Abstract
In this article, the soliton solutions of the Gilson–Pickering equation have been constructed using the sinh-Gordon function method (ShGFM) and (G′/G, 1/G)-expansion method, which are applied to obtain exact solutions of nonlinear partial differential equations. A solution function different from the solution function in the classical (G′/G, 1/G)-expansion method has been considered which are based on complex trigonometric, hyperbolic, and rational solutions. By invoking ShGFM and (G′/G, 1/G)-expansion methods, different traveling wave solutions have been investigated. For the sake of avoiding the complex calculations, the ready package program has been tackled. The comparative analysis of sinh-Gordon function and (G′/G, 1/G)-expansion methods has shown several differences and similarities. A comparative analysis of ShGFM and (G′/G, 1/G)-expansion methods assures that the (G′/G, 1/G)-expansion method has been found to be more intensive, powerful, reliable and effective method for the Gilson–Pickering equation. The graphical illustrations of two-, three-dimensional, and contour graphs have been depicted as well.
Similar content being viewed by others
References
H.M. Baskonus, H. Bulut, A. Atangana, On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Mater. Struct. 25(3), 035022 (2016)
T.A. Sulaiman, H. Bulut, A. Yokus, H.M. Baskonus, On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering. Indian J. Phys. 93(5), 647–656 (2019)
A. Yokus, H.M. Baskonus, T.A. Sulaiman, H. Bulut, Numerical simulation and solutions of the two-component second order KdV evolutionarysystem. Numer. Methods Partial Differ. Equ. 34(1), 211–227 (2018)
H. Durur, M. Şenol, A. Kurt, O. Taşbozan, Zaman-Kesirli Kadomtsev–Petviashvili Denkleminin Conformable Türev ile Yaklaşık Çözümleri. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 12(2), 796–806 (2020)
D.G. Prakasha, P. Veeresha, H.M. Baskonus, Residual power series method for fractional Swift-Hohenberg equation. Fract. Fract. 3(1), 9 (2019)
K.K. Ali, R. Yilmazer, A. Yokus, H. Bulut, Analytical solutions for the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation in plasma physics. Phys. A Stat. Mech. İts Appl. 548(C), 124327 (2020)
A. Yokuş, H. Durur, Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G′) expansion method for nonlinear dynamic theory. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21(2), 590–599 (2019)
H. Durur, A. Yokuş, (1/G′)-Açılım Metodunu Kullanarak Sawada-Kotera Denkleminin Hiperbolik Yürüyen Dalga Çözümleri. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi 19(3), 615–619 (2019)
R. Silambarasan, H.M. Baskonus, H. Bulut, Jacobi elliptic function solutions of the double dispersive equation in the Murnaghan’s rod. Eur. Phys. J. Plus 134(3), 125 (2019)
H. Durur, Different types analytic solutions of the (1 + 1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method. Mod. Phys. Lett. B 34(03), 2050036 (2020)
H. Khan, S. Barak, P. Kumam, M. Arif, Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G′/G)-Expansion Method. Symmetry 11(4), 566 (2019)
I. Aziz, B. Šarler, The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets. Math. Comput. Model. 52(9–10), 1577–1590 (2010)
I. Aziz, M. Asif, Haar wavelet collocation method for three-dimensional elliptic partial differential equations. Comput. Math. Appl. 73(9), 2023–2034 (2017)
D. Kaya, A. Yokus, A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations. Math. Comput. Simul. 60(6), 507–512 (2002)
D. Kaya, A. Yokus, A decomposition method for finding solitary and periodic solutions for a coupled higher-dimensional Burgers equations. Appl. Math. Comput. 164(3), 857–864 (2005)
B. Faraj, M. Modanli, Using difference scheme method for the numerical solution of telegraph partial differential equation. J. Garmian Univ. 3, 157–163 (2017)
O. Tasbozan, A. Kurt, H. Durur, Implementation of New Sub Equation Method To Time Fractional Partial Differential Equations. Int. J. Eng. Math. Phys. 1, 1–12 (2019)
A. Kurt, O. Tasbozan, H. Durur, The exact solutions of conformable fractional partial differential equations using new sub equation method. Fundam. J. Math. Appl. 2(2), 173–179 (2020)
M. Yavuz, N. Ozdemir, Numerical inverse Laplace homotopy technique for fractional heat equations. Therm. Sci. 22(Suppl. 1), 185–194 (2018)
A. Yokus, H. Durur, H. Ahmad, S.W. Yao, Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation. Mathematics 8(6), 908 (2020)
H. Ahmad, M. Rafiq, C. Cesarano, H. Durur, Variational iteration algorithm-I with an auxiliary parameter for solving boundary value problems. Earthline J. Math. Sci. 3(2), 229–247 (2020)
H. Durur, O. Taşbozan, A. Kurt, M. Şenol, New wave solutions of time fractional Kadomtsev–Petviashvili equation arising in the evolution of nonlinear long waves of small amplitude. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 12(2), 807–815 (2020)
M.A. Shallal, K.K. Ali, K.R. Raslan, H. Rezazadeh, A. Bekir, Exact solutions of the conformable fractional EW and MEW equations by a new generalized expansion method. J. Ocean Eng. Sci. (2020). https://doi.org/10.1016/j.joes.2019.12.004
H. Durur, A. Kurt, O. Tasbozan, New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method. Appl. Math. Nonlinear Sci. 5(1), 455–460 (2020)
S.M. Mirhosseini-Alizamini, H. Rezazadeh, K. Srinivasa, A. Bekir, New closed form solutions of the new coupled Konno-Oono equation using the new extended direct algebraic method. Pramana 94(1), 1–12 (2020)
H. Ahmad, A.R. Seadawy, T.A. Khan, P. Thounthong, Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. J. Univ. Sci. 14(1), 346–358 (2020)
A.R. Seadawy, A.H. Arnous, A. Biswas, M. Belic, Optical solitons with Sasa-Satsuma equation by F-expansion scheme. Optoelectron. Adv. Mater. Rapid Commun. 13(1–2), 31–36 (2019)
A. Yokus, H. Durur, H. Ahmad, Hyperbolic Type Solutions For The Couple Boıtı-Leon-Pempinelli System. Facta Univ. Ser. Math. Inform. 35(2), 523–531 (2020)
H. Durur, A. Yokuş, Analytical solutions of Kolmogorov–Petrovskii–Piskunov equation. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22(2), 628–636 (2020)
H. Ahmad, T.A. Khan, H. Durur, G.M. Ismail, A. Yokus, Analytic approximate solutions of diffusion equations arising in oil pollution. J. Ocean Eng. Sci. (2020). https://doi.org/10.1016/j.joes.2020.05.002
H. Durur, O. Tasbozan, A. Kurt, New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations. Appl. Math. Nonlinear Sci. 5(1), 447–454 (2020)
M. Yavuz, A. Yokus, Analytical and numerical approaches to nerve impulse model of fractional-order. Numer. Methods Partial Differ. Equ. (2020). https://doi.org/10.1002/num.22476
A. Chen, W. Huang, S. Tang, Bifurcations of travelling wave solutions for the Gilson-Pickering equation. Nonlinear Anal. Real World Appl. 10(5), 2659–2665 (2009)
X. Fan, S. Yang, D. Zhao, Travelling wave solutions for the Gilson-Pickering equation by using the simplified G/G-expansion method. Int. J. Nonlinear Sci. 8, 368–373 (2009)
T. Ak, A. Saha, S. Dhawan, Performance of a hybrid computational scheme on traveling waves and its dynamic transition for Gilson-Pickering equation. Int. J. Mod. Phys. C 30(04), 1950028 (2019)
H.M. Baskonus, Complex soliton solutions to the Gilson-Pickering model. Axioms 8(1), 18 (2019)
K.K. Ali, R. Yilmazer, S. Noeiaghdam, Wave solutions of Gilson–Pickering Equation (2019). arXiv preprint arXiv:1907.06254
G. Ebadi, A.H. Kara, M.D. Petković, A. Biswas, Soliton solutions and conservation laws of the Gilson-Pickering equation. Waves Random Complex Media 21(2), 378–385 (2011)
F. Zabihi, M. Saffarian, A not-a-knot meshless method with radial basis functions for numerical solutions of Gilson-Pickering equation. Eng. Comput. 34(1), 37–44 (2018)
T. Muhammad, U.A. Aziz, S.O. Mohamed, B. Dumitru, M.A. Maysaa, Abundant periodic wave solutions for fifth-order Sawada-Kotera equations. Results Phys. 17, 103105 (2020)
N. Raza, S. Arshed, Chiral bright and dark soliton solutions of Schrödinger’s equation in (1+2)-dimensions. Ain Shams Eng. J. 20, 5–8 (2020). https://doi.org/10.1016/j.asej.2020.03.018
A.A. Kashif, J.F. Gomez-Aguilar, A comparison of heat and mass transfer on a Walter’s-B fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the Fox-H function. Eur. Phys. J. Plus 134, 101–113 (2019). https://doi.org/10.1140/epjp/i2019-12507-4
Ghanbari. Behzad, Raza N, An analytical method for soliton solutions of perturbed Schrödinger’s equation with quadratic-cubic nonlinearity. Mod. Phys. Lett. B 33(3), 1950018 (2019)
A.A. Kashif, A. Abdon, A comparative study of convective fluid motion in rotating cavity via Atangana-Baleanu and Caputo-Fabrizio fractal–fractional differentiations. Eur. Phys. J. Plus 135, 226–242 (2020). https://doi.org/10.1140/epjp/s13360-020-00136-x
N. Raza, A. Zubair, Bright, dark and dark-singular soliton solutions of nonlinear Schrödinger’s equation with spatio-temporal dispersion. J. Mod. Opt. 65, 1975–1982 (2018)
K.A. Abro, A. Yildirim, An analytic and mathematical synchronization of micropolar nanofluid by Caputo–Fabrizio approach. Sci. Iran. Int. J. Sci. Technol. 26(6), 3917–3927 (2019). https://doi.org/10.24200/sci.2019.52437.2717
U.A. Aziz, T. Muhammad, U.R. Hamood, Singular and bright singular combo optical solitons in birefringent to the Biswas-Arshed equation. Optik 210, 164489 (2020)
N. Raza, A. Javid, Optical dark and dark-singular soliton solutions of (1+2)-dimensional chiral nonlinear Schrodinger’s equation. Waves Rand. Compl. Med. (2018). https://doi.org/10.1080/17455030.2018.1451009
N. Raza, I. Murtaza, S. Sial, M. Younis, On solitons: the biomolecular nonlinear transmission line models with constant and time variable coefficients. Waves Random Complex Media 28, 553–569 (2017)
A.A. Kashif, A. Abdon, Role of non-integer and integer order differentiations on the relaxation phenomena of viscoelastic fluid. Phys. Scr. 95, 035228 (2020). https://doi.org/10.1088/1402-4896/ab560c
N. Raza, S. Sial, M. Kaplan, Exact periodic and explicit solutions of higher dimensional equations with fractional temporal evolution. Optik 156, 628–634 (2018)
K.A. Abro, S. Ambreen, A. Abdon, Thermal stratification of rotational second-grade fluid through fractional differential operators. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-09312-8
A. Javid, N. Raza, Singular and dark optical solitons to the well posed Lakshmanan–Porsezian–Daniel model. Optik 171, 120–129 (2018)
L. Bhojraj, A.A. Kashif, W.S. Abdul, Thermodynamical analysis of heat transfer of gravity-driven fluid flow via fractional treatment: an analytical study. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-09429-w
N. Raza, A. Javid, Optical dark and singular solitons to the Biswas-Milovic equation in nonlinear optics with spatio-temporal dispersion. Optik 158, 1049–1057 (2018)
K.A. Abro, F.G.A. Jose, Role of Fourier sine transform on the dynamical model of tensioned carbon nanotubes with fractional operator. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6655
A. Javid, N. Raza, M.S. Osman, Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets. Commun. Theor. Phys. 71, 362–366 (2019)
C. Gilson, A. Pickering, Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations. J. Phys. A Math. Gen. 28(10), 2871 (1995)
B. Fornberg, G.B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 289(1361), 373–404 (1978)
G.B. Whitham, Variational methods and applications to water waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 299(1456), 6–25 (1967)
G. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974)
P. Rosenau, J.M. Hyman, Compactons: solitons with finite wavelength. Phys. Rev. Lett. 70(5), 564 (1993)
R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661 (1993)
B. Fuchssteiner, A.S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4(1), 47–66 (1981)
M.J. Ablowitz, M.A. Ablowitz, P.A. Clarkson, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 (Cambridge University Press, Cambridge, 1991)
A. Yokus, B. Kuzu, U. Demiroğlu, Investigation of solitary wave solutions for the (3 + 1)-dimensional Zakharov-Kuznetsov equation. Int. J. Mod. Phys. B 33(29), 1950350 (2019)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yokuş, A., Durur, H., Abro, K.A. et al. Role of Gilson–Pickering equation for the different types of soliton solutions: a nonlinear analysis. Eur. Phys. J. Plus 135, 657 (2020). https://doi.org/10.1140/epjp/s13360-020-00646-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-020-00646-8