New solitary wave solutions and stability analysis of the Benney-Luke and the Phi-4 equations in mathematical physics

Behzad Ghanbari; Mustafa Inc; Abdullahi Yusuf; Dumitru Baleanu; Behzad Ghanbari; Mustafa Inc; Abdullahi Yusuf; Dumitru Baleanu

doi:10.3934/math.2019.6.1523

AIMS Mathematics

2019, Volume 4, Issue 6: 1523-1539. doi: 10.3934/math.2019.6.1523

Previous Article Next Article

Research article Special Issues

New solitary wave solutions and stability analysis of the Benney-Luke and the Phi-4 equations in mathematical physics

1.
Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran
2.
Firat University, Science Faculty, Department of Mathematics, 23119 Elazig, Turkey
3.
Federal University Dutse, Science Faculty, Department of Mathematics, 7156 Dutse, Nigeria
4.
Cankaya University, Science Faculty, Department of Mathematics, 06530 Ankara, Turkey
5.
Institute of Space Sciences, Magurele, Romania

Received: 28 June 2019 Accepted: 28 August 2019 Published: 24 September 2019
MSC : 35C08, 34K20, 32W50

In this paper, we present new solitary wave solutions for the Benney-Luke equation (BLE) and Phi-4 equation (PE). The new generalized rational function method (GERFM) is used to reach such solutions. Moreover, the stability for the governing equations is investigated via the aspect of linear stability analysis. It is proved that, both the governing equations are stable. We can also solve other nonlinear system of PDEs which are involve in mathematical physics and many other branches of physical sciences with the help of this new method.
- Benney-Luke equation,
- Phi-4 equation,
- GERFM,
- stability analysis
Citation: Behzad Ghanbari, Mustafa Inc, Abdullahi Yusuf, Dumitru Baleanu. New solitary wave solutions and stability analysis of the Benney-Luke and the Phi-4 equations in mathematical physics[J]. AIMS Mathematics, 2019, 4(6): 1523-1539. doi: 10.3934/math.2019.6.1523

Related Papers:

Abstract

In this paper, we present new solitary wave solutions for the Benney-Luke equation (BLE) and Phi-4 equation (PE). The new generalized rational function method (GERFM) is used to reach such solutions. Moreover, the stability for the governing equations is investigated via the aspect of linear stability analysis. It is proved that, both the governing equations are stable. We can also solve other nonlinear system of PDEs which are involve in mathematical physics and many other branches of physical sciences with the help of this new method.

References

[1]	M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, Cambridge, 1990.
[2]	F. Tchier, A. I. Aliyu, A. Yusuf, et al. Dynamics of solitons to the ill-posed Boussinesq equation, Eur. Phys. J. Plus, 132 (2017), 136.
[3]	F. Tchier, A. Yusuf, A. I. Aliyu, et al. Soliton solutions and conservation laws for lossy nonlinear transmission line equation, Superlattices Microstruct, 107 (2017), 320-336. doi: 10.1016/j.spmi.2017.04.003
[4]	W. X. Ma, A soliton hierarchy associated with so (3,R), Appl. Math. Comput., 220 (2013), 117-122.
[5]	E. Bas, B. Acay, R. Ozarslan, The price adjustment equation with different types of conformable derivatives in market equilibrium, AIMS Mathematics, 4 (2019), 805-820. doi: 10.3934/math.2019.3.805
[6]	B. Acay, E. Bas, T. Abdeljawad, Non-local fractional calculus from different viewpoint generated by truncated M-derivative, J. Comput. Appl. Math., 366 (2020), 112410.
[7]	S. Ali, M. Younis, M. O. Ahmad, et al. Rogue wave solutions in nonlinear optics with coupled Schrodinger equations, Opt. Quant. Electron., 50 (2018), 266.
[8]	N. Raza, I. G. Murtaza, S. Sial, et al. On solitons: the biomolecular nonlinear transmission line models with constant and time variable coefficients, Wave. Random Complex, 28 (2018), 553-569. doi: 10.1080/17455030.2017.1368734
[9]	M. Younis, S. T. R. Rizvi, S. Ali, Analytical and soliton solutions: Nonlinear model of nano-bioelectronics transmission lines, Appl. Math. Comput., 265 (2015), 994-1002.
[10]	S. Ali, S. T. R. Rizvi, M. Younis, Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients, Nonlinear Dynam., 82 (2015), 1755-1762. doi: 10.1007/s11071-015-2274-z
[11]	B. Younas, M. Younis, M. O. Ahmed, et al. Chirped optical solitons in nanofibers, Mod. Phys. Lett. B, 32 (2018), 1850320.
[12]	K. Ali, S. T. R. Rizvi, A. Khalil, et al. Chirped and dipole soliton in nonlinear negative-index materials, Optik, 172 (2018), 657-661. doi: 10.1016/j.ijleo.2018.06.063
[13]	K. U. Tariq, M. Younis, Bright, dark and other optical solitons with second order spatiotemporal dispersion, Optik, 142 (2017), 446-450. doi: 10.1016/j.ijleo.2017.06.003
[14]	M. Younis, Optical solitons in (n+1) dimensions with Kerr and power law nonlinearities, Mod. Phys. Lett. B, 31 (2017), 1750186.
[15]	M. Younis, U. Younas, S. ur Rehman, et al. Optical bright-dark and Gaussian soliton with third order dispersion, Optik, 134 (2017), 233-238. doi: 10.1016/j.ijleo.2017.01.053
[16]	E. Bas, B. Acay, R. Ozarslan, Fractional models with singular and non-singular kernels for energy efficient buildings, Chaos, 29 (2019), 023110.
[17]	J. H. He, Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons and Fractals, 19 (2004), 847-851. doi: 10.1016/S0960-0779(03)00265-0
[18]	G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994.
[19]	K. Khan, M. A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method, Ain Shams Eng. J., 4 (2013), 903-909. doi: 10.1016/j.asej.2013.01.010
[20]	K. Khan, M. A. Akbar, Traveling wave solutions of the (2+1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method, Ain Shams Eng. J., 5 (2014), 247-256. doi: 10.1016/j.asej.2013.07.007
[21]	A. Bekir, A. Boz, Exact solutions for nonlinear evolution equation using Exp-function method, Phys. Lett. A, 372 (2008), 1619-1625. doi: 10.1016/j.physleta.2007.10.018
[22]	H. O. Roshid, N. Rahman, M. A. Akbar, Traveling waves solutions of nonlinear Klein Gordon equation by extended (G/G)-expasion method, Ann. Pure Appl. Math., 3 (2013), 10-16.
[23]	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 (2019), 362.
[24]	M. S. Osman, One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation, Nonlinear Dyn., 96 (2019), 1491-1496. doi: 10.1007/s11071-019-04866-1
[25]	M. S. Osman, New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics, Pramana-J. Phys., 93 (2019), 26.
[26]	M. S. Osman, D. Lu, M. M. A. Khater, et al. Complex wave structures for abundant solutions related to the complex Ginzburg-Landau model, Optik, 192 (2019), 162927.
[27]	D. Lu, K. U. Tariq, M. S. Osman, et al. New analytical wave structures for the (3 + 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq models and their applications, Results phys., 14 (2019), 102491.
[28]	H. I. Abdel-Gawad, N. S. Elazab, M. Osman, Exact Solutions of Space Dependent Korteweg-de Vries Equation by The Extended Unified Method, J. Phys. Soc. Jpn, 82 (2013), 044004.
[29]	M. Osman, Multi-soliton rational solutions for some nonlinear evolution equations, Open Phys., 14 (2016), 26-36.
[30]	H. I. Abdel-Gawad and M. Osman, On shallow water waves in a medium withtime-dependent dispersion and nonlinearitycoefficients, J. Adv. Res., 6 (2015), 593-599. doi: 10.1016/j.jare.2014.02.004
[31]	B. Ghanbari, M. S. Osman, D. Baleanu, Generalized exponential rational function method for extended Zakharov-Kuzetsov equation with conformable derivative, Mod. Phys. Lett. A, 34 (2019), 1950155.
[32]	M. S. Osman, A. M. Wazwaz, A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Mathematical Methods in the Applied Sciences.
[33]	U. Khan, R. Ellahi, R. Khan, et al. Extracting new solitary wave solutions of Benny-Luke equation and Phi-4 equation of fractional order by using (G'/G)-expansion method, Opt. Quant. Electron., 49 (2017), 362.
[34]	B. Ghanbari, M. Inc, A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation, Eur. Phys. J. Plus, 133 (2018), 142.
[35]	M. Saha, A. K. Sarma, Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2420-2425. doi: 10.1016/j.cnsns.2012.12.028
[36]	A. R. Seadawy, M. Arshad, D. Lu, Stability analysis of new exact traveling wave solutions of new coupled KdV and new coupled Zakharov-Kuznetsov systems, Eur. Phys. J. Plus, 132 (2017), 162.
[37]	M. Inc, A. Yusuf, A. I. Aliyu, et al. Soliton solutions and stability analysis for some conformable nonlinear partial differential equations in mathematical physics, Opt. Quant. Electron., 50 (2018), 190.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)