Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations

Huitzilin Yépez-Martínez; Mir Sajjad Hashemi; Ali Saleh Alshomrani; Mustafa Inc; Huitzilin Yépez-Martínez; Mir Sajjad Hashemi; Ali Saleh Alshomrani; Mustafa Inc

doi:10.3934/math.2023852

AIMS Mathematics

2023, Volume 8, Issue 7: 16655-16690. doi: 10.3934/math.2023852

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Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations

1.
Universidad Autónoma de la Ciudad de México, Prolongación San Isidro 151, Col. San Lorenzo Tezonco, Del. Iztapalapa, C.P. 09790 México D.F., Mexico
2.
Biruni University, Department of Computer Engineering, Istanbul, Turkey
3.
Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4.
Firat University, Science Faculty, Department of Mathematics, Elazig 23119, Turkey
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 21 March 2023 Revised: 02 May 2023 Accepted: 08 May 2023 Published: 11 May 2023
MSC : 83C15, 74J35, 35A20

In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier have been solved using analytical methods, which impose certain limitations on the interaction parameters and the coefficients of the guess solutions. However, in the case of the double sub-equation guess solution, analytic solutions were allowed. The soliton solutions that were obtained through this method display real positive values for the wave phase transformation, which is a novel result in the application of the generalized projective Riccati method. In previous applications of this method, the real positive properties of the solutions were not thoroughly investigated.
Citation: Huitzilin Yépez-Martínez, Mir Sajjad Hashemi, Ali Saleh Alshomrani, Mustafa Inc. Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations[J]. AIMS Mathematics, 2023, 8(7): 16655-16690. doi: 10.3934/math.2023852

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Abstract

In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier have been solved using analytical methods, which impose certain limitations on the interaction parameters and the coefficients of the guess solutions. However, in the case of the double sub-equation guess solution, analytic solutions were allowed. The soliton solutions that were obtained through this method display real positive values for the wave phase transformation, which is a novel result in the application of the generalized projective Riccati method. In previous applications of this method, the real positive properties of the solutions were not thoroughly investigated.

References

[1]	J. G. Liu, W. H. Zhu, L. Zhou, Breather wave solutions for the Kadomtsev-Petviashvili equation with variable coefficients in a fluid based on the variable-coefficient three-wave approach, Math. Methods Appl. Sci., 43 (2020), 458–465. https://doi.org/10.1002/mma.5899 doi: 10.1002/mma.5899
[2]	J. G. Liu, Q. Ye, Stripe solitons and lump solutions for a generalized Kadomtsev-Petviashvili equation with variable coefficients in fluid mechanics, Nonlinear Dyn., 96 (2019), 23–29. https://doi.org/10.1007/s11071-019-04770-8 doi: 10.1007/s11071-019-04770-8
[3]	J. G. Liu, M. S. Osman, Nonlinear dynamics for different nonautonomous wave structures solutions of a 3D variable-coefficient generalized shallow water wave equation, Chin. J. Phys., 77 (2022), 1618–1624. https://doi.org/10.1016/j.cjph.2021.10.026 doi: 10.1016/j.cjph.2021.10.026
[4]	J. G. Liu, A. M. Wazwaz, W. H. Zhu, Solitary and lump waves interaction in variable-coefficient nonlinear evolution equation by a modified ansätz with variable coefficients, J. Appl. Anal. Comput., 12 (2022), 517–532. https://doi.org/10.11948/20210178 doi: 10.11948/20210178
[5]	M. L. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, 199 (1995), 169–172. https://doi.org/10.1016/0375-9601(95)00092-H doi: 10.1016/0375-9601(95)00092-H
[6]	M. L. Wang, Y. B. Zhou, Z. B. Li, Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 216 (1996), 67–75. https://doi.org/10.1016/0375-9601(96)00283-6 doi: 10.1016/0375-9601(96)00283-6
[7]	E. J. Parkes, B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun., 98 (1996), 288–300. https://doi.org/10.1016/0010-4655(96)00104-X doi: 10.1016/0010-4655(96)00104-X
[8]	E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212–218. https://doi.org/10.1016/S0375-9601(00)00725-8 doi: 10.1016/S0375-9601(00)00725-8
[9]	S. K. Liu, Z. T. Fu, S. D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69–74. https://doi.org/10.1016/S0375-9601(01)00580-1 doi: 10.1016/S0375-9601(01)00580-1
[10]	Z. T. Fu, S. K. Liu, S. D. Liu, Q. Zhao, The JEFE method and periodic solutions of two kinds of nonlinear wave equations, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 67–75. https://doi.org/10.1016/S1007-5704(02)00082-5 doi: 10.1016/S1007-5704(02)00082-5
[11]	K. A. Gepreel, Explicit Jacobi elliptic exact solutions for nonlinear partial fractional differential equations, Adv. Differ. Equations, 2014 (2014), 286. https://doi.org/10.1186/1687-1847-2014-286 doi: 10.1186/1687-1847-2014-286
[12]	Sirendaoreji, New exact travelling wave solutions for the Kawahara and modified Kawahara equations, Chaos Solitons Fract., 19 (2004), 147–150. https://doi.org/10.1016/S0960-0779(03)00102-4 doi: 10.1016/S0960-0779(03)00102-4
[13]	Y. M. Chu, M. Inc, M. S. Hashemi, S. Eshaghi, Analytical treatment of regularized Prabhakar fractional differential equations by invariant subspaces, Comp. Appl. Math, 41 (2022), 271. https://doi.org/10.1007/s40314-022-01977-1 doi: 10.1007/s40314-022-01977-1
[14]	M. S. Hashemi, Invariant subspaces admitted by fractional differential equations with conformable derivatives, Chaos Solitons Fract., 107 (2018), 161–169. https://doi.org/10.1016/j.chaos.2018.01.002 doi: 10.1016/j.chaos.2018.01.002
[15]	M. S. Hashemi, M. Mirzazadeh, Optical solitons of the perturbed nonlinear Schrödinger equation using Lie symmetry method, Optik, 281 (2023), 170816. https://doi.org/10.1016/j.ijleo.2023.170816 doi: 10.1016/j.ijleo.2023.170816
[16]	M. S. Hashemi, D. Baleanu, Lie symmetry analysis of fractional differential equations, New York: Chapman and Hall/CRC, 2020. https://doi.org/10.1201/9781003008552
[17]	M. L. Wang, X. Z. Li, J. L. Zhang, The $(\dfrac{G'}{G})$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2007), 417–423. https://doi.org/10.1016/j.physleta.2007.07.051 doi: 10.1016/j.physleta.2007.07.051
[18]	H. O. Roshid, M. R. Kabir, R. C. Bhowmik, B. K. Datta, Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp$\left(-\varphi\left(\varphi\right)\right)$-expansion method, SpringerPlus, 3 (2014), 692. https://doi.org/10.1186/2193-1801-3-692 doi: 10.1186/2193-1801-3-692
[19]	H. O. Roshid, M. Azizur Rahman, The exp$\left(-\Phi\left(\varphi\right)\right)$-expansion method with application in the (1+1)-dimensional classical Boussinesq equations, Results Phys., 4 (2014), 150–155. https://doi.org/10.1016/j.rinp.2014.07.006 doi: 10.1016/j.rinp.2014.07.006
[20]	M. B. Hossen, H. O. Roshid, M. Zulfikar, Modified double sub-equation method for finding complexiton solutions to the (1+1) dimensional nonlinear evolution equations, Int. J. Appl. Comput. Math., 3 (2017), 679–697. https://doi.org/10.1007/s40819-017-0377-6 doi: 10.1007/s40819-017-0377-6
[21]	H. O. Roshid, Novel solitary wave solution in shallow water and ion acoustic plasma waves in-terms of two nonlinear models via MSE method, J. Ocean Eng. Sci., 2 (2017), 196–202. https://doi.org/10.1016/j.joes.2017.07.004 doi: 10.1016/j.joes.2017.07.004
[22]	Y. Yıldırım, A. Biswas, M. Asma, M. Ekici, B. P. Ntsime, E. M. E. Zayed, et al., Optical soliton perturbation with Chen-Lee-Liu equation, Optik, 220 (2020), 165177. https://doi.org/10.1016/j.ijleo.2020.165177 doi: 10.1016/j.ijleo.2020.165177
[23]	M. C. Nucci, P. G. L. Leach, The determination of nonlocal symmetries by the technique of reduction of order, J. Math. Anal. Appl., 251 (2000), 871–884. https://doi.org/10.1006/jmaa.2000.7141 doi: 10.1006/jmaa.2000.7141
[24]	S. Martini, N. Ciccoli, M. C. Nucci, Group analysis and heir-equations of a mathematical model for thin liquid films, J. Nonlinear Math. Phys., 16 (2009), 77–92. https://doi.org/10.1142/S1402925109000078 doi: 10.1142/S1402925109000078
[25]	M. S. Hashemi, A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative, Chaos Solitons Fract., 152 (2021), 111367. https://doi.org/10.1016/j.chaos.2021.111367 doi: 10.1016/j.chaos.2021.111367
[26]	A. Akbulut, M. Mirzazadeh, M. S. Hashemi, K. Hosseini, S. Salahshour, C. Park, Triki-Biswas model: its symmetry reduction, Nucci's reduction and conservation laws, Int. J. Mod. Phys. B, 37 (2022), 2350063. https://doi.org/10.1142/S0217979223500637 doi: 10.1142/S0217979223500637
[27]	F. L. Xia, F. Jarad, M. S. Hashemi, M. B. Riaz, A reduction technique to solve the generalized nonlinear dispersive mK(m, n) equation with new local derivative, Results Phys., 38 (2022), 105512. https://doi.org/10.1016/j.rinp.2022.105512 doi: 10.1016/j.rinp.2022.105512
[28]	E. M. E. Zayed, K. A. E. Alurrfi, The generalized projective Riccati equations method for solving nonlinear evolution equations in mathematical physics, Abstr. Appl. Anal., 2014 (2014), 259190. https://doi.org/10.1155/2014/259190 doi: 10.1155/2014/259190
[29]	E. Yomba, The general projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations, Chin. J. Phys., 43 (2005), 991–1003.
[30]	G. Akram, S. Arshed, M. Sadaf, F. Sameen, The generalized projective Riccati equations method for solving quadratic-cubic conformable time-fractional Klein-Fock-Gordon equation, Ain Shams Eng. J., 13 (2022), 101658. https://doi.org/10.1016/j.asej.2021.101658 doi: 10.1016/j.asej.2021.101658
[31]	E. M. E. Zayed, K. A. E. Alurrfi, The generalized projective Riccati equations method and its applications to nonlinear PDEs describing nonlinear transmission lines, Commun. Appl. Electron., 3 (2015), 1–8. https://doi.org/10.5120/cae2015651924 doi: 10.5120/cae2015651924
[32]	T. Y. Wu, J. E. Zhang, L. P. Cook, V. Roythurd, M. Tulin, On modeling nonlinear long waves, In: Mathematics is for solving problems, SIAM, 1996,233–249.
[33]	H. Q. Sun, A. H. Chen, Exact solutions of the classical Boussinesq system, Arab J. Basic Appl. Sci., 25 (2018), 85–91. https://doi.org/10.1080/25765299.2018.1449416 doi: 10.1080/25765299.2018.1449416
[34]	N. H. Aljahdaly, Some applications of the modified $(G'/G^2)$-expansion method in mathematical physics, Results Phys., 13 (2019), 102272. https://doi.org/10.1016/j.rinp.2019.102272 doi: 10.1016/j.rinp.2019.102272
[35]	E. M. E. Zayed, S. Al-Joudi, Applications of an extended $(G'/G)$-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Math. Prob. Eng., 2010 (2010), 768573. https://doi.org/10.1155/2010/768573 doi: 10.1155/2010/768573
[36]	A. H. Arnous, M. Mirzazadeh, Bäcklund transformation of fractional Riccati equation and its applications to the spacetime FDEs, Math. Methods Appl. Sci., 38 (2015), 4673–4678. https://doi.org/10.1002/mma.3371 doi: 10.1002/mma.3371
[37]	M. Eslami, B. Fathi Vajargah, M, Mirzazadeh, A. Biswas, Application of first integral method to fractional partial differential equations, Indian J. Phys., 88 (2014), 177–184. https://doi.org/10.1007/s12648-013-0401-6 doi: 10.1007/s12648-013-0401-6
[38]	H. Jafari, H. Tajadodi, D. Baleanu, A. Al-Zahrani, Y. Alhamed, A. Zahid, Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma Tasso Olver equation, Cent. Eur. J. Phys., 11 (2013), 1482–1486. https://doi.org/10.2478/s11534-013-0203-7 doi: 10.2478/s11534-013-0203-7
[39]	S. A. Elwakil, S. K. El-labany, M. A. Zahran, R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A, 299 (2002), 179–188. https://doi.org/10.1016/S0375-9601(02)00669-2 doi: 10.1016/S0375-9601(02)00669-2
[40]	M. Alquran, I. Jaradat, D. Baleanu, M. Syam, The Duffing model endowed with fractional time derivative and multiple pantograph time delays, Rom. J. Phys., 64 (2019), 107.
[41]	K. Zhang, Z. Zhang, T. Yuwen, Phase portraits and traveling wave solutions of a fractional generalized reaction Duffing equation, Adv. Pure Math., 12 (2022), 465–477. https://doi.org/10.4236/apm.2022.127035 doi: 10.4236/apm.2022.127035
[42]	A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput., 195 (2008), 24–33. https://doi.org/10.1016/j.amc.2007.04.066 doi: 10.1016/j.amc.2007.04.066
[43]	A. EL Achab, On the integrability of the generalized Pochhammer-Chree (PC) equations, Phys. A: Stat. Mech. Appls., 545 (2020), 123576. https://doi.org/10.1016/j.physa.2019.123576 doi: 10.1016/j.physa.2019.123576
[44]	A. Ali, A. R. Seadawy, D. Baleanu, Propagation of harmonic waves in a cylindrical rod via generalized Pochhammer-Chree dynamical wave equation, Results Phys., 17 (2020), 103039. https://doi.org/10.1016/j.rinp.2020.103039 doi: 10.1016/j.rinp.2020.103039
[45]	Y. Liu, Existence and blow up of solutions of a nonlinear Pochhammer-Chree equation, Indiana Univ. Math. J., 45 (1996), 797–816.
[46]	H. Triki, A. Benlalli, A. M. Wazwaz, Exact solutions of the generalized Pochhammer-Chree equation with sixth-order dispersion, Rom. J. Phys., 60 (2015), 935–951.
[47]	J. Li, L. Zhang, Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation, Chaos Solitons Fract., 14 (2002), 581–593. https://doi.org/10.1016/S0960-0779(01)00248-X doi: 10.1016/S0960-0779(01)00248-X
[48]	B. Li, Y. Chen, H. Zhang, Travelling wave solutions for generalized pochhammer-chree equations, Z. Naturforschung A, 57 (2002), 874–882. https://doi.org/10.1515/zna-2002-1106 doi: 10.1515/zna-2002-1106
[49]	H. Yépez-Martínez, H. Rezazadeh, M. Inc, M. A. Akinlar, New solutions to the fractional perturbed Chen-Lee-Liu equation with a new local fractional derivative, Waves Random Complex Media, 2021. https://doi.org/10.1080/17455030.2021.1930280

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