Abstract
The novel Φ6-model expansion method is employed to the Kundu–Mukherjee–Naskar model of the nonlinear partial differential equation that plays a significant role in optical fiber. A variety of nonlinear dynamical exact and optical solitons are extracted in several forms like rational, hyperbolic, trigonometric function solutions by the utilization of a sound computational integration tool. Besides, we also secure mixed combined solitons and singular periodic wave solutions with unknown parameters. For the validation of the solutions constraint conditions are also emerged. The outcomes elucidate that the governing model theoretically possesses significantly rich structures of optical solutions. We also present the modulation instability analysis of the governing model. Moreover, under the suitable choice of involved parameters 3-D, 2-D, and their corresponding contour plots are also sketched.






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Alam, M.N., Seadawy, A.R., Baleanu, D.: Closed-form solutions to the solitary wave equation in an unmagnetized dusty plasma. Alex. Eng. J. 59(3), 1505–1514 (2020)
Ali, K., Rizvi, S.T.R., Nawaz, B., Younis, M.: Optical solitons for paraxial wave equation in Kerr media. Mod. Phys. Lett. B 33(03), 1950020 (2019)
Ali, K.K., Rezazadeh, H., Talarposhti, R.A., Bekir, A.: New soliton solutions for resonant nonlinear Schrödinger’s equation having full nonlinearity. Int. J. Mod. Phys. B 34(06), 2050032 (2020)
Aslan, E.C., Inc, M., Baleanu, D.: Optical solitons and stability analysis of the NLSE with anti-cubic nonlinearity. Superlattices Microstruct. 109, 784–793 (2017)
Baleanu, D., Inc, M., Aliyu, I.A., Yusuf, A.: Optical solitons, nonlinear self-adjointness and conservation laws for the cubic nonlinear Shrödinger’s equation with repulsive delta potential. Superlattices Microstruct. 111, 546–555 (2017a)
Baleanu, D., Inc, M., Yusuf, A, Aliyu, I.A.: Optical solitons, nonlinear self-adjointness and conservation laws for Kundu–Eckhaus equation. Chin. J. Phys. 55(06), 2341–2355 (2017b)
Baskonus, H.M.: New acoustic wave behaviors to the Davey–Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 86(1), 177–183 (2016)
Baskonus, H.M., Sulaiman, T.A., Bulut, H.: Dark, bright and other optical solitons to the decoupled nonlinear Schrödinger equation arising in dual-core optical fibers. Opt. Quantum Electron. 50(4), 1–12 (2018)
Baskonus, H.M., Gòmez-Aguilar, J.F.: New singular soliton solutions to the longitudinal wave equation in a magneto-electro-elastic circular rod with M-derivative. Mod. Phys. Lett. B 33(21), 1950251 (2019a)
Baskonus, H.M., Sulaiman, T.A., Bulut, H.: On the new wave behavior to the Klein–Gordon–Zakharov equations in plasma physics. Indian J. Phys. 93(3), 393–399 (2019b)
Bilal, M., Seadawy, A.R., Younis, M., Rizvi, S.T.R., El-Rashidy, K., Mahmoud, S.F.: Analytical wave structures in plasma physics modelled by Gilson–Pickering equation by two integration norms. Results Phys. 23, 103959 (2021a)
Bilal, M., Seadawy, A.R., Younis, M., Rizvi, S.T.R., Zahed, H.: Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis. Math. Methods Appl. Sci. 44(05), 4094–4104 (2021b)
Biswas, A., Mirzazadeh, M., Eslami, M., Zhou, Q., Bhrawy, A., Belic, M.: Optical solitons in nano-fibers with spatio-temporal dispersion by trial solution method. Optik 127(18), 7250–7257 (2016)
Ebaida, A., El-Zahar, E.R., Aljohani, A.F., Salah, B., Krid, M., Machado, J.T.: Exact solutions of the generalized nonlinear Fokas–Lennells equation. Results Phys. 14, 102472 (2019)
Ekici, M., Sonmezoglu, A., Biswas, A., Belic, M.R.: Optical solitons in (2+1)-dimensions with Kundu–Mukherjee–Naskar equation by extended trial function scheme. Chin. J. Phys. 57, 72–77 (2019)
Farah, N., Seadawy, A.R., Ahmad, S., Rizvi, S.T.R., Younis, M.: Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model. Opt. Quantum Electron. 52(7), 1–15 (2020)
Gao, W., Yel, G., Baskonus, H.M., Cattani, C.: Complex solitons in the conformable (2+1)-dimensional Ablowitz–Kaup–Newell–Segur equation. AIMS Math. 5(1), 507–521 (2020)
Ghanbari, B., Yusuf, A., Baleanu, D., Bayram, M.: Families of exact solutions of Biswas–Milovic equation by an exponential rational function method. Tbilisi Math. J. 13(2), 39–65 (2020)
Guirao, J.L.G., Baskonus, H.M., Kumar, A.: Regarding new wave patterns of the newly extended nonlinear (2+1)-dimensional Boussinesq equation with fourth order. Mathematics 8(3), 341 (2020)
Guo, D., Tian, S.F., Zhang, T.T., Li, J.: Modulation instability analysis and soliton solutions of an integrable coupled nonlinear Schrödinger system. Nonlinear Dyn. 94(4), 2749–2761 (2018)
He, J.H., Yusry, O.E.D.: Periodic property of the time-fractional Kundu–Mukherjee–Naskar equation. Results Phys. 19, 103345 (2020)
Ilhan, O.A., Manafian, J., Alizadeh, A., Baskonus, H.M.: New exact solutions for nematicons in liquid crystals by the \(\tan (\frac{\phi }{2})\)-expansion method arising in fluid mechanics. Eur. Phys. J. Plus 135(3), 1–19 (2020)
Inc, M., Aliyu, I.A., Yusuf, A.: Optical solitons to the nonlinear Shrödinger’s equation with spatio-temporal dispersion using complex amplitude ansatz. J. Mod. Opt. 64(21), 2273–2280 (2017a)
Inc, M., Aliyu, I.A., Yusuf, A., Baleanu, D.: Optical solitons and modulation instability analysis of an integrable model of (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Superlattices Microstruct. 112, 628–638 (2017b)
Inc, M., Aliyu, I.A., Yusuf, A., Baleanu, D., Nuray, E.: Complexiton and solitary wave solutions of the coupled nonlinear Maccari’s system using two integration schemes. Mod. Phys. Lett. B 32(02), 1850014 (2018a)
Inc, M., Aliyu, I.A., Yusuf, A., Baleanu, D.: Optical solitary waves, conservation laws and modulation instability analysis to the nonlinear Schrödinger’s equation in compressional dispersive Alvèn waves. Optik 155, 319–327 (2018b)
Inc, M., Aliyu, I.A., Yusuf, A., Baleanu, D.: Dispersive optical solitons and modulation instability analysis of Schrödinger-Hirota equation with spatio-temporal dispersion and Kerr law nonlinearity. Superlattices Microstruct. 113, 319–327(2018c)
Inc, M., Aliyu, I.A., Yusuf, A., Baleanu, D.: Optical solitons, conservation laws and modulation instability analysis for the modified nonlinear Schrödinger’s equation for Davydov solitons. J. Electromagn. Waves Appl. 32(07), 858–873 (2018d)
Kundu, A., Mukherjee, A., Naskar, T.: Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents. Proc. R. Soc. A 470, 20130576 (2014)
Miah, M.M., Ali, H.M.S., Akbar, M.A., Seadawy, A.R.: New applications of the two variable \((\frac{G^{\prime }}{G}, \frac{1}{G})\)-expansion method for closed form traveling wave solutions of integro-differential equations. JOES 4(2), 132–143 (2019)
Mirhosseni-Alizamini, S.M., Rezazadeh, H., Srinivasa, K., Bekir, A.: New closed form solutions of the new coupled Konno–Oono equation using the new extended direct algebraic method. Pramana 94, 52 (2020)
Nestor, S., Houwe, A., Betchewe, G., Inc, M., Doka, S.Y.: A series of abundant new optical solitons to the conformable space-time fractional perturbed nonlinear Schrödinger equation. Phys. Scr. 95(08), 085108 (2020)
Osman, M.S.: New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics. Pramana 93, 1–10 (2019)
Rehman, S.U., Ahmad, J.: Modulation instability analysis and optical solitons in birefringent fibers to RKL equation without four wave mixing. Alex. Eng. J. 60, 1339–1354 (2020)
Rehman, S.U., Seadawy, A.R., Younis, M., Rizvi, S.T.R., Sulaiman, T.A., Yusuf, A.: Modulation instability analysis and optical solitons of the generalized model for description of propagation pulses in optical fiber with four non-linear terms. Mod. Phys. Lett. B 35(06), 2150112 (2021)
Seadawy, A.R., Lu, D., Nasreen, N.: Construction of solitary wave solutions of some nonlinear dynamical system arising in nonlinear water wave models. Indian J. Phys. 94, 1785–1794 (2020a)
Seadawy, A.R., Nasreen, N., Lu, D., Arshad, M.: Arising wave propagation in nonlinear media for the (2+ 1)-dimensional Heisenberg ferromagnetic spin chain dynamical model. Physica A 538, 122846 (2020b)
Seadawy, A.R., Rehman, S.U., Younis, M., Rizvi, S.T.R., Althobaiti, S., Makhlouf, M.M.: Modulation instability analysis and longitudinal wave propagation in an elastic cylindrical rod modelled with Pochhammer–Chree equation. Physica scr. 96(4), 045202 (2021a)
Seadawy, A.R., Bilal, M., Younis, M., Rizvi, S.T.R., Althobaiti, S., Makhlouf, M.M.: Analytical mathematical approaches for the double-chain model of DNA by a novel computational technique. Chaos Solitons Fractals 144, 110669 (2021b)
Sirisubtawee, S., Koonprasert, S.: Exact traveling wave solutions of certain nonlinear partial differential equations using the \((\frac{G^{^{\prime }}}{G^2})\)-expansion method. Adv. Math. Phys. 2018, 7628651 (2018)
Sulaiman, T.A., Bulut, H., Yokus, A., Baskonus, H.M.: On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering. Indian J. Phys. 93(5), 647–656 (2019)
Wen, X.: Higher-order rational solutions for the (2+1)-dimensional KMN equation. Proc. Rom. Acad. Ser. A 18(3), 191–198 (2017)
Yildirim, Y.: Optical solitons to Kundu–Mukherjee–Naskar model with trial equation approach. Optik 183, 1061–1065 (2019)
Younis, M., Younas, U., Rehman, S.U., Bilal, M., Waheed, A.: Optical bright-dark and Gaussian soliton with third order dispersion. Optik 134, 233–238 (2017)
Yokus, A., Baskonus, H.M., Sulaiman, T.A., Bulut, H.: Numerical simulations and solutions of the two component second order KdV evolutionary system. Numer. Methods Partial Differ. Equ. 34(1), 211–227 (2018)
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(06), 065001 (2020a)
Younis, M., Bilal, M., Rehman, S.U., Younas, U., Rizvi, S.T.R.: Investigation of optical solitons in birefringent polarization preserving fibers with four-wave mixing effect. Int. J. Mod. Phys. B 34(11), 2050113 (2020b)
Yue, C., Lu, D., Khater, M.M.A., Abdel-Aty, A.H., Alharbi, W., Attia, R.A.M.: On explicit wave solutions of the fractional nonlinear DSW system via the modified Khater method. Fractals 28(08), 2040034 (2020)
Zayed, E.M.E., Nowehy, A.G.A., Elshater, M.E.M.: New \(\Phi ^6\)-model expansion method and its applications to the resonant nonlinear Schrodinger equation with parabolic law nonlinearity. Eur. Phys. J. Plus 133, 417 (2018)
Zhang, Q.M., Xiong, M., Chen, L.W.: The first integral method for solving exact solutions of two higher order nonlinear Schrödinger equations. Adv. Appl. Math. 4(1), 1–9 (2019)
Zulfiqar, A., Ahmad, J.: Soliton solutions of fractional modified unstable Schrödinger equation using exp-function method. Results Phys. 19, 103476 (2020)
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Bilal, M., Shafqat-Ur-Rehman & Ahmad, J. Investigation of optical solitons and modulation instability analysis to the Kundu–Mukherjee–Naskar model. Opt Quant Electron 53, 283 (2021). https://doi.org/10.1007/s11082-021-02939-3
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DOI: https://doi.org/10.1007/s11082-021-02939-3