Research Article

Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation

Number: 56 September 30, 2026

Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation

Abstract

The Wazwaz–Kaur equation is a significant nonlinear partial differential equation. This equation is used to describe nonlinear wave propagation in fields including fluid dynamics, plasma physics, and other areas of mathematical physics. In this study, we investigate analytical solutions of the $(3+1)$-dimensional Wazwaz–Kaur Boussinesq Equation (WKBE). These solutions play a significant role in describing complex wave structures and in understanding the underlying physical mechanisms. For the construction of exact analytical solutions, we utilize the Kumar–Malik Method (KMM), which offers a systematic procedure for addressing nonlinear partial differential equations. Using this approach, a wide class of exact solutions is obtained, including Jacobi elliptic solutions as well as their hyperbolic, trigonometric, and exponential limiting forms. All solutions are validated through symbolic computations performed in Maple Software. The obtained solutions are visualized through graphical representations to demonstrate how different physical parameters affect the wave profiles. These findings are of great use in understanding the nonlinear behavior of the $(3+1)$-dimensional WKBE and in extending it to a wider range of physical systems with multidimensional wave interactions. Overall, the results of this research confirm the effectiveness of the KMM and introduce new exact solutions that may provide a useful reference for future analytical and numerical studies of nonlinear evolution equations.

Keywords

Kumar-Malik method, analytical solutions, nonlinear partial differential equations, $(3+1)$-dimensional Wazwaz-Kaur Boussinesq equation

References

  1. H. Liu, C.-L. Bai, X. Xin, Improved equivalent transformation method for reduction NLPDEs with time-dependent variables, Applied Mathematics Letters 120 (2021) 107290.
  2. S. R. Choudhury, Solitary wave families of NLPDES via reversible systems theory, Mathematics and Computers in Simulation 80 (1) (2009) 37–45.
  3. B. Abraham-Shrauner, Exact solutions of nonlinear partial differential equations, Discrete and Continuous Dynamical Systems - Series S 11 (4) (2018) 577–582.
  4. T. A. Sulaiman, U. Younas, M. Younis, J. Ahmad, S. U. Rehman, M. Bilal, A. Yusuf, Modulation instability analysis, optical solitons and other solutions to the $(2+1)$-dimensional hyperbolic nonlinear Schrodinger's equation, Computational Methods for Differential Equations 10 (1) (2022) 179–190.
  5. U. Demirbilek, S. A. Baloch, I. Siddique, H. Bulut, T. Radwan, Exploring lump solutions, bifurcation, sensitivity, and chaotic dynamics in the extended KP-Boussinesq equation, Journal of Nonlinear Mathematical Physics 32 (1) (2025) 49.
  6. M. Shakeel, Attaullah, N. A. Shah, J. D. Chung, Application of modified exp-function method for strain wave equation for finding analytical solutions, Ain Shams Engineering Journal 14 (3) (2023) 101883.
  7. S. Zhang, Exp-function method for constructing explicit and exact solutions of a lattice equation, Applied Mathematics and Computation 199 (1) (2008) 242–249.
  8. X.-W. Zhou, Exp-function method for solving Huxley equation}, Mathematical Problems in Engineering 2008 (1) (2008) 538489.
  9. A.-M. Wazwaz, The} tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Applied Mathematics and Computation 188 (2) (2007) 1467–1475.
  10. F. M. Al-Askar, W. W. Mohammed, A. M. Albalahi, M. El-Morshedy, The impact of the Wiener process on the analytical solutions of the stochastic $(2+1)$-dimensional breaking soliton equation by using tanh-coth method, Mathematics 10 (5) (2022) 817.
APA
Duka, M., & Akbulut, A. (2026). Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation. Journal of New Theory, 56, 23-42. https://doi.org/10.53570/jnt.1953752
AMA
1.Duka M, Akbulut A. Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation. JNT. 2026;(56):23-42. doi:10.53570/jnt.1953752
Chicago
Duka, Mersin, and Arzu Akbulut. 2026. “Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation”. Journal of New Theory, nos. 56: 23-42. https://doi.org/10.53570/jnt.1953752.
EndNote
Duka M, Akbulut A (September 1, 2026) Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation. Journal of New Theory 56 23–42.
IEEE
[1]M. Duka and A. Akbulut, “Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation”, JNT, no. 56, pp. 23–42, Sept. 2026, doi: 10.53570/jnt.1953752.
ISNAD
Duka, Mersin - Akbulut, Arzu. “Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation”. Journal of New Theory. 56 (September 1, 2026): 23-42. https://doi.org/10.53570/jnt.1953752.
JAMA
1.Duka M, Akbulut A. Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation. JNT. 2026;:23–42.
MLA
Duka, Mersin, and Arzu Akbulut. “Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation”. Journal of New Theory, no. 56, Sept. 2026, pp. 23-42, doi:10.53570/jnt.1953752.
Vancouver
1.Mersin Duka, Arzu Akbulut. Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation. JNT. 2026 Sep. 1;(56):23-42. doi:10.53570/jnt.1953752