The investigation of exact solutions of Korteweg-de Vries equation with dual power law nonlinearity using the \(\exp_a\) and \(\exp(-\Phi (\xi))\) methods. (English) Zbl 1499.35533
Summary: In this work, the methodologies of the \(\exp_a\) and \(\exp(-\Phi (\xi))\) methods are used to investigate the solutions of Korteweg-de Vries (KdV) equation with dual power law nonlinearity. The obtained solutions consist of kink, anti-kink, logarithmic, trigonometric, hyperbolic and rational functions. Moreover, 3D and 2D graphics of some of these exact solutions have been plotted to visualize the underlying dynamics of proposed results.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Keywords:
the \(\exp_a\) method; the exp \(\exp(-\Phi (\xi))\)-expansion method; KdV equation; dual power law nonlinearity; exact solutionsReferences:
| [1] | Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, Vol. 149 (1991), New York: Cambridge University Press · Zbl 0762.35001 |
| [2] | Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform (1981), SIAM: SIAM, Philadelphia · Zbl 0472.35002 |
| [3] | Akram, G.; Batool, F., A class of traveling wave solutions for space-time fractional biological population model in mathematical physics, Indian J. Phys., 91, 1145-1148 (2017) |
| [4] | Akram, G.; Batool, F., Solitary wave solutions of the Schafer-Wayne short-pulse equation using two reliable methods, Opt. Quant. Electr., 49, 1, 1-9 (2017) |
| [5] | Ali, A. T.; Hassan, E. R., General \(####\) function method for nonlinear evolution equations, Appl. Math. Comput., 217, 2, 451-459 (2010) · Zbl 1201.65180 |
| [6] | Bai, C. L.; Zhao, H., Generalized extended tanh-function method and its application, Chaos Solitons Fractals, 27, 4, 1026-1035 (2006) · Zbl 1088.35534 |
| [7] | Bandyopadhyay, S., A new class of solutions of combined KdV-mKdV equation, Math. Phys., 1-4 (2014) |
| [8] | Batool, F.; Akram, G., Solitary wave solutions of \(####\)-dimensional soliton equation arising in mathematical physics, Optik, 144, 156-162 (2017) |
| [9] | Biswas, A., Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients, Nonlinear Dyn., 58, 1-2, 345-348 (2009) · Zbl 1183.35241 |
| [10] | Biswas, A.; Alqahtani, R. T.; Abdelkawy, M. A., Optical soliton perturbation with parabolic and dual-power law nonlinearities by semi-inverse variational principle, Optik, 147, 82-87 (2017) |
| [11] | Biswas, A.; Mirzazadeh, M., Dark optical solitons with power law nonlinearity using \(####\)-expansion method, Optik, 125, 4603-4608 (2014) |
| [12] | Biswas, A.; Yildirim, Y.; Yaser, E.; Triki, H.; Alshomrani, A. S.; Ullah, M. Z.; Zhou, Q.; Moshokoa, S. P.; Belic, M., Optical soliton perturbation with full nonlinearity by trial equation method, Optik, 157, 1366-1375 (2018) |
| [13] | Bulut, H.; Sulaiman, T. A.; Baskonus, H. M., New solitary and optical wave structures to the Korteweg-de vries equation with dual power law nonlinearity, Opt. Quantum Electr., 48, 12, 564 (2016) |
| [14] | Chen, Y.; Wang, Q., Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation, Chaos Solitons Fract., 24, 745-757 (2005) · Zbl 1072.35509 |
| [15] | Güner, Ö.; Bekir, A., Solving nonlinear space-time fractional differential equations via ansatz method, Comput. Methods Differ. Equ., 6, 1-11 (2018) · Zbl 1438.35429 |
| [16] | Hosseini, K.; Ansari, R., New exact solutions of nonlinear comformable time-fractional Bossinesq equations using the modified Kudryashov method, Waves Random Complex Media, 27, 628-636 (2017) · Zbl 07659364 |
| [17] | Hosseini, K.; Ayati, Z.; Ansari, R., New exact solutions of the Tzitzéica-type equations in non-linear optics using the \(####\) function method, J. Mod. Opt., 65, 7, 847-851 (2018) |
| [18] | Hosseini, K.; Bejarbaneh, Y. E.; Mayeli, P.; Zhou, Q., Traveling wave solutions of the Korteweg-de vries equation with dual-power law nonlinearity using the improved tan \(####\)-expansion method, Optik, 156, 498-504 (2018) |
| [19] | Hosseini, K.; Matinfar, M.; Mirzazadeh, M., Soliton solutions of high-order nonlinear Schrödinger equations with different laws of nonlinearities, Regular Chaotic Dyn., 26, 1, 105-112 (2021) · Zbl 1473.35506 |
| [20] | Hosseini, K.; Samavat, M.; Mirzazadeh, M.; Ma, W. X.; Hammouch, Z., A new \(####\)-dimensional Hirota bilinear equation: Its Bácklund transformation and rational-type solutions, Regular Chaotic Dyn, 25, 4, 383-391 (2020) · Zbl 1450.37065 |
| [21] | Hosseini, K.; Zabihi, A.; Samadani, F.; Ansari, R., New explicit exact solutions of the unstable nonlinear Schrödinger’s equation using the \(####\) and hyperbolic function methods, Opt. Quant. Electr., 50, 2, 82 (2018) |
| [22] | Kaplan, M.; Ozer, M. N., Multiple-soliton solutions and analytical solutions to a nonlinear evolution equation, Opt. Quant. Electron, 50, 1, 1-10 (2018) |
| [23] | Kumar, V. S.; Rezazadeh, H.; Eslami, M.; Izadi, F.; Osman, M. S., Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity, Int. J. Appl. Comput. Math., 5, 5, 1-10 (2019) · Zbl 1431.35155 |
| [24] | Luo, X. G.; Wu, Q. B.; Zhang, B. Q., Revisit on partial solutions in the adomian decomposition method: Solving heat and wave equations, J. Math. Anal. Appl., 321, 353-363 (2006) · Zbl 1103.65106 |
| [25] | Maimistov, A. I., Completely integrable models of nonlinear optics, Pramana J. Phys., 57, 953-968 (2001) |
| [26] | Mirzazadeh, M.; Eslami, M.; Zerrad, E.; Mahmood, M. F.; Biswas, A.; Belic, M., Optical solitons in nonlinear directional couplers by sine-cosine function method and Bernoullis equation approach, Nonlinear Dyn., 81, 1933-1949 (2015) · Zbl 1348.78023 |
| [27] | Postolache, M.; Gurefe, Y.; Sommezoglu, A.; Ekici, M.; Misirli, E., Extended trial equation method and application to some nonlinear problems, Bull. Univ. Politeh. Bucharest, Ser. A, 76, 2, 3-12 (2014) · Zbl 1313.35275 |
| [28] | Raza, N.; Rafiq, M. H.; Kaplan, M.; Kumar, S.; Chu, Y. M., The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations, Resul. Phys., 22 (2021) |
| [29] | Sajid, N.; Akram, G., The application of the exp \(####\)-expansion method for finding the exact solutions of two integrable equations, Math. Probl. Eng., 2018, 1-10 (2018) · Zbl 1427.35010 |
| [30] | Sajid, N.; Akram, G., Optical solitons with full nonlinearity for the conformable space-time fractional Fokas-Lenells equation, Optik, 196, 16 (2019) |
| [31] | Sajid, N.; Akram, G., Solitary dynamics of longitudinal wave equation arises in magneto-electro-elastic circular rod, Modern Phys. Lett. B, 35 (2020) |
| [32] | Sajid, N.; Akram, G., Novel solutions of Biswas-Arshed equation by newly \(####\)-model expansion method, Optik, 211 (2020) |
| [33] | Sajid, N.; Akram, G., Dark, singular, bright, rational and periodic solutions of the space-time fractional Fokas-Lenells equation by the \(####\)-model expansion method, Optik, 228 (2021) |
| [34] | Salas, A.; Kumer, S.; Yildririm, A.; Biswas, A., Cnoidal waves, solitary waves and painleve analysis of the 5th order KdV equation with dual-power law nonlinearity, Proc. Roman. Acad. Ser. A, 14, 1, 28-34 (2013) |
| [35] | Sulaiman, T. A., Three-component coupled nonlinear Schrödinger equation: Optical soliton and modulation instability analysis, Phys. Scrip., 95, 6 (2020) |
| [36] | Sulaiman, T. A.; Yusuf, A.; Atangana, A., New lump, lump-kink, breather waves and other interaction solutions to the \(####\)-dimensional soliton equation, Commun. Theor. Phys., 72, 8 (2020) · Zbl 1451.35052 |
| [37] | Triki, H.; Azzouzi, F.; Biswas, A.; Moshokoa, S. P.; Belic, M., Bright optical solitons with Kerr law nonlinearity and fifth order dispersion, Optik, 128, 172-177 (2017) |
| [38] | Triki, H.; Biswas, A.; Moshokoa, S. P.; Belic, M., Optical solitons and conservation laws with quadratic-cubic nonlinearity, Optik, 128, 63-70 (2017) |
| [39] | Triki, H.; Wazwaz, A. M., Bright and dark soliton solutions for a \(####\) equation with t-dependent coefficients, Phys. Lett. A, 373, 25, 2162-2165 (2009) · Zbl 1229.35232 |
| [40] | Triki, H.; Wazwaz, A. M., Soliton solutions for \(####\)-dimensional and \(####\)-dimensional \(####\) equations, Appl. Math. Comput., 217, 1733-1740 (2010) · Zbl 1203.35200 |
| [41] | Whithan, G. B., Linear and Nonlinear Waves (1974), Wiley: Wiley, New York · Zbl 0373.76001 |
| [42] | Younis, M.; Sulaiman, T. A.; Bilal, M.; Rehman, S. U.; Younas, U., Modulation instability analysis, optical and other solutions to the modified nonlinear Schrödinger equation, Commun. Theor. Phys., 72, 6 (2020) · Zbl 1451.82022 |
| [43] | Yusuf, A.; Sulaiman, T. A.; Bayram, M., Breather wave, lump-periodic solutions and some other interaction phenomena to the Caudrey-Dodd-Gibbon equation, Eur. Phys. J. Plus, 135, 7, 1-8 (2020) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
