Abstract
The present study suggests a large family of optical solutions to the Kundu–Eckhaus model. Modified form of an auxiliary equation approach have been used to set the solution families. Compatible variable transform reduces the governing equation to a non linear ordinary differential equation. The parameters of the predicted solution in finite series form are determined by substitution of the solution into this ODE. The results cover a large family of optical solutions that are determined by implementation of various techniques in the literature. In this perspective, the modifed form of the a auxiliary equation approach is evaluated as a generalization of different methods widely used to construct the solutions.
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Arnous, A.H., Ullah, M.Z., Moshokoa, S.P., Zhou, Q., Triki, H., Mirzazadeh, M., Biswas, A.: Optical solitons in nonlinear directional couplers with trial function scheme. Nonlinear Dyn. 88(3), 1891–1915 (2017)
Bekir, A.: Application of the (G′/G)-expansion method for nonlinear evolution equations. Phys. Lett. A 372(19), 3400–3406 (2008)
Bibi, S., Mohyud-Din, S.T., Khan, U., Ahmed, N.: Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order. Results Phys. 7, 4440–4450 (2017)
Biswas, A., Zhou, Q., Moshokoa, S.P., Triki, H., Belic, M., Alqahtani, R.T.: Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations. Opt. Int. J. Light Electron Opt. 145, 14–17 (2017a)
Biswas, A., Kara, A.H., Ullah, M.Z., Zhou, Q., Triki, H., Belic, M.: Conservation laws for cubic–quartic optical solitons in Kerr and power law media. Opt. Int. J. Light Electron Opt. 145, 650–654 (2017b)
Biswas, A., Ullah, M.Z., Zhou, Q., Moshokoa, S.P., Triki, H., Belic, M.: Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle. Opt. Int. J. Light Electron Opt. 145, 18–21 (2017c)
Biswas, A., Triki, H., Zhou, Q., Moshokoa, S.P., Ullah, M.Z., Belic, M.: Cubic–quartic optical solitons in Kerr and power law media. Opt. Int. J. Light Electron Opt. 144, 357–362 (2017d)
Biswas, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons having weak non-local nonlinearity by two integration schemes. Opt. Int. J. Light Electron Opt. 164, 380–384 (2018a)
Biswas, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Ekici, M., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with Fokas-Lenells equation using three exotic and efficient integration schemes. Opt. Int. J. Light Electron Opt. 165, 288–294 (2018b)
Ebadi, G., Biswas, A.: The G′/G method and 1-soliton solution of the Davey–Stewartson equation. Math. Comput. Model. 53(5–6), 694–698 (2011)
Ekici, M., Mirzazadeh, M., Sonmezoglu, A., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Dark and singular optical solitons with Kundu–Eckhaus equation by extended trial equation method and extended G′/G-expansion scheme. Opt. Int. J. Light Electron Opt. 127(22), 10490–10497 (2016)
El-Wakil, S.A., Abdou, M.A.: New exact travelling wave solutions using modified extended tanh-function method. Chaos Solitons Fractals 31(4), 840–852 (2007)
Elwakil, S.A., El-Labany, S.K., Zahran, M.A., Sabry, R.: Modified extended tanh-function method for solving nonlinear partial differential equations. Phys. Lett. A 299(2–3), 179–188 (2002)
Hosseini, K., Bekir, A., Ansari, R.: Exact solutions of nonlinear conformable time-fractional Boussinesq equations using the exp (-Φ (ξ))-expansion method. Opt. Quant. Electron. 49(4), 131 (2017)
Khater, M.M., Seadawy, A.R., Lu, D.: Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti–Leon–Pempinelli equations system and Time-fractional Cahn–Allen equation. Results Phys. 7, 2325–2333 (2017)
Khater, M.M., Seadawy, A.R., Lu, D.: Dispersive optical soliton solutions for higher order nonlinear Sasa-Satsuma equation in mono mode fibers via new auxiliary equation method. Superlattices Microstruct. 113, 346–358 (2018)
Korkmaz, A.: Exact solutions of space-time fractional EW and modified EW equations. Chaos Solitons Fractals 96, 132–138 (2017a)
Korkmaz, A.: Exact solutions to (3 + 1) conformable time fractional Jimbo–Miwa, Zakharov–Kuznetsov and modified Zakharov–Kuznetsov equations. Commun. Theor. Phys. 67(5), 479–482 (2017b)
Korkmaz, A.: Complex wave solutions to mathematical biology models I: Newell–Whitehead-Segel and Zeldovich equations. J. Comput. Nonlinear Dyn. 13(8), 081004 (2018)
Korkmaz, A., Hosseini, K.: Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods. Opt. Quant. Electron. 49(8), 278 (2017)
Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)
Li, W., Ma, G., Yu, W., Zhang, Y., Liu, M., Yang, C., Liu, W.: Soliton structures in the (1 + 1)-dimensional Ginzburg-Landau equation with a parity-time-symmetric potential in ultrafast optics. Chin. Phys. B 27(3), 030504 (2018)
Liu, W., Liu, M., OuYang, Y., Hou, H., Ma, G., Lei, M., Wei, Z.: Tungsten diselenide for mode-locked erbium-doped fiber lasers with short pulse duration. Nanotechnology 29(17), 174002 (2018a)
Liu, W., Zhu, Y.N., Liu, M., Wen, B., Fang, S., Teng, H., Lei, M., Liu, L.M., Wei, Z.: Optical properties and applications for MoS 2-Sb 2 Te 3-MoS 2 heterostructure materials. Photon. Res. 6(3), 220–227 (2018b)
Liu, M., Liu, W., Pang, L., Teng, H., Fang, S., Wei, Z.: Ultrashort pulse generation in mode-locked erbium-doped fiber lasers with tungsten disulfide saturable absorber. Opt. Commun. 406, 72–75 (2018c)
Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via tan (ϕ (ξ))-expansion method. Opt. Int. J. Light Electron Opt. 127(14), 5543–5551 (2016)
Mirzazadeh, M.: Analytical study of solitons to nonlinear time fractional parabolic equations. Nonlinear Dyn. 85(4), 2569–2576 (2016)
Rezazadeh, H., Manafian, J., Khodadad, F.S., Nazari, F.: Traveling wave solutions for density-dependent conformable fractional diffusion–reaction equation by the first integral method and the improved tan (ϕ (ξ))-expansion method. Opt. Quant. Electron. 50(3), 121 (2018a)
Rezazadeh, H., Osman, M.S., Eslami, M., Ekici, M., Sonmezoglu, A., Asma, M., Othman, W.A.M., Wong, B.R., Mirzazadeh, M., Zhou, Q., Biswas, A., Belic, M.: Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic–cubic nonlinearity. Opt. Int. J. Light Electron Opt. 164, 84–92 (2018b)
Rezazadeh, H., Korkmaz, A., Eslami, M., Vahidi, J., Asghari, R.: Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method. Opt. Quant. Electron. 50(3), 150 (2018c)
Soliman, A.A.: The modified extended tanh-function method for solving Burgers-type equations. Physica A 361(2), 394–404 (2006)
Sonmezoglu, A., Yao, M., Ekici, M., Mirzazadeh, M., Zhou, Q.: Explicit solitons in the parabolic law nonlinear negative-index materials. Nonlinear Dyn. 88(1), 595–607 (2017)
Taghizadeh, N., Mirzazadeh, M., Tascan, F.: The first-integral method applied to the Eckhaus equation. Appl. Math. Lett. 25(5), 798–802 (2012)
Triki, H., Alqahtani, R.T., Zhou, Q., Biswas, A.: New envelope solitons for Gerdjikov-Ivanov model in nonlinear fiber optics. Superlattices Microstruct. 111, 326–334 (2017)
Wang, M., Li, X., Zhang, J.: The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008)
Yang, X.F., Deng, Z.C., Wei, Y.: A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv. Differ. Equ. 2015(1), 117–133 (2015)
Yang, C., Li, W., Yu, W., Liu, M., Zhang, Y., Ma, G., Lei, M., Liu, W.: Amplification, reshaping, fission and annihilation of optical solitons in dispersion-decreasing fiber. Nonlinear Dyn. 92(2), 203–213 (2018)
Zhang, J., Jiang, F., Zhao, X.: An improved (G′/G)-expansion method for solving nonlinear evolution equations. Int. J. Comput. Math. 87(8), 1716–1725 (2010)
Zhou, Q.: Analytic study on optical solitons in a Kerr-law medium with an imprinted parity-time-symmetric mixed linear-nonlinear lattice. Proc. Rom. Acad. Ser. A 18, 223–230 (2017)
Zhou, Q., Biswas, A.: Optical solitons in parity-time-symmetric mixed linear and nonlinear lattice with non-Kerr law nonlinearity. Superlattices Microstruct. 109, 588–598 (2017)
Zhou, Q., Mirzazadeh, M., Ekici, M., Sonmezoglu, A.: Analytical study of solitons in non-Kerr nonlinear negative-index materials. Nonlinear Dyn. 86(1), 623–638 (2016)
Zhou, Q., Ekici, M., Mirzazadeh, M., Sonmezoglu, A.: The investigation of soliton solutions of the coupled sine-Gordon equation in nonlinear optics. J. Mod. Opt. 64(16), 1677–1682 (2017)
Zuo, J.M., Zhang, Y.M.: Application of the (G′/G)-expansion method to solve coupled MKdV equations and coupled Hirota-Satsuma coupled KdV equations. Appl. Math. Comput. 217(12), 5936–5941 (2011)
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Rezazadeh, H., Korkmaz, A., Eslami, M. et al. A large family of optical solutions to Kundu–Eckhaus model by a new auxiliary equation method. Opt Quant Electron 51, 84 (2019). https://doi.org/10.1007/s11082-019-1801-4
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DOI: https://doi.org/10.1007/s11082-019-1801-4