The Functional Variable Method to Find New Exact Solutions of the Nonlinear Evolution Equations with Dual-Power-Law Nonlinearity

References

[1] M. Eslami, F. S. Khodadad, F. Nazari and H. Rezazadeh, The first integral method applied to the Bogoyavlenskii equations by means of conformable fractional derivative, Opt. Quantum Electron. 49(12) (2017), 391.10.1007/s11082-017-1224-zSearch in Google Scholar

[2] H. Aminikhah, A. R. Sheikhani and H. Rezazadeh, Exact solutions for the fractional differential equations by using the first integral method, Nonlinear Eng. 4(1) (2015), 15–22.10.1515/nleng-2014-0018Search in Google Scholar

[3] M. Mirzazadeh and M. Eslami, Exact solutions of the Kudryashov-Sinelshchikov equation and nonlinear telegraph equation via the first integral method, Nonlinear Anal. Modell. Control. 17(4) (2012), 481–488.10.15388/NA.17.4.14052Search in Google Scholar

[4] M. Eslami, M. Mirzazadeh and A. Neirameh, New exact wave solutions for Hirota equation, Pramana. 84(1) (2015), 3–8.10.1007/s12043-014-0837-zSearch in Google Scholar

[5] Z.-Y. Zhang, J. Zhong, S. S. Dou, J. Liu, D. Peng and T. Gao, First integral method and exact solutions to nonlinear partial differential equations arising in mathematical physics, Rom. Rep. Phys. 65(4) (2013), 1155–1169.Search in Google Scholar

[6] N. Taghizadeh, M. Mirzazadeh, M. Rahimian and M. Akbari, Application of the simplest equation method to some time-fractional partial differential equations, Ain Shams Eng. J. 4(4) (2013), 897–902.10.1016/j.asej.2013.01.006Search in Google Scholar

[7] K. U. Tariq, M. Younis, H. Rezazadeh, S. T. R. Rizvi and M. S. Osman, Optical solitons with quadratic–cubic nonlinearity and fractional temporal evolution, Mod. Phys. Lett. B. 32(26) (2018), 1850317.10.1142/S0217984918503177Search in Google Scholar

[8] M. Mirzazadeh, A. H. Arnous, M. F. Mahmood, E. Zerrad and A. Biswas, Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach, Nonlinear Dyn. 81(1–2) (2015), 277–282.10.1007/s11071-015-1989-1Search in Google Scholar

[9] H. Rezazadeh, M. S. Osman, M. Eslami, M. Ekici and M. Belic, Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity, Optik. 164 (2018), 84–92.10.1016/j.ijleo.2018.03.006Search in Google Scholar

[10] M. Eslami, Trial solution technique to chiral nonlinear Schrodinger’s equation in (1+2)-dimensions, Nonlinear Dyn. 85(2) (2016), 813–816.10.1007/s11071-016-2724-2Search in Google Scholar

[11] M. Eslami and M. Mirzazadeh, Exact solutions of modified Zakharov–Kuznetsov equation by the homogeneous balance method, Ain Shams Eng. J. 5(1) (2014), 221–225.10.1016/j.asej.2013.06.005Search in Google Scholar

[12] H. Rezazadeh, A. Korkmaz, M. Eslami, J. Vahidi and R. Asghari, Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method, Opt. Quantum Electron. 50(3) (2018), 150.10.1007/s11082-018-1416-1Search in Google Scholar

[13] H. Rezazadeh, J. Manafian, F. S. Khodadad and F. Nazari, Traveling wave solutions for density-dependent conformable fractional diffusion–reaction equation by the first integral method and the improved tan(φ(ξ)/2)-expansion, Opt. Quantum Electron. 51 (2018), 121.10.1007/s11082-018-1388-1Search in Google Scholar

[14] R. A. Talarposhti, S. E. Ghasemi, Y. Rahmani and D. D. Ganji, Application of Exp-function method to wave solutions of the Sine-Gordon and Ostrovsky equations, Acta Math. Appl. Sinica, English Series. 32(3) (2016), 571–578.10.1007/s10255-016-0571-zSearch in Google Scholar

[15] M. Eslami and A. Neirameh, New exact solutions for higher order nonlinear Schrödinger equation in optical fibers, Opt. Quantum Electron. 50(1) (2018), 47.10.1007/s11082-017-1310-2Search in Google Scholar

[16] A. Korkmaz, O. E. Hepson, K. Hosseini, H. Rezazadeh and M. Eslami, Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class, J. King Saud Univ. Sci. (2018). doi:10.1016/j.jksus.2018.08.013.Search in Google Scholar

[17] H. Rezazadeh, New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity, Optik. 167 (2018), 218–227.10.1016/j.ijleo.2018.04.026Search in Google Scholar

[18] H. Rezazadeh, H. Tariq, M. Eslami, M. Mirzazadeh and Q. Zhou, New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chin. J. Phys. 56(6) (2018), 2805–2816.10.1016/j.cjph.2018.08.001Search in Google Scholar

[19] V. S. Kumar, H. Rezazadeh, M. Eslami, F. Izadi and M. S. Osman, Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity, Int. J. Appl. Comput. Math. 5 (2019), 127.10.1007/s40819-019-0710-3Search in Google Scholar

[20] Z.-Y. Hang, X.-Y. Gan, D.-M. Yu, Y.-H. Zhang and X.-P. Li, A note on exact traveling wave solutions of the perturbed nonlinear Schrodingers equation with Kerr law nonlinearity, Commun. Theor. Phys. 57 (2012), 764–770.10.1088/0253-6102/57/5/05Search in Google Scholar

[21] Z.-Y. Zhang, J. Zhong, S. S. Dou, J. Liu, D. Peng and T. Gao, Abundant exact traveling wave solutions for the Klein-Gordon-Zakharov equations via the tanh-coth expansion method and and Jacobi elliptic function expansion method, Rom. J. Phys. 58(7-8) (2013), 749–765.Search in Google Scholar

[22] Z.-Y. Zhang, Jacobi elliptic function expansion method for the mKdV-ZK and the Hirota equations, Rom. J. Phys. 60(9–10) (2015), 1384–1394.Search in Google Scholar

[23] A. Biswas, M. Mirzazadeh and M. Eslami, Dispersive dark optical soliton with Schödinger-Hirota equation by (G′/G)-expansion approach in power law medium, Optik. 125(16) (2014), 4215–4218.10.1016/j.ijleo.2014.03.039Search in Google Scholar

[24] M. Eslami, A. Neyrame and M. Ebrahimi, Explicit solutions of nonlinear (2+1)-dimensional dispersive long wave equation, J. King Saud Univ. Sci. 24(1) (2012), 69–71.10.1016/j.jksus.2010.08.003Search in Google Scholar

[25] M. Mirzazadeh, M. Eslami and A. Biswas, Soliton solutions of the generalized Klein-Gordon equation by using (G’/G)-expansion method, Comput. Appl. Math. 33(3) (2014), 831–839.10.1007/s40314-013-0098-3Search in Google Scholar

[26] J. G. Liu, M. S. Osman, W. H. Zhu, L. Zhou and G.-P. Ai, Different complex wave structures described by the Hirota equation with variable coefficients in inhomogeneous optical fibers, Appl. Phys. B. 125 (2019), 175.10.1007/s00340-019-7287-8Search in Google Scholar

[27] X.-J. Miao and Z.-Y. Zhang, The modified (G′/G)-expansion method and traveling wave solutions of nonlinear the perturbed nonlinear Schrodingers equation with Kerr law nonlinearity, Commun. Nonlinear Sci. Numer. Simul. 16(11) (2011), 4259–4267.10.1016/j.cnsns.2011.03.032Search in Google Scholar

[28] Z.-Y. Zhang, J. Huang, J. Zhong, S. S. Dou, J. Liu, D. Peng and T. Gao, The extended (G′/G)-expansion method and travelling wave solutions for the perturbed nonlinear Schrodingers equation with Kerr law nonlinearity, Pramana. 82(6) (2014), 1011–1029.10.1007/s12043-014-0747-0Search in Google Scholar

[29] Z.-Y. Zhang and J. Wu, Generalized (G′/G)-expansion method and exact traveling wave solutions of the perturbed nonlinear Schrodinger’s equation with Kerr law nonlinearity in optical fiber materials, Opt. Quantum Electron. 49 (2017), 52.10.1007/s11082-016-0884-4Search in Google Scholar

[30] Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao and Y.-Z. Chen, Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrodingers equation with Kerr law nonlinearity, Phys. Lett. A. 375 (2011), 1275–1280.10.1016/j.physleta.2010.11.070Search in Google Scholar

[31] Z.-Y. Zhang, X.-Y. Gan and -M.-M. Yu, Bifurcation behavior of the traveling wave solutions of nonlinear the perturbed nonlinear Schrodingers equation with Kerr law nonlinearity, Zeitschrift fur Naturforschung A. 66a (2011), 721–727.10.5560/zna.2011-0041Search in Google Scholar

[32] Z.-Y. Zhang, F.-L. Xia and X.-P. Li, Bifurcation analysis and the travelling wave solutions of the Klein-Gordon-Zakharov equations, Pramana. 80(1) (2013), 41–59.10.1007/s12043-012-0357-7Search in Google Scholar

[33] Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao and Y.-Z. Chen, New exact solutions to the perturbed nonlinear Schrodingers equation with Kerr law nonlinearity, Appl. Math. Comput. 216 (2010), 3064–3072.10.1016/j.amc.2010.04.026Search in Google Scholar

[34] Z.-Y. Zhang, New exact traveling wave solutions for the nonlinear Klein-Gordon equation, Turkish J. Phys. 32 (2008), 235–240.Search in Google Scholar

[35] Z.-Y. Zhang, Y.-X. Li, Z.-H. Liu and X.-J. Miao, New exact solutions to the perturbed nonlinear Schrodingers equation with Kerr law nonlinearity via modified trigonometric function series method, Commun. Nonlinear Sci. Numer. Simul. 16(8) (2011), 3097–3106.10.1016/j.cnsns.2010.12.010Search in Google Scholar

[36] Z.-Y. Zhang, Y.-H. Zhang, X.-Y. Gan and D.-M. Yu, A note on exact traveling wave solutions for the Klein-Gordon-Zakharov equations, Zeitschrift fur Naturforschung. 67a (2012), 167–172.10.5560/zna.2012-0007Search in Google Scholar

[37] M. Eslami, Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations, Appl. Math. Comput. 285 (2016), 141–148.10.1016/j.amc.2016.03.032Search in Google Scholar

[38] M. Mirzazadeh, M. Eslami and A. Biswas, Dispersive optical solitons by Kudryashov’s method, Opt. Int. J. Light Electron Opt. 125(23) (2014), 6874–6880.10.1016/j.ijleo.2014.02.044Search in Google Scholar

[39] D. Lu, K. U. Tariq, M. S. Osman, D. Baleanu, M. Younis and M. M. A. Khater, New analytical wave structures for the (3 + 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq models and their applications, Results Phys. 14 (2019), 102491.10.1016/j.rinp.2019.102491Search in Google Scholar

[40] M. S. Osman, D. Lu, M. M. A. Khater and R. A. M. Attia, Complex wave structures for abundant solutions related to the complex Ginzburg–Landau model, Optik. 192 (2019), 162927.10.1016/j.ijleo.2019.06.027Search in Google Scholar

[41] H. I. Abdel-Gawad and M. S. Osman, On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients, J. Adv. Res. 6(4) (2015), 593–599.10.1016/j.jare.2014.02.004Search in Google Scholar PubMed PubMed Central

[42] Z.-Y. Zhang, Exact traveling wave solutions of the perturbed Klein-Gordon equation with quadratic nonlinearity in (1+1)-dimension, Part I-without local inductance and dissipation effect, Turk. J. Phys. 37 (2013), 259–267.10.3906/fiz-1205-13Search in Google Scholar

[43] B. Ghanbari, M. S. Osman and D. Baleanu, Generalized exponential rational function method for extended Zakharov–Kuzetsov equation with conformable derivative, Mod. Phys. Lett. A. 34(20) (2019), 1950155.10.1142/S0217732319501554Search in Google Scholar

[44] J.-G. Liu, M. Eslami, H. Rezazadeh and M. Mirzazadeh, Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev-Petviashvili equation, Nonlinear Dyn. 95(2) (2019), 1027–1033.10.1007/s11071-018-4612-4Search in Google Scholar

[45] M. S. Osman and A. M. Wazwaz, A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Math. Methods Appl. Sci. (in press).10.1002/mma.5721Search in Google Scholar

[46] M. S. Osman and H. I. Abdel-Gawad, Multi-wave solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equations with variable coefficients, Eur. Phys. J. Plus. 130 (2015), 215.10.1140/epjp/i2015-15215-1Search in Google Scholar

[47] H. I. Abdel-Gawad and M. S. Osman, On the variational approach for analyzing the stability of solutions of evolution equations, Kyungpook Math. J. 53(4) (2013), 661–680.10.5666/KMJ.2013.53.4.680Search in Google Scholar

[48] Z.-Y. Zhang, J. Zhong, S. S. Dou, J. Liu, D. Peng and T. Gao, A new method to construct traveling wave solutions for the Klein-Gordon Zakharov equations, Rom. Journ. Phys. 58(7-8) (2013), 766–777.Search in Google Scholar

[49] A. Zerarka, S. Ouamane and A. Attaf, On the functional variable method for finding exact solutions to a class of wave equations, Appl. Math. Comput. 217(7) (2010), 2897–2904.10.1016/j.amc.2010.08.070Search in Google Scholar

[50] A. Zerarka and S. Ouamane, Application of the functional variable method to a class of nonlinear wave equations, World J. Modell. Simul. 6(2) (2010), 150–160.Search in Google Scholar

[51] M. Eslami and M. Mirzazadeh, Functional variable method to study nonlinear evolution equations, Open Eng. 3(3) (2013), 451–458.10.2478/s13531-013-0104-ySearch in Google Scholar

[52] W. Djoudi and A. Zerarka, Exact structures for the KdV–mKdV equation with variable coefficients via the functional variable method, Optik. 127(20) (2016), 9621–9626.10.1016/j.ijleo.2016.07.045Search in Google Scholar

[53] A. Nazarzadeh, M. Eslami and M. Mirzazadeh, Exact solutions of some nonlinear partial differential equations using functional variable method, Pramana. 81(2) (2013), 225–236.10.1007/s12043-013-0565-9Search in Google Scholar

[54] A. C. Çevikel, A. Bekir, M. Akar and S. San, A procedure to construct exact solutions of nonlinear evolution equations, Pramana J. Phys. 79(3) (2012), 337–344.10.1007/s12043-012-0326-1Search in Google Scholar

[55] H. Aminikhah, A. H. R. Sheikhani and H. Rezazadeh, Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method, Boletim da Sociedade Paranaense de Matemática. 34(2) (2015), 213–229.10.5269/bspm.v34i2.25501Search in Google Scholar

[56] M. Eslami, H. Rezazadeh, M. Rezazadeh and S. S. Mosavi, Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation, Opt. Quantum Electron. 49(8) (2017), 279.10.1007/s11082-017-1112-6Search in Google Scholar

[57] H. Aminikhah, A. R. Sheikhani and H. Rezazadeh, Exact solutions of some nonlinear systems of partial differential equations by using the functional variable method, Mathematica. 56(2) (2016), 103–116.Search in Google Scholar

[58] A. Bekir, Ö. Güner, E. Aksoy and Y. Pandır (2015). Functional variable method for the nonlinear fractional differential equations. AIP Conference Proceedings, 1648, 73000110.1063/1.4912955Search in Google Scholar

[59] N. Hongsit, M. A. Allen and G. Rowlands, Growth rate of transverse instabilities of solitary pulse solutions to a family of modified Zakharov-Kuznetsov equations, Phys. Lett. A. 372(14) (2008), 2420–2422.10.1016/j.physleta.2007.12.005Search in Google Scholar

[60] A. Biswas and E. Zerrad, 1-soliton solution of the Zakharov-Kuznetsov equation with dual-power law nonlinearity, Commun. Nonlinear Sci. Numer. Simul. 14(9) (2009), 3574–3577.10.1016/j.cnsns.2008.10.004Search in Google Scholar

[61] A. Bekir, Application of the (G′/G)-expansion method for nonlinear evolution equations, Phys. Lett. A. 372(19) (2008), 3400–3406.10.1016/j.physleta.2008.01.057Search in Google Scholar

[62] G. Ebadi, Solitons and other solutions to Zakharov-Kuznetsov equation with dual-power law nonlinearity, Int. J. Nonlinear Sci. 16(3) (2013), 248–254.Search in Google Scholar

[63] X. L. Yang, J. S. Tang and Z. Qiao, Traveling wave solutions of the generalized BBM equation, Pacific J. Appl. Math. 1(3) (2009), 221–234.Search in Google Scholar

[64] A. El Achab and A. Bekir, Travelling wave solutions to the generalized Benjamin-Bona-Mahony (BBM) equation using the first integral method, Int. J. Nonlinear Sci. 19(1) (2015), 40–46.Search in Google Scholar

[65] A. Biswas, 1-Soliton solution of Benjamin–Bona–Mahoney equation with dual-power law nonlinearity, Commun. Nonlinear Sci. Numer. Simul. 15(10) (2010), 2744–2746.10.1016/j.cnsns.2009.10.023Search in Google Scholar

[66] X. Liu, L. Tian and Y. Wu, Exact solutions of the generalized Benjamin-Bona-Mahony equation, Math. Prob. Eng. 2010 (2010), 796398.10.1155/2010/796398Search in Google Scholar

[67] C. M. Khalique, Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities, Pramana. 80(3) (2013), 413–427.10.1007/s12043-012-0489-9Search in Google Scholar

[68] D. B. Belobo and T. Das, Solitary and Jacobi elliptic wave solutions of the generalized Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci. Numer. Simul. 48 (2017), 270–277.10.1016/j.cnsns.2017.01.001Search in Google Scholar

[69] W. Zhang, Q. Chang and B. Jiang, Explicit exact solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order, Chaos, Solitons Fractals. 13(2) (2002), 311–319.10.1016/S0960-0779(00)00272-1Search in Google Scholar

[70] M. Postolache, Y. Gurefe, A. Sonmezoglu, M. Ekici and E. Misirli, Extended trial equation method and applications to some nonlinear problems, UPB Sci. Bull. 76(2) (2014), 1223–7027.Search in Google Scholar

[71] Y. Gurefe, A. Sonmezoglu and E. Misirli, Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics, Pramana. 77(6) (2011), 1023–1029.10.1007/s12043-011-0201-5Search in Google Scholar

[72] B. Li, Y. Chen and H. Zhang, Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order, Phys. Lett. A. 305(6) (2002), 377–382.10.1016/S0375-9601(02)01515-3Search in Google Scholar

[73] H. Bulut, T. A. Sulaiman and H. M. Baskonus, New solitary and optical wave structures to the Korteweg-de Vries equation with dual-power law nonlinearity, Opt. Quantum Electron. 48(12) (2016), 564.10.1007/s11082-016-0831-4Search in Google Scholar

[74] B. Li, Y. Chen and H. Zhang, Explicit exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order, Chaos, Solitons Fractals. 15(4) (2003), 647–654.10.1016/S0960-0779(02)00152-2Search in Google Scholar

[75] M. Wadati, Wave propagation in nonlinear lattice. I, J. Phys. Soc. Jpn. 38(3) (1975), 673–680.10.1143/JPSJ.38.673Search in Google Scholar

[76] M. Wadati, Wave propagation in nonlinear lattice. II, J. Phys. Soc. Jpn. 38(3) (1975), 681–686.10.1143/JPSJ.38.681Search in Google Scholar

[77] B. Dey, Domain wall solutions of KdV like equations with higher order nonlinearity, J. Phys. A: Math. Gen. 19(1) (1986), L9.10.1088/0305-4470/19/1/003Search in Google Scholar

[78] M. W. Coffey, On series expansions giving closed-form solutions of Korteweg–de Vries-like equations, SIAM J. Appl. Math. 50(6) (1990), 1580–1592.10.1137/0150093Search in Google Scholar

[79] Z. I. Al-Muhiameed and E. A. B. Abdel-Salam, Generalized hyperbolic function solution to a class of nonlinear Schrödinger-type equations, J. Appl. Math. 2012 (2012), 265348.10.1155/2012/265348Search in Google Scholar

[80] E. Yomba, Generalized hyperbolic functions to find soliton-like solutions for a system of coupled nonlinear Schrödinger equations, Phys. Lett. A. 372(10) (2008), 1612–1618.10.1016/j.physleta.2007.10.015Search in Google Scholar