Approximate symmetries of the perturbed KdV-KS equation

Iranian Journal of Numerical Analysis and Optimization

مقاله 3، دوره 15، Issue 3 - شماره پیاپی 34، آذر 2025، صفحه 914-929   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.90854.1551

نویسندگان

A. Mohammadpouri^{* 1}؛ M.S. Hashemi²؛ R. Abbasi¹؛ R. Abbasi¹

¹Faculty of Mathematics, Statistics and Computer Sciences, University of Tabriz, Tabriz, Iran.

²Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran.

چکیده

The analysis of approximate symmetries in perturbed nonlinear partial differential equations $(PDEs)$ stands as a cornerstone for unraveling  complex physical behaviors and solution patterns. This paper delves into the investigation of approximate symmetries inherent in the perturbed Korteweg-de Vries and Kuramoto-Sivashinsky $(KdV-KS)$ equation, fundamental models in the realm of fluid dynamics and wave phenomena. Our study commences by detailing the method to derive approximate vector Lie symmetry generators that underpin the approximate symmetries of the perturbed $KdV-KS$ equation. These generators, while not exact, provide invaluable insights into the equation’s dynamics and solution characteristics under perturbations. A comprehensive approximate commutator table is subsequently constructed, elucidating the relationships and interplay between these approximate symmetries and shedding light on their algebraic structure. Leveraging the power of the adjoint representation, we examine the stability of these approximate symmetries when subjected to perturbations. This analysis enables us to discern the most resilient symmetries, instrumental in identifying intrinsic features that persist even in the face of disturbances. Furthermore, we harness the concept of approximate symmetry reductions, a pioneering technique that allows us to distill crucial dynamics from the complexity of the perturbed equation. Through this methodology, we uncover invariant solutions and reduced equations that serve as effective surrogates for the original system, capturing its essential behavior and facilitating analytical and numerical investigations. In summary, our exploration into the approximate symmetries of the perturbed $KdV-KS$ equation not only advances our comprehension of the equation’s intricate dynamics but also offers a comprehensive framework for studying the impact of perturbations on approximate symmetries, all while opening new avenues for tackling nonlinear $PDEs$ in diverse scientific disciplines.

کلیدواژه‌ها

Approximate Lie symmetries؛ Commutator table؛ Adjoint representation؛ Reductions

مراجع

[1] Abbagari, S., Houwe, A., Saliou, Y., Douvagaï, D., Chu, Y.M., Inc, M., Rezazadeh, H. and Doka, S.Y. Analytical survey of the predator– prey model with fractional derivative order, AIP Advances, 11 (3) (2021) 035127.
[2] Akbulut, A.R.Z.U., Mirzazadeh, M., Hashemi, M.S., Hosseini, K., Salahshour, S. and Park, C. Triki–biswas model: Its symmetry reduction, Nucci’s reduction and conservation laws, Int. J. Mod. Phys. B. 37 (07) (2023) 2350063.
[3] Arnold, L., Jones, C.K., Mischaikow, K., Raugel, G. and Jones, C.K.,Geometric singular perturbation theory, Dynamical Systems: Lec-tures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13–22, 1994 (1995): 44–118.

[4] Cheng, X., Hou, J. and Wang, L. Lie symmetry analysis, invariant sub-space method and q-homotopy analysis method for solving fractional system of single-walled carbon nanotube, Comput. Appl. Math. 40 (2021) 1–17.
[5] Chentouf, B. and Guesmia, A. Well-posedness and stability results for the Korteweg–de vries–Burgers and Kuramoto–Sivashinsky equations with infinite memory: A history approach, Nonlinear Analysis: Real World Applications 65 (2022) 103508.
[6] Chu, Y., Khater, M.M. and Hamed, Y. Diverse novel analytical and semi-analytical wave solutions of the generalized (2+ 1)-dimensional shallow water waves model, AIP Advances ,11 (1) (2021) 015223.
[7] Chu, Y., Shallal, M.A., Mirhosseini-Alizamini, S.M., Rezazadeh, H., Javeed, S. and Baleanu, D. Application of modified extended tanh tech-nique for solving complex ginzburg-landau equation considering Kerr law nonlinearity, Comput. Mater. Contin. 66 (2) (2021) 1369–1377.
[8] Chu, Y.-M., Inc, M., Hashemi, M.S., and Eshaghi, S. Analytical treatment of regularized prabhakar fractional differential equations by invariant sub-spaces, Comput. Appl. Math. 41 (6) (2022) 271.
[9] Cinar, M., Secer, A., Ozisik, M. and Bayram, M. Optical soliton solutions of (1+ 1)-and (2+ 1)-dimensional generalized Sasa–Satsuma equations using new kudryashov method, Int. J. Geom. Methods Mod. Phys. 20 (02) (2023) 2350034.
[10] Du, Z. and Li, J. Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation, J. Differ. Equ.306 (2022) 418–438. 
[11] Euler, N., Shul’ga, M.W. and Steeb, W.-H. Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation, J. Phys. A Math. Gen. 25 (18) (1992) L1095.
[12] Fan, X.and Tian, L. The existence of solitary waves of singularly per-turbed MKdv–KS equation, Chaos Solit. Fract. 26 (4) (2005) 1111–1118.

[13] Grebenev, V. and Oberlack, M. Approximate lie symmetries of the Navier-stokes equations, J. Nonlinear Math. Phys. 14 (2) (2007) 157–163.
[14] Hashemi, M.S. and Mirzazadeh, M. Optical solitons of the perturbed nonlinear Schrödinger equation using lie symmetry method, Optik 281 (2023) 170816.
[15] Kara, A., Mahomed, F. and Unal, G. Approximate symmetries and con-servation laws with applications, Int. J. Theor. Phys. 38 (9) (1999) 2389–2399.
[16] Malik, S., Hashemi, M.S., Kumar, S., Rezazadeh, H., Mahmoud, W. and Osman, M. Application of new Kudryashov method to various nonlinear partial differential equations, Opt. Quantum Electron. 55 (1) (2023) 8.
[17] Mohammadpouri, A., Hasannejad, S. and Haji Badali, A. The study of maximal surfaces by Lie symmetry, Comput. Method. Differ. Equ. (2024) 1–8.
[18] Mohammadpouri, A., Hashemi, M.S. and Samaei, S. Noether symmetries and isometries of the area-minimizing Lagrangian on vacuum classes of pp-waves, Eur. Phys. J. Plus, 138 (2) (2023) 1–7.
[19] Mohammadpouri, A., Hashemi, M.S., Samaei, S. and Salar Anvar, S. Symmetries of the minimal Lagrangian hypersurfaces on cylindrically symmetric static space-times, Comput. Method. Differ. Equ. 13(1) (2025) 249–257.
[20] Ozisik, M., Secer, A., Bayram, M., Sulaiman, T.A. and Yusuf, A. Ac-quiring the solitons of inhomogeneous Murnaghan’s rod using extended Kudryashov method with Bernoulli–Riccati approach, Int. J. Modern Phys. B 36 (30) (2022) 2250221.
[21] Prakash, P., Priyendhu, K. and Lakshmanan, M. Invariant subspace method for (m+ 1)-dimensional non-linear time-fractional partial differential equations, Commun. Nonlinear Sci. Numer. Simul. 111 (2022) 106436.

[22] Sahoo, S., Saha Ray, S., Abdou, M.A.M., Inc, M. and Chu, Y.M. New soliton solutions of fractional Jaulent-Miodek system with symmetry analysis, Symmetry 12 (6) (2020) 1001.
[23] Triki, H., Mirzazadeh, M., Ahmed, H.M., Samir, I. and Hashemi, M.S., Higher-order Sasa–Satsuma equation: Nucci’s reduction and soliton solutions, Eur. Phys. J. Plus, 138 (5) (2023) 1–10.
[24] Xia, F.L., Jarad, F., Hashemi, M.S. and Riaz, M.B. A reduction technique to solve the generalized nonlinear dispersive mk (m, n) equation with new local derivative, Results Phys. 38 (2022) 105512.
[25] Yen, T.C., Lang, R.A. and Izmaylov, A.F. Exact and approximate symmetry projectors for the electronic structure problem on a quantum computer, J. Chem. Phys. 151 (16) (2019) 164111 .