New exact solutions for the Boiti-Leon-Manna-Pempinelli equation

Year 2024, Volume: 42 Issue: 1, 225 - 234, 27.02.2024

Rasime Kübra Yılmaz Durmuş Dağhan

Abstract

Phenomena in physics, plasma physics, optical fibers, chemical physics, fluid mechanics, and many fields are often described by the nonlinear evolution equations. The analytical solutions of these equations are very important to understand the evaluation of the physical models. In this paper, the Boiti-Leon-Manna-Pempinelli (BLMP) nonlinear partial differential equation, which can be used to describe the incompressible fluid flow, is analytically studied by using the five different techniques which are direct integration, (G' / G)-expansion method, different form of the (G' / G)-expansion method, two variable (G' / G, 1 / G)-expansion method, and (1 / G')- expansion method. Hyperbolic, trigonometric and rotational forms of solutions are obtained. Our solutions are reduced to the well-known solutions found in the literature by as-signing the some special values to the constants appeared in the analytic solutions. Moreover, we have also obtained the new analytic solutions of the BLMP equation.

Keywords

Evolution Equations, Exact Solution, (G, 1 / G)- Expansion Method

References

• REFERENCES
• [1] Darvishi MT, Najafi M, Kavitha L, Menkates M. Stair and step soliton solutions of the integrable (2+1)-dimensional and (3+1)-dimensional Boiti-Leaon-Manna-Pempinelli equations. Commun Theor Phys 2013;58:785–794. [CrossRef]
• [2] Arbabi S, Najafi M. Soliton solutions of nonlinear evolution equations in mathematical Physics. Optic 2016;127:4270–4274. [CrossRef]
• [3] Wang ML, Li X, Zhang J. The (G'/G)-Expansion Method and Traveling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Phys Lett A 2008;372:417–423. [CrossRef]
• [4] Li LX, Wang ML. The (G'/G)-Expansion Method and Travelling Wave Solutions for a Higher-Order Nonlinear Schrödinger Equation. Appl Math Comput 2009;208:440–445. [CrossRef]
• [5] Li LX, Li EQ, Wang ML. The (G'/G,1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations. Appl Math A J Chin Univ 2010;25:454–462. [CrossRef]
• [6] Yokus A. Solutions of some nonlinear partial differential equa-tions and comparison of their solutions (Doctorial Thesis). Elazig: Firat University; 2011.
• [7] Biswas A, Milovic DM, Kumar S, Yildirim A. Perturbation of shallow water waves by semi-inverse variational principle. Indian J Phys 2013;87:567–569. [CrossRef]
• [8] Biswas A. Soliton solutions of the perturbed resonant nonlinear Schrödinger's equation with full nonlinearity by semi-inverse variational principle. Quant Phys Lett 2013;1:79–83.
• [9] Biswas A, Milovic D. Chiral solitons with Bohm potential by He’s variational principle. Phys At Nucl 2011;74:755–757. [CrossRef]
• [10] He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006;20:1141–1199. [CrossRef]
• [11] Gao XY. Mathematical view with observational experimental consideration on certain (2+1)-dimensional waves in the cosmic laboratory dusty plasmas. Appl Math Lett 2019;91:165–172. [CrossRef]
• [12] Liu L, Tian B, Yuan YQ, Du Z. Dark-bright solitons and semirational rogue waves for the coupled Sasa-Satsuma equations. Phys Rev E 2018;97:052217. [CrossRef]
• [13] Wu XY, Tian B, Liu L, Sun Y. Rogue waves for a variable-coefficient Kadomtsev Petviashvili equation in fluid mechanics. Comput Math Appl 2018;76:215–223. [CrossRef]
• [14] Zhao XH, Tian B, Xie XY, Wu XY, Sun Y, Guo YJ. Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional Davey-Stewartson system on surface waves of finite depth. Waves Random Complex Media 2018;28:356–366. [CrossRef]
• [15] Yuan YQ, Tian B, Liu L, Wu XY, Sun Y. Solitons for the (2+1)-dimensional Konopelchenko Dubrovsky equations. J Math Anal Appl 2018;460:476–486. [CrossRef]
• [16] Du Z, Tian B, Chai HP, Sun Y, Zhao XH. Rogue waves for the coupled variable-coefficient fourth-order nonlinear Schrödinger equations in an inhomogeneous optical fiber. Chaos Solitons Fractals 2018;109:90–98. [CrossRef]
• [17] Zhang CR, Tian B, Wu XY, Yuan YQ, Du XX. Rogue waves and solitons of the coherently-coupled nonlinear Schrödinger equations with the positive coherent coupling. Phys Scr 2018;93:095202. [CrossRef]
• [18] Du XX, Tian B, Wu XY, Yin HM, Zhang CR. Lie group analysis, analytic solutions, and conservation laws of the (3+1)-dimensional Zakharov-Kuznetsov-Burgers equation in a collisionless magnetized electron-positron-ion plasma. Eur Phys J Plus 2018;133:378–391. [CrossRef]
• [19] Hu CC, Tian B, Wu XY, Yuan YQ, Du Z. Mixed lump-kink and rogue wave-kink solutions for a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics. Eur Phys J Plus 2018;133:40–47. [CrossRef]
• [20] Daghan D, Donmez O. Analytic solutions and parametric studies of the Schamel equation for two different ion-acoustic waves in plasmas. J Appl Mech Tech Phys 2018;59:321–333. [CrossRef]
• [21] Donmez O, Daghan D. Analytic solutions of the schamel-kdv equation by using different methods: Application to a dusty space plasma. Suleyman Demirel Univ J Nat Appl Sci 2017;21:208–215. [CrossRef]
• [22] Daghan D, Esen RK. Exact solutions for two different non-linear partial differential equations. New Trends Math Sci 2018;3:83–93. [CrossRef]
• [23] Yildiz G, Daghan D. New exact solutions of a nonlinear integrable equation. Math Meth Appl Sci 2020;43:6761–6770. [CrossRef]
• [24] Turkyilmazoglu M. Exact solutions corresponding to the viscous incompressible and conducting fluid flow due to a porous rotating disk. J Heat Transf 2009;131:091701. [CrossRef] [25] Turkyilmazoglu M. Exact solutions corresponding to the viscous incompressible and conducting fluid flow due to a rotating disk. Z Angew Math Mech 2009;89:490–503. [CrossRef]
• [26] Turkyilmazoglu M. Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow. Int J Non-Linear Mech 2011;46:306–311. [CrossRef]
• [27] Turkyilmazoglu M. Maximum wave run-up over beaches of convex/concave bottom profiles. Cont Shelf Res 2022;232:104610. [CrossRef]
• [28] Farhan M, Omar Z, Mebarek-Oudina F, Raza J, Shah Z, Choudhari RV, et al. Implementation of the one-step one-hybrid block method on the nonlinear equation of a circular sector oscillator. Comput Math Model 2020;31:116132. [CrossRef]
• [29] Djebali R, Mebarek-Oudina F, Rajashekhar C. Similarity solution analysis of dynamic and thermal boundary layers: further formulation along a vertical flat plate. Phys Scr. 2021;96:085206. [CrossRef]
• [30] Alkasassbeh M, Omar Z, Mebarek-Oudina F, Raza J, Chamkha A. Heat transfer study of convective fin with temperature dependent internal heat generation by hybrid block method. Heat Transf Asian Res 2019;48:1225–1244. [CrossRef]
• [31] Qu CZ. Symmetry Algebras of Generalized (2+1)-Dimensional KdV Equation. Commun Theor Phys 1996;25:369–372. [CrossRef]
• [32] Gilson CR, Nimmo JJC, Willox R. A (2+1)-Dimensional Generalization of the AKNS shallow water wave equation. Phys Lett A 1993;180:337–345. [CrossRef]
• [33] Boiti M, Leon JPJ, Manna M, Pempinelli F. On a spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Probl 1986;2:271–279. [CrossRef]
• [34] Boiti M, Leon JPL, Manna M, Pempinelli F. On a spectral transform of a KdV-like equation related to the Schrödinger operator in the plane. Inverse Probl 1987;3:25–36. [CrossRef]
• [35] Song-Hua M, Jian-Ping F. Multi dromion-solitoff and fractal excitations for (2+1)-dimensional boiti-leon-manna-pempinelli system. Commun Theor Phys (Beijing,China) 2009;52:641–645. [CrossRef]
• [36] Lin L. Quasi-Periodic Waves and Asymptotic Property for Boiti-Leon-Manna-Pempinelli Equation. Commun Theor Phys (Beijing, China) 2010;54:208–214. [CrossRef]
• [37] Luo L. New Exact Solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli Equation. Phys Lett A 2011;375:1059–1063. [CrossRef]
• [38] Li Y, Li D. New Exact Solutions for the (2+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation. Appl Math Sci 2012;6:576–587. [CrossRef]
• [39] Delisle L, Mosaddeghi M. Classical and SUSY solutions of the Boiti-Leon-Manna-Pempinelli Equation. J Phys A Math Theor 2013;46:115203. [CrossRef]
• [40] Najafi M, Arbabi S. Wronskian Determinant Solutions of the (2+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation. Int J Adv Math Sci 2013;1:8–11. [CrossRef]
• [41] Dong H, Zhang Y, Zhang Y, Yin B. Generalized bilinear differential operators, binary bell polynomials, and exact periodic wave solution of boiti-leon-manna-pempinelli equation. Abstr Appl Anal 2014;738609. [CrossRef]
• [42] Asadi N, Nadjafikhah M. Geometry of Boiti-Leon-Manna-Pempinelli Equation. Indian J Sci Technol 2015;8:1–7. [CrossRef]
• [43] Zamiri A. The (G'/G)-expansion method for the (2+1)-dimensional boiti-leon-manna-pempinelli equation. J Basic Appl Sci Res 2013;3:522–527. [CrossRef]
• [44] Alofi AS, Abdelkawy MA. New exact solutions of Boiti-Leon-Manna-Pempinelli equation using extended F-expansion method. Life Sci J 2012;9:3995–4002.
• [45] Tang Y, Zai W. New periodic-wave solutions for (2+1)-and (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equations. Nonlinear Dyn 2015;81:249–255. [CrossRef]
• [46] Daghan D, Donmez O, Tuna A. Explicit solutions of the nonlinear partial differential equations. Nonlinear Anal Real World Appl 2010;11:2152–2163. [CrossRef]
• [47] Daghan D, Yildiz O, Toros S. Comparison of (G'/G)-methods for finding exact solutions of the Drinfeld-Sokolov system. Math Slovaca 2015;65:607–632. [CrossRef]
• [48] Daghan D, Donmez O. Investigating the effect of integration constants and various plasma parameters on the dynamics of the soliton in different physical plasmas. Phys Plasmas 2015;22:072114. [CrossRef]
• [49] Daghan D, Donmez O. Exact solutions of the gardner equation and their applications to the different physical plasmas. Braz J Phys 2016;46:321–333. [CrossRef]
• [50] Kudryashov NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simul 2009;14:3507–3529. [CrossRef]