Research Article

Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System

Volume: 16 Number: 3 September 8, 2026
TR EN

Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System

Abstract

This study investigates the symmetry structure, invariant solutions, and conservation properties of a nonlinear dispersive wave system describing coupled nonlinear interactions. The Lie group method is employed to determine the complete set of Lie point symmetries admitted by the governing equations. Based on the obtained symmetry algebra, the commutation relations and adjoint representation are constructed, which enable the derivation of optimal systems of one– and two–dimensional subalgebras. These optimal systems are used to obtain similarity reductions that transform the original nonlinear partial differential equations into ordinary differential equations. In particular, traveling–wave reductions are derived and integrated to obtain exact solitary wave solutions. Furthermore, conservation laws are systematically constructed using the multiplier method, yielding several conserved densities associated with mass, momentum, and energy invariants. To illustrate the physical behavior of the obtained solutions, numerical simulations are performed. Two-dimensional, three-dimensional, and contour plots of the solitary waves are presented.

Keywords

Lie point symmetries, Optimal system, Similarity reduction, Conserved Quantities, Multiplier

References

  1. Badshah, F., Tariq, K. U., Inc, M., & Zeeshan, M. (2024). On the solitonic structures for the fractional Schrödinger–Hirota equation. Optical and Quantum Electronics, 56(5), 848. https://doi.org/10.1007/s11082-024-06447-y
  2. Biswas, A. (2008). 1-soliton solution of the K (m, n) equation with generalized evolution. Physics Letters A, 372(25), 4601-4602
  3. Biswas, A., & Milovic, D. (2010). Bright and dark solitons of the generalized nonlinear Schrödinger’s equation. Communications in Nonlinear Science and Numerical Simulation, 15(6), 1473-1484.
  4. Bluman, G. W., & Anco, S. C. (2002). Symmetry and integration methods for differential equations. Springer. https://doi.org/10.1007/b97380
  5. Bogoyavlenskij, O. (2010). Restricted Lie point symmetries and reductions for ideal magnetohydrodynamics equilibria. Journal of Engineering Mathematics, 66(1), 141-152. https://doi.org/10.1007/s10665-009-9326-7
  6. Fan, E. G., & Yuen, M. (2014). Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations. Physics Letters A, 378(7-8), 623-626. https://doi.org/10.1016/j.physleta.2013.12.045
  7. Feireisl, E., Gwiazda, P., Swierczewska-Gwiazda, A., et al. (2017). Regularity and energy conservation for the compressible Euler equations. Archive for Rational Mechanics and Analysis, 223(3), 1375-1395. https://doi.org/10.1007/s00205-016-1060-5
  8. Gulsen, S., Hashemi, M. S., Alhefthi, R., Inc, M., & Bicer, H. (2023). Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients. Computational & Applied Mathematics. https://doi.org/10.1007/s40314-023-02358-y
  9. Hashemi, M. S., Haji-Badali, A., Alizadeh, F., & Inc, M. (2023). Classical and non-classical Lie symmetry analysis, conservation laws and exact solutions of the time-fractional Chen-Lee-Liu equation. Computational & Applied Mathematics. https://doi.org/10.1007/s40314-023-02217-w
  10. Hashemi, M. S., Haji-Badali, A., Alizadeh, F., & Yang, X.-J. (2022). Non-classical Lie symmetries for nonlinear time-fractional Heisenberg equations. Mathematical Methods in the Applied Sciences. https://doi.org/10.1002/mma.8353
APA
Biçer, H. (2026). Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System. Karadeniz Fen Bilimleri Dergisi, 16(3), 1383-1396. https://doi.org/10.31466/kfbd.1914276
AMA
1.Biçer H. Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System. KFBD. 2026;16(3):1383-1396. doi:10.31466/kfbd.1914276
Chicago
Biçer, Harun. 2026. “Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System”. Karadeniz Fen Bilimleri Dergisi 16 (3): 1383-96. https://doi.org/10.31466/kfbd.1914276.
EndNote
Biçer H (September 1, 2026) Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System. Karadeniz Fen Bilimleri Dergisi 16 3 1383–1396.
IEEE
[1]H. Biçer, “Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System”, KFBD, vol. 16, no. 3, pp. 1383–1396, Sept. 2026, doi: 10.31466/kfbd.1914276.
ISNAD
Biçer, Harun. “Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System”. Karadeniz Fen Bilimleri Dergisi 16/3 (September 1, 2026): 1383-1396. https://doi.org/10.31466/kfbd.1914276.
JAMA
1.Biçer H. Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System. KFBD. 2026;16:1383–1396.
MLA
Biçer, Harun. “Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System”. Karadeniz Fen Bilimleri Dergisi, vol. 16, no. 3, Sept. 2026, pp. 1383-96, doi:10.31466/kfbd.1914276.
Vancouver
1.Harun Biçer. Optimal Systems, and Solitary Wave Dynamics of a Nonlinear Dispersive Wave System. KFBD. 2026 Sep. 1;16(3):1383-96. doi:10.31466/kfbd.1914276