Research Article

Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation

Number: 56 September 30, 2026

Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation

Abstract

In this study, a nonlinear evolution equation is used to model the dynamics of continuous wave motion and soliton bursts, mainly within the fields of fluid mechanics and wave dynamics. Dynamic analytical approaches, such as the enhanced modified extended $\tanh$-expansion method (EMETEM), the $(m+1/G')$-expansion method, and the new Kudryashov method, are used to obtain new soliton solutions, such as dark, bright, singular, periodic, and lump soliton solutions for the equation. An analysis of stability was conducted on several derived wave solutions, and the stability characteristics of these solutions were explored using the Hamiltonian criterion. The energy balance method (EBM) is employed to obtain an approximate periodic solution by using the Hamiltonian invariant to maintain the constancy of kinetic and potential energy. Ultimately, to explore the physical characteristics of several of the derived wave solutions, they have been represented through three-dimensional (3D), contour, and two-dimensional (2D) graphics at particular parameter values.

Keywords

Enhanced modified extended $\tanh$-expansion method, $(m+1/G')$-expansion method, new Kudryashov method, stability analysis, energy balance method

References

  1. D. Ghosh,Nonlinear evolution equations: A brief review, International Journal of Engineering and Applied Physics 5 (2) (2025) 1214–1222.
  2. A. Raza,Nonlinear dynamics in complex systems: A mathematical approach, Frontiers in Applied Physics and Mathematics 1 (1) (2024) 42–56.
  3. B. Mohan,Structural analysis of multi-dimensional pulse propagation of the KP-Boussinesq model in optical fibers, Modern Physics Letters B 40 (16) (2026) 2650118.
  4. M. Wadati,Introduction to solitons, Pramana 57 (5-6) (2001) 841–847.
  5. B. Mohan, S. Kumar,Painlevé analysis, restricted bright-dark N-solitons, and N-rogue waves of a $(4+1)$-dimensional variable-coefficient generalized KP equation in nonlinear sciences, Nonlinear Dynamics 113 (10) (2025) 11893–11906.
  6. B. Mohan, S. Kumar, R. Kumar,Higher-order rogue waves and dispersive solitons of a novel P-type $(3+1)$-D evolution equation in soliton theory and nonlinear waves, Nonlinear Dynamics 111 (21) (2023) 20275–20288.
  7. S. Kumar, S. K. Dhiman,Lie symmetry analysis, optimal system, exact solutions and dynamics of solitons of a $(3+1)$-dimensional generalized BKP–Boussinesq equation, Pramana 96 (1) (2022) 31.
  8. T. Mathanaranjan, R. Myrzakulov, Integrable Akbota equation: Conservation laws, optical soliton solutions and stability analysis, Optical and Quantum Electronics 56 (4) (2024) 564.
  9. Y.-H. Yin, X. Lü, W.-X. Ma,Bäcklund transformation, exact solutions and diverse interaction phenomena to a $(3+1)$-dimensional nonlinear evolution equation, Nonlinear Dynamics 108 (4) (2022) 4181–4194.
  10. B. Mohan, S. Kumar, Generalization and analytic exploration of soliton solutions for nonlinear evolution equations via a novel symbolic approach in fluids and nonlinear sciences, Chinese Journal of Physics 92 (2024) 10–21.
APA
Demirbilek, U., & Kızıl, İ. (2026). Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation. Journal of New Theory, 56, 43-61. https://doi.org/10.53570/jnt.1992043
AMA
1.Demirbilek U, Kızıl İ. Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation. JNT. 2026;(56):43-61. doi:10.53570/jnt.1992043
Chicago
Demirbilek, Ulviye, and İlknur Kızıl. 2026. “Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation”. Journal of New Theory, nos. 56: 43-61. https://doi.org/10.53570/jnt.1992043.
EndNote
Demirbilek U, Kızıl İ (September 1, 2026) Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation. Journal of New Theory 56 43–61.
IEEE
[1]U. Demirbilek and İ. Kızıl, “Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation”, JNT, no. 56, pp. 43–61, Sept. 2026, doi: 10.53570/jnt.1992043.
ISNAD
Demirbilek, Ulviye - Kızıl, İlknur. “Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation”. Journal of New Theory. 56 (September 1, 2026): 43-61. https://doi.org/10.53570/jnt.1992043.
JAMA
1.Demirbilek U, Kızıl İ. Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation. JNT. 2026;:43–61.
MLA
Demirbilek, Ulviye, and İlknur Kızıl. “Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation”. Journal of New Theory, no. 56, Sept. 2026, pp. 43-61, doi:10.53570/jnt.1992043.
Vancouver
1.Ulviye Demirbilek, İlknur Kızıl. Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation. JNT. 2026 Sep. 1;(56):43-61. doi:10.53570/jnt.1992043