracted by utilizing two effective analytical techniques, i.e the generalized trial equation scheme (GTES) and enhance modified extended tanh function method (eMETFM). The governing equation undergoes transformation into an ordinary differential equation (ODE) through a well-suited wave transformation. The achieved solutions are structured in the forms of the exponential, trigonometric, hyperbolic, and Jacobi elliptic function solutions. Different soliton solutions like, dark, bright, singular bright, singular periodic, W-M-shaped soliton solutions are obtained.

    Additionally, the reliability and effectiveness of the applied methodologies, some of the achieved solutions are visualized in 2D, 3D and density representation by choosing well parameters values. In particular, four forms of solution functions including soliton, bright soliton, singular soliton, periodic wave solutions are investigated. To achieve this, an illustrative example of the Helmholtz equation is provided to demonstrate the feasibility and reliability of the procedure used in this study. A multiplier technique is then used to derive conservation laws. The effect of the free parameters on the behavior of acquired figures using the obtained solutions to investigate the proposed model also analyzed due to the nature of nonlinearities. Moreover, some selected solutions are illustrated graphically to show the physical nature of obtained solutions. These techniques offer a strong foundation for resolving nonlinear partial differential equations, which are crucial for simulating intricate cubic–quintic physical processes. This model may be significant in the investigate of the propagation of ultrashort optical pulses in the non-paraxial domain.

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