Unified symmetry classification, conservation laws and soliton dynamics for a class of variable-coefficient higher-order nonlinear evolution equations
Abstract
Variable-coefficient higher-order nonlinear evolution equations constitute an important class of mathematical models for the description of nonlinear dispersive wave propagation in inhomogeneous media, including shallow water flows, plasmas, optical systems and elastic structures. This paper develops a unified analytical framework for a broad family of such equations incorporating nonlinear convection, third-order dispersion, fifth-order dispersion, BBM-type mixed dispersion and nonlinear dispersive modulation. Admissible equivalence transformations are first constructed and used to reduce the general class, under non-vanishing fifth-order dispersion, to a canonical normalized form. The normalization yields a simplified group-classification problem by establishing an explicit compatibility condition for the temporal component of every admitted Lie point symmetry. Complete Lie symmetry classifications are obtained for the kernel, power-law, exponential and constant-coefficient subclasses, followed by the construction of optimal systems and similarity reductions. Conservation laws are derived from the conservative structure of the governing equation, including a universal mass invariant and a regularized energy conservation law for an important canonical subclass. Furthermore, an exact bright-soliton solution is derived for the constant-coefficient equation, rigorously verified by direct substitution, and employed to validate numerical simulations. The proposed framework unifies equivalence transformations, Lie symmetry analysis, conservation laws, similarity reductions and soliton dynamics within a single parameterized setting, providing a systematic methodology for the analytical investigation of higher-order nonlinear dispersive wave equations.
Downloads
References
P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer, New York, 1993).
G. W. Bluman, and S. Kumei, Symmetries and Differential Equations, (Springer, Berlin, 1989).
N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, (Reidel, Dordrecht, 1985).
L. V. Ovsiannikov, Group Analysis of Differential Equations, (Academic Press, New York, 1982).
E. Noether, Invariante Variationsprobleme, in: Nachrichten von der Gesellschaft der Wissenschaften zu G¨ottingen, Mathematisch-Physikalische Klasse, pp. 235–257 (1918).
R. Hirota, The Direct Method in Soliton Theory (Cambridge University Press, Cambridge, 2004).
M. J. Ablowitz, and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991).
H. Schamel, and S. Bujarbarua, ”Analytical double layers,” Phys. Fluids, 26, 190–193 (1983). https://doi.org/10.1063/1.864006
H. Schamel, ”Weak double layers: Existence, stability, evidence,” Z. Naturforsch. A, 38, 1170–1183 (1983). https://doi.org/10.1515/zna-1983-1102
H. Schamel, and V. I. Maslov, ”Adiabatic growth of electron holes in current-carrying plasmas,” Phys. Scr. T50, 42 (1994). https://doi.org/10.1088/0031-8949/1994/T50/006
V. I. Maslov, and H. Schamel, ”Growing electron holes in drifting plasmas,” Phys. Lett. A, 178, 171–174 (1993). https://doi.org/10.1016/0375-9601(93)90746-M
H. Schamel, and V. I. Maslov, ”Langmuir wave contraction caused by electron holes,” Phys. Scr. T82, 122 (1999). https://doi.org/10.1238/Physica.Topical.082a00122
V. I. Maslov, ”Electron beam excitation of a potential well in a magnetized plasma waveguide,” Phys. Lett. A, 165, 63–68 (1992). https://doi.org/10.1016/0375-9601(92)91055-V
A. P. M´arquez, M. L. Gandarias, and S. C. Anco, ”Conservation laws, symmetries, and line solitons of a Kawahara–KP equation,” Journal of Computational and Applied Mathematics, 436, 115412 (2023). https://doi.org/10.1016/j.cam.2023.115412
A.P. Marquez, T. M. Garrido, C. M. Khalique, andM. L. Gandarias, ”Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with p-power,” Discrete and Continuous Dynamical Systems Series S, 18, 1024–1035 (2025). https://doi.org/10.3934/dcdss.2024130
W. Hereman, ”Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions,” International Journal of Quantum Chemistry, 106, 278–299 (2006). https://doi.org/10.1002/qua.20727
F. Afzal, and A. A. Lupas, ”Lie Symmetry Analysis, Optimal Systems and Physical Interpretation of Solutions for the KdV-Burgers Equation,” Symmetry, 17, 1981 (2025). https://doi.org/10.3390/sym17111981
S. T. R. Rizvi, et al., ”Lie symmetry analysis, conservation laws and soliton solutions by complete discrimination system for polynomial approach of Landau Ginzburg Higgs equation along with its stability analysis,” Optik, 300, 171675 (2024). https://doi.org/10.1016/j.ijleo.2024.171675
A. Hussain, M. K. Zia, K. S. Nisar, V. Vijayakumar, and I. Khan, ”Lie analysis, conserved vectors, nonlinear self-adjoint classification and exact solutions of generalized (N + 1)-dimensional nonlinear Boussinesq equation,” AIMS Mathematics, 7, 13139–13168 (2022). https://doi.org/10.3934/math.2022725
S. Kumar, et al., ”Lie symmetries, modulation instability, conservation laws, and the dynamic waveform patterns of several invariant solutions to a (2+1)-dimensional Hirota bilinear equation,” Discrete and Continuous Dynamical Systems - Series S, 18, 1054–1074 (2025). https://doi.org/10.3934/dcdss.2024136
S. O. Mbusi, A. R. Adem, and B. Muatjetjeja, ”Lie symmetry analysis, multiple exp-function method and conservation laws for the (2+1)-dimensional Boussinesq equation,” Optical and Quantum Electronics, 56, 670 (2024). https://doi.org/10.1007/s11082-024-06339-1
F. S. Costa, J. C. A. Soares, J. V. C. Sousa, R. R. Luz, and J. A. R. Santos, ”Lie point symmetries and conservation laws to fractional ζ(t)-KdV equation,” International Journal of Theoretical Physics, 65, 11 (2026). https://doi.org/10.1007/s10773-025-06221-0
S. Miao, et al., ”An Analysis of the Lie Symmetry and Conservation Law of a Variable-Coefficient Generalized Calogero–Bogoyavlenskii–Schiff Equation in Nonlinear Optics and Plasma Physics,” Mathematics, 12, 3619 (2024). https://doi.org/10.3390/math12223619
M. Junaid-U-Rehman, G. Kudra, and J. Awrejcewicz, ”Conservation laws, solitary wave solutions, and Lie analysis for the nonlinear chains of atoms,” Scientific Reports, 13, 11537 (2023). https://doi.org/10.1038/s41598-023-38658-w
H. Almusawa, and A. Jhangeer, ”Nonlinear self-adjointness, conserved quantities and Lie symmetry of dust size distribution on a shock wave in quantum dusty plasma,” Communications in Nonlinear Science and Numerical Simulation, 114, 106660 (2022). https://doi.org/10.1016/j.cnsns.2022.106660
M. A. Kawser, M. A. Akbar, and M. A. Khan, ”Exploring variable coefficient models: Insights into nonlinear wave behavior and soliton solutions in physical systems,” Results in Physics, 56, 107242 (2024). https://doi.org/10.1016/j.rinp.2023.107242
H. Wang Z. Zou, and X. Shen, ”Lie symmetry analysis, self-adjointness and conservation law for a type of nonlinear equation,” Mathematics, 9, 1313 (2021). https://doi.org/10.3390/math9121313
E. Yas¸ar, and E. Yas¸ar, ”Exact wave solutions and local conservation laws of a third-order Boussinesq system,” Discrete and Continuous Dynamical Systems - Series S, 18, 1075–1092 (2025). https://doi.org/10.3934/dcdss.2024148
M. Kumar, S. Anand, and D. V. Tanwar, ”Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV–mKdV equation for internal gravity waves,” Mathematical Methods in the Applied Sciences, 47, 6909–6927 (2024). https://doi.org/10.1002/mma.9949
Copyright (c) 2026 Suares Clovis Oukouomi Noutchie
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
