A finite generating set of differential invariants for Lie symmetry group of the fifth-order KdV types

Haghighatdoost, Ghorbanali; Bazghandi, Mostafa; Pashaei, Firooz

doi:10.22034/cmde.2023.53967.2262

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	A finite generating set of differential invariants for Lie symmetry group of the fifth-order KdV types
Computational Methods for Differential Equations
مقاله 10، دوره 11، شماره 4، مهر 2023، صفحه 803-810 اصل مقاله (299.96 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2023.53967.2262
نویسندگان
Ghorbanali Haghighatdoost^* ¹؛ Mostafa Bazghandi¹؛ Firooz Pashaei²
¹Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.
²Department of Mathematics, University of Maragheh, P.O.Box 55181-83111, Maragheh, Iran.
چکیده
In this paper, we study the algebraic structure of differential invariants of a fifth-order KdV equation, known as the Kawahara KdV equation. Using the moving frames method, we locate a finite generating set of differential invariants, recurrence relations, and syzygies among the differential invariants generators of the equation. We prove that the differential invariant algebra of the equation can be generated by two first-order differential invariants.
کلیدواژه‌ها
Differential invariants؛ Moving frames؛ KdV equations؛ Kawaharara equation

مراجع
[1] A. Akbulut, M. Kaplan, and M. K. A. Kaabar, New conservation laws and exact solutions of the special case of the fifth-order KdV equation, J. Ocean Eng. Sci., 7(4) (2022), 377–382. [2] N. H. Aljahdaly, et al. The analysis of the fractional-order system of third-order KdV equation within different operators, Alex. Eng. J., 61(12) (2022), 11825–11834. [3] O¨ . Arnaldsson, Involutive moving frames, Differential Geom. Appl., 69 (2020), 101603. [4] O¨ . Arnaldsson, Involutive moving frames II; The Lie-Tresse theorem, Differential Geom. Appl. 79 (2021), 101802. [5] M. Bazghandi, Lie Symmetries and Similarity Solutions of Phi-Four Equation, Indian J. Math. 61(2) (2019), 187–197. [6] A. Bihlo, J. Jackaman, and F. Valiquette, Invariant variational schemes for ordinary differential equations, Stud. Appl. Math., 148(3) (2022), 991–1020. [7] Y. A. Chirkunov, Generalized equivalence transformations and group classification of systems of differential equa- tions, J. Appl. Mech. Tech. Phys. 53 (2012), 147–155. [8] E. Cartan, Les probl´emes d´equivalence, in Oeuvres completes, Part II, Vol. 2, Gauthiers-Villars, Paris, (1953), 1311–1334. [9] W. Chen, J. Li, C. Miao, and J. Wu, Low regularity solutions of two fifth-order KdV type equations, J. Anal. Math., 107 (2009), 221–238. [10] J. Diehl, R. Preiß, M. Ruddy, and N. Tapia, The Moving-Frame Method for the Iterated-Integrals Signature: Orthogonal Invariants, Found. Comput. Math., (2022), 1–61. [11] M. Fels and P. J. Olver, Moving coframes. I. A practical algorithm, Acta Appl. Math., 51 (1998), 161–213. [12] M. Fels and P. J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., 55 (1999), 127–208. [13] P. A. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J., 41 (1974), 775–814. [14] Gh. Haghighatdoost, M. Bazghandi, and F. Pashaie, Differential Invariants of coupled Hirota-Satsuma KDV Equations, Kragujev. J. Math., 49(5) (2025), 793–805. [15] G. H. Halphen, Sur les invariant diff´erentiels, in : Oeuvres, Vol. 2, Gauthiers-Villars, Paris, (1913), 197–253. [16] Y. Hu, F. Zhang, and X. Xin, Lie symmetry analysis, optimal system and exact solutions of variable-coefficients Sakovich equation, J. Geom. Phys., 184 (2023), 104712. [17] P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, Cambridge, UK, 2000. [18] D. Kaya and S. M. El-Sayed, On a generalized fifth order KdV equations, Phys. Lett. A, 310(1) (2003), 44–51. [19] T. Kawaharara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan., 33(1) (1972), 260–264. [20] F. Klein, A comparative review of recent researches in geometry, Bull. Am. Math., 2(10) (1893), 215–249. [21] I. A. Kogan, Inductive Approach to Cartan’s Moving Frame Method with Applications to Classical Invariant Theory, 2000, University of Minnesota, Phd dissertation. [22] B. Kruglikov and V. Lychagin, Global Lie–Tresse theorem, Sel. Math., 22 (2016), 1357–1411. [23] R. Li, X. Yong, Y. Chen, and Y. Huang. Equivalence transformations and differential invariants of a generalized cubic–quintic nonlinear Schr¨odinger equation with variable coefficients, Nonlinear Dyn., 102 (2020), 339–348. [24] S. Lie, and G. Scheffers, Vorlesungen u¨ber continuierliche Gruppen mit geometrischen und anderen Anwendungen, B.G. Teubner, Leipzig, 1893. [25] H. Liu, J. Li, and L. Liu, Lie symmetry analysis, optimal systems and exact solutions to the fifth-order KdV types of equations, J. Math. Anal. Appl., 368(2) (2010), 551–558. [26] Q. Meng and C. Zhang, A third-order KdV solution for internal solitary waves and its application in the numerical wave tank, J. Ocean Eng. Sci., 1(2) (2016), 93–108. [27] F. Oliveri, Lie symmetries of differential equations: classical results and recent contributions, Symmetry, 2(2) (2010), 658–706. [28] P. J. Olver, Equivalence, invariants and symmetry, Cambridge University Press, 1995. [29] P. J. Olver, Applications of Lie groups to differential equations, Vol. 107. Springer Science & Business Media, 2000. [30] P. J. Olver and J. Pohjanpelto, Maurer-Cartan forms and the structure of Lie pseudo-groups, Selecta Math., 11 (2005), 99–126. [31] P. J. Olver and J. Pohjanpelto, Moving frames for Lie pseudo-groups, Canadian J. Math., 60 (2008), 1336–1386. [32] P. J. Olver and F. Valiquette, Recursive moving frames for Lie pseudo-groups, Results Math., 73(2) (2018), 1–64. [33] P. J. Olver, Generating differential invariants, J. Math. Anal. Appl., 333 (2007), 450–471. [34] P. J. Olver Differential invariants of surfaces, Differential Geom. Appl., 27(2) (2009), 230–239. [35] P. J. Olver and J. Pohjanpelto. Differential invariant algebras of Lie pseudo-groups, Adv. Math., 222(5) (2009), 1746–1792. [36] G. G. Polat and P. J. Olver, Joint differential invariants of binary and ternary forms, Port. Math., 76(2) (2020), 169–204. [37] M. Sabzevari, Convergent Normal Form for Five Dimensional Totally Nondegenerate CR Manifolds in C4, J. Geom. Anal., 31(8) (2021), 7900–7946. [38] A. Tresse, Sur les invariants diff´erentiels des groupes continus de transformations, Acta Math., 18 (1894), 1–88. [39] A. M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math. Comput., 184(2) (2007), 1002-1014. [40] G. Wang, X. Liu, and Y. Zhang, Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Commun. Nonlinear Sci. Numer. Simul., 18(9) (2013), 2321–2326. [41] G. Wang, T. Z. Xu, and X. Q. Liu, New explicit solutions of the fifth-order KdV equation with variable coefficients, Bull. Malays. Math. Sci. Soc, 37(3) (2014), 769–778. [42] X. Yan, J. Liu, J. Yang, and X. Xin, Lie symmetry analysis, optimal system and exact solutions for variable- coefficients (2+ 1)-dimensional dissipative long-wave system J. Math. Anal. Appl., 518(1) (2023), 126671. [43] X. Yong, X. Yang, L. Wu, and J. Gao, Equivalence transformations of a fifth-order partial differential equation with variable-coefficients, Appl. Math. Lett., 123 (2022), 107564.
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A finite generating set of differential invariants for Lie symmetry group of the fifth-order KdV types