Optimal system of 1-d subalgebras and conserved quantities of a nonlinear wave equation in three dimensions arising in engineering physics

Adeyemo, Oke Davies; Khalique, Chaudry Masood

doi:10.2478/ama-2024-0022

Artykuł - szczegóły

Tytuł artykułu

Optimal system of 1-d subalgebras and conserved quantities of a nonlinear wave equation in three dimensions arising in engineering physics

Autorzy

Adeyemo Oke Davies , Khalique Chaudry Masood

Treść / Zawartość

Pełne teksty:

Pobierz

Identyfikatory

DOI

10.2478/ama-2024-0022

Warianty tytułu

Języki publikacji

Abstrakty

The construction of explicit structures of conserved vectors plays diverse crucial roles in the study of nonlinear science inclusive of the fact that they are invoked in developing appropriate numerical schemes and for other mathematical analyses. Therefore, in this paper, we examine the conserved quantities of a nonlinear wave equation, existing in three dimensions, and highlight their applications in physical sciences. The robust technique of the Lie group theory of differential equations (DEs) is invoked to achieve analytic solutions to the equation. This technique is used in a systematic way to generate the Lie point symmetries of the equation under study. Consequently, an optimal system of one-dimensional (1-D) Lie subalgebras related to the equation is obtained. Thereafter, we engage the formal Lagrangian of the nonlinear wave equation in conjunction with various gained subalgebras to construct conservation laws of the equation under study using Ibragimov’s theorem for conserved vectors.

Słowa kluczowe

nonlinear wave equation in three dimensions Lie group theory differential equations 1-D optimal system of subalgebras formal Lagrangian conserved vectors

Wydawca

Oficyna Wydawnicza Politechniki Białostockiej

Czasopismo

Acta Mechanica et Automatica

Rocznik

2024

Tom

Vol. 18, no. 2

Strony

177--192

Opis fizyczny

Bibliogr. 83 poz., rys., tab.

Twórcy

autor

Adeyemo Oke Davies

adeyemodaviz@gmail.com

Material Science, Innovation and Modelling Research Focus Area, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa

https://orcid.org/0000-0002-8745-5387

autor

Khalique Chaudry Masood

masood.khalique@nwu.ac.za

Material Science, Innovation and Modelling Research Focus Area, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa

https://orcid.org/0000-0002-1986-4859

Bibliografia

1. Gao XY. Mathematical view with observational/experimental consid-eration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas. Appl Math Lett. 2019;91:65–172.
2. Adeyemo OD, Motsepa T, Khalique CM. A study of the generalized nonlinear advection-diffusion equation arising in engineering scienc-es. Alex Eng J. 2022;61(1):185--194.
3. Khalique CM, Adeyemo OD. A study of (3+1)-dimensional general-ized Korteweg-de Vries-Zakharov-Kuznetsov equation via Lie sym-metry approach. Results Phys. 2020;18: 103197.
4. Du XX, Tian B, Qu QX, Yuan YQ, Zhao XH. Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magneto-plasma. Chaos Solitons Fract., 2020;134:109709.
5. Zhang CR, Tian B, Qu QX, Liu L, Tian HY. Vector bright solitons and their interactions of the couple FokasLenells system in a birefringent optical fiber. Z. Angew Math Phys. 2020;71:1–19.
6. Gao XY, Guo YJ, Shan WR. Water-wave symbolic computation for the Earth, Enceladus and Titan: The higher-order Boussinesq-Burgers system, auto-and non-auto-Bcklund transformations. Appl Math Lett. 2020;104:106170.
7. Hussain A, Usman M, Al-Sinan BR, Osman WM. Symmetry analysis and closed-form invariant solutions of the nonlinear wave equations in elasticity using optimal system of Lie subalgebra. Chin J Phys. 2023;83:1–13.
8. Usman M, Hussain A, Zaman FD, Eldin SM. Group invariant solu-tions of wave propagation in phononic materials based on the re-duced micromorphic model via optimal system of Lie subalgebra. Results Phys. 2023;48:106413.
9. Kumar S, Rani S. Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+ 1)-dimensional dis-sipative long wave system. Phys Scr. 2021;96:125202.
10. Rani S, Kumar S, Kumar R. Invariance analysis for determining the closed-form solutions, optimal system, and various wave profiles for a (2+1)- dimensional weakly coupled B-Type Kadomtsev-Petviashvili equations. J Ocean Eng Sci. 2023;8:133–144.
11. Adeyemo OD, Khalique CM. Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional nonlinear wave equation of engineering physics with applications in signal processing, electro-magnetism and complex media. Alex Eng J. 2023;73:751–769.
12. Akgu¨l A, Ahmad A. Reproducing kernel method for Fangzhu’s oscillator for water collection from air, Mathematical Methods in the Applied Sciences (2020).
13. Yusuf A, Sulaiman TA, Khalil EM, Bayram M, Ahmad H. Construction of multi-wave complexiton solutions of the Kadomtsev-Petviashvili equation via two efficient analyzing techniques. Results Phys., 2021;21:103775.
14. Kumar S, Niwas M, Wazwaz AM. Lie symmetry analysis, exact analytical solutions and dynamics of solitons for (2+1)-dimensional NNV equations. Phys Scr. 2020;95:095204.
15. Adeyemo OD, Khalique CM. Lie group theory, stability analysis with dispersion property, new soliton solutions and conserved quantities of 3D generalized nonlinear wave equation in liquid containing gas bubbles with applications in mechanics of fluids, biomedical sciences and cell biology. Commun Nonlinear Sci Numer Simul. 2023;123:107261.
16. Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press; Cambridge, UK, 1991.
17. Zheng CL, Fang JP. New exact solutions and fractional patterns of generalized Broer-Kaup system via a mapping approach. Chaos Soli-ton Fract. 2006;27:1321–1327.
18. Akbar MA, Ali NHM. Solitary wave solutions of the fourth-order Boussinesq equation through the exp(−Φ(η))-expansion method. SpringerPlus. 2014;3:344.
19. Weiss J, Tabor M, Carnevale G. The Painl´ev´e property and a partial differen- tial equations with an essential singularity. Phys Lett A. 1985;109:205–208.
20. Zhang L, Khalique CM. Classification and bifurcation of a class of second- order ODEs and its application to nonlinear PDEs. Discrete Contin Dyn Syst - S. 2018;11:777–790.
21. Biswas A, Jawad AJM, Manrakhan WN. Optical solitons and com-plexitons of the Schr¨odinger-Hirota equation. Opt Laser Technol. 2012;44:2265–2269.
22. Chun C, Sakthivel R. Homotopy perturbation technique for solving two-point boundary value problems-comparison with other methods. Comput Phys Commun. 2010;181:1021–1024.
23. Darvishi MT, Najafi M. A modification of extended homoclinic test approach to solve the (3+1)-dimensional potential-YTSF equation. Chin Phys Lett. 2011;28:040202
24. Wazwaz AM. Traveling wave solution to (2+1)-dimensional nonlinear evolution equations. J Nat Sci Math. 2007;1:1–13.
25. Osman MS. One-soliton shaping and inelastic collision between double soli- tons in the fifth-order variable-coefficient Sawada-Kotera equation. Nonlinear Dynam. 2019;96:1491–1496.
26. Wazwaz AM. Partial Differential Equations. CRC Press; Boca Raton, Florida, USA, 2002.
27. Salas AH, Gomez CA. Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation. Math Probl Eng. 2010;2010.
28. Gu CH. Soliton Theory and Its Application. Zhejiang Science and Technology Press; Zhejiang, China, 1990.
29. Zhou Y, Wang M, Wang Y. Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys Lett A. 2003;308:31–36.
30. Zeng X, Wang DS. A generalized extended rational expansion meth-od and its application to (1+1)-dimensional dispersive long wave equation. Appl Math Comput. 2009;212:296–304.
31. Jawad AJM, Mirzazadeh M, Biswas A. Solitary wave solutions to nonlinear evolution equations in mathematical physics. Pramana. 2014;83:457–471.
32. Ovsiannikov LV. Group Analysis of Differential Equations. Academic Press; New York, USA, 1982.
33. Olver PJ. Applications of Lie Groups to Differential Equations. sec-ond ed., Springer-Verlag; Berlin, Germany, 1993.
34. Hirota R. The Direct Method in Soliton Theory. Cambridge University Press; Cambridge, UK, 2004.
35. Kudryashov NA, Loguinova NB. Extended simplest equation method for nonlinear differential equations. Appl Math Comput. 2008;205:396-402.
36. Matveev VB, Salle MA. Darboux Transformations and Solitons. Springer; New York, USA, 1991.
37. Wang M, Li X, Zhang J. The (Gj/G)−expansion method and travelling wave solutions for linear evolution equations in mathematical phys-ics. Phys Lett A. 2005;24:1257–1268.
38. Wazwaz AM. The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations. Appl Math Comput. 2005;169:321–338.
39. Kudryashov NA. Simplest equation method to look for exact solutions of non-linear differential equations. Chaos Solitons Fract. 2005;24:1217–1231
40. Chen Y, Yan Z. New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method. Cha-os Solitons Fract. 2005;26:399–406.
41. He JH, Wu XH. Exp-function method for nonlinear wave equations. Chaos Solitons Fract. 2006;30:700–708.
42. Ma WX. Comment on the (3+1)-dimensional Kadomtsev-Petviashvili equations. Commun Nonlinear Sci Numer. Simulat. 2011;16:2663–2666.
43. Kadomtsev BB, Petviashvili VI. On the stability of solitary waves in weakly dispersing media. Sov Phys Dokl. 1970;192:753–756.
44. You F; Xia T, Chen D. Decomposition of the generalized KP, cKP and mKP and their exact solutions. Phys Lett A. 2008;372:3184–3194.
45. Kuznetsov EA, Turitsyn SK. Two- and three-dimensional solitons in weakly dispersive media. Zh Ebp Teor Fa. 1982;82:1457–1463.
46. Ablowitz MJ, Segur H. On the evolution of packets of water waves. J Fluid Mech. 1979;92:691–715.
47. Infeld E, Rowlands G. Three-dimensional stability of Korteweg-de Vries waves and solitons II. Acta Phys Polon A. 1979;56:329–332.
48. Xu G, Li Z. Symbolic computation of the Painlev´e test for nonlinear partial differential equations using Maple. Comput Phys Commun. 2004;161:65–75.
49. Ma WX, Fan E. Linear superposition principle applying to Hirota bilinear equations. Comput Math Appl. 2011;61: 950–959.
50. Senatorski A, Infeld E. Simulations of two-dimensional Kadomtsev- Petviashvili soliton dynamics in three-dimensional space. Phys Rev Lett. 1996;77:2855–2858.
51. Alagesan T, Uthayakumar A, Porsezian K. Painlev´e analysis and B¨acklund transformation for a three-dimensional Kadomtsev-Petviashvili equation. Chaos Soliton Fract. 1997;8:893–895.
52. Ma WX, Abdeljabbar A, Asaad MG. Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation. Appl Math Comput. 2011;217:10016–10023.
53. Wazwaz AM. Multiple-soliton solutions for a (3 +1)-dimensional generalized KP equation. Commun Nonlinear Sci Numer Simulat. 2012;17:491–495.
54. Wazwaz AM, El-Tantawy SA. A new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation. Nonlinear Dyn. 2016;84:1107–1112.
55. Liu JG, Tian Y, Zeng ZF. New exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in multi- temperature electron plasmas. AIP Adv. 2017;7:2158–3226.
56. Kadomtsev BB, Petviashvili VI. On the stability of solitary waves in weakly dispersive media. Sov Phys Dokl. 1970;15:539–541.
57. Gai L, Bilige S, Jie Y. The exact solutions and approximate analytic solutions of the (2+1)-dimensional KP equation based on symmetry method. Springer- Plus. 2016;5:1267.
58. Ebadi G, Fard NY, Bhrawy AH, Kumar S, Triki H, Yildirim A, Biswas A. Solitons and other solutions to the (3+1)-dimensional extended kadomtsev- petviashvili equation with power law nonlinearity. Rom Rep Phys. 2013;65:27–62.
59. Zakharov VE. Shock waves propagated on solitons on the surface of a fluid. Radiophys Quantum Electron. 1986;29:813–817.
60. Saha A, Pa N. Chatterjee P. Bifurcation and quasiperiodic behaviors of ion acoustic waves in magnetoplasmas with nonthermal electrons featuring tsallis distribution. Braz J Phys. 2015;45:325–333.
61. Borhanifar A, Jafari H, Karimi SA. New solitons and periodic solu-tions for the Kadomtsev-Petviashvili equation. J Nonlinear Sci Appl. 2008;1:224–229.
62. Khan K, Akbar MA. Exact traveling wave solutions of Kadomtsev-Petviashvili equation. J Egypt Math Soc. 2015;23: 278–281.
63. Zhang X, Chen Y, Tang X. Rogue wave and a pair of resonance stripe solitons to KP equation. Comput Math Appl. 2018;76:1938–1949.
64. Khalique CM. On the solutions and conservation laws of a coupled kadomtsev- petviashvili equation. J Appl Math. 2013;2013.
65. Zhao HQ, Ma WX. Mixed lump-kink solutions to the KP equation. Comput Math Appl. 2017;74:1399–1405.
66. Ma Z, Chen J, Fei J. Lump and line soliton pairs to a (2+1)-dimensional integrable Kadomtsev-Petviashvili equation. Comput Math Appl. 2018;76:1130–1138.
67. Lu¨ X, Ma WX, Zhou Y, Khalique CM. Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic compu-tation. Comput Math Appl. 2016;71:1560–1567.
68. Ma WX. Bilinear equations, Bell polynomials and linear superposition principle. J Phys Conf Ser. 2013;411:2013.
69. Adeyemo OD, Khalique CM. Symmetry solutions and conserved quantities of an extended (1+3)-dimensional Kadomtsev-Petviashvili-like equation. Appl Math Inf Sci. 2021;15:1–12.
70. Adeyemo OD, Khalique CM. Dynamics of soliton waves of group-invariant solutions through optimal system of an extended KP-like equation in higher dimensions with applications in medical sciences and mathematical physics. J Geom Phys. 2022;177:104502.
71. Ibragimov NH. CRC Handbook of Lie Group Analysis of Differential Equations. Vols 1–3, CRC Press; Boca Raton, Florida, 1994–1996.
72. Hu X, Li Y, Chen Y. A direct algorithm of one-dimensional optimal system for the group invariant solutions. J Math Phys. 2015;56:053504.
73. Ibragimov NH. Integrating factors, adjoint equations and Lagrangi-ans. J Math Anal Appl. 2006;318:742–757.
74. Ibragimov NH. A new conservation theorem. J Math Anal Appl. 2007;333:311–328.
75. Khalique CM, Adeyemo OD. Closed-form solutions and conserved vectors of a generalized (3+1)-dimensional breaking soliton equation of engineering and nonlinear science. Mathematics. 2020;8:1692.
76. Anco SC. Generalization of Noethers theorem in modern form to non- variational partial differential equations. In: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computa-tional Science. Fields Institute Communications. 2017;79:119--182.
77. Conservation laws: https://www.britannica.com/science/conservation-law
78. https://selftution.com/different-forms-or-types-of-energy-in-physics-and-examples-mechanical-potential-kinetic-heat-chemical-light-sound-magnetic- electrical-atomic-nuclear-thermal/
79. https://byjus.com/physics/law-of-conservation-of-energy/
80. https://studiousguy.com/conservation-of-momentum-examples/
81. https://sciencequery.com/laws-of-conservation-of-momentum/
82. https://www.usgs.gov/media/images/a-turbine-connected-a-generator- produces-power-inside-a-dam
83. https://www.usgs.gov/media/images/flow-water-produces-hydroelectricity

Typ dokumentu

Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-a36872a9-2bcb-4a0d-9d59-35b141ccccef