# American Institute of Mathematical Sciences

March  2020, 13(3): 925-936. doi: 10.3934/dcdss.2020054

## Optical solitons to the fractional perturbed NLSE in nano-fibers

 1 Firat University, Faculty of Science, 23119 Elazig, Turkey 2 Federal University Dutse, Faculty of Science, 7156 Jigawa, Nigeria 3 Final International University, Faculty of Education, Kyrenia, Cyprus 4 Harran University, Faculty of Education, 63290 Sanliurfa, Turkey

* Corresponding author: Haci Mehmet Baskonus

Received  June 2018 Revised  August 2018 Published  March 2019

In this paper, we study the space-time fractional perturbed nonlinear Schr$\bf{\ddot o}$dinger equation under the Kerr law nonlinearity by using the extended sinh-Gordon equation expansion method. The perturbed nonlinear Schr$\bf{\ddot o}$dinger equation is a nonlinear model which arises in nano-fibers. Some family of optical solitons and singular periodic wave solutions are successfully revealed. The parametric conditions for the existence of valid solitons are stated. Under the choice of suitable values of the parameters, the 3-dimensional and 2-dimensional graphs to some of the reported solutions are plotted.

Citation: Tukur Abdulkadir Sulaiman, Hasan Bulut, Haci Mehmet Baskonus. Optical solitons to the fractional perturbed NLSE in nano-fibers. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 925-936. doi: 10.3934/dcdss.2020054
##### References:
 [1] A. Abdon and B. Dumitru, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.   Google Scholar [2] M. A. Akinlar and M. Kurulay, A novel method for analytical solutions of fractional partial differential equations, Mathematical Problems in Engineering, 2013 (2013), Art. ID 195708, 4 pp. doi: 10.1155/2013/195708.  Google Scholar [3] K. K. Ali, R. I. Nuruddeen and K. R. Raslan, New structures for the space-time fractional simplified MCH and SRLW equations, Chaos, Solitons and Fractals, 106 (2018), 304-309.  doi: 10.1016/j.chaos.2017.11.038.  Google Scholar [4] S. Arbabi and M. Najafi, Exact solitary wave solutions of the complex nonlinear Schr$\bf{\ddot o}$dinger equations, Optik, 127 (2016), 4682-4688.   Google Scholar [5] A. H. Arnous, M. Z. Ullah, M. Asma, S. P. Moshokoa, Q. Zhou, M. Mirzazadeh, A. Biswas and M. Belic, Dark and singular dispersive optical solitons of Schr$\bf{\ddot o}$dinger-Hirota equation by modified simple equation method, Optik, 136 (2017), 445-450.   Google Scholar [6] M. Arshad, A. R. Seadawy and D. Lu, Elliptic function and solitary wave solutions of the higher-order nonlinear Schr$\bf{\ddot o}$dinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability, The European Physical Journal Plus, 132 (2017), 371.   Google Scholar [7] A. Atangana and D. Baleanu, Application of Fixed Point Theorem for Stability Analysis of a Nonlinear Schrodinger with Caputo-Liouville Derivatives, Filomat, 31 (2017), 2243-2248.  doi: 10.2298/FIL1708243A.  Google Scholar [8] E. Bas, R. Yilmaza and E. Panakhov, Fractional solutions of bessel equation with $N$-method, The Scientific World Journal, 2013 (2013), Article ID 685695, 8 pages. doi: 10.1155/2013/685695.  Google Scholar [9] H. M. Baskonus, T. A. Sulaiman, H. Bulut and T. Akturk, Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schr$\bf{\ddot o}$dinger equation with $\delta$-potential, Superlattices and Microstructures, 115 (2016), 19-29.   Google Scholar [10] H. M. Baskonus, H. Bulut and T. A. Sulaiman, Investigation of various travelling wave solutions to the extended (2+1)-dimensional quantum ZK equation, The European Physical Journal Plus, 132 (2017), 482.  doi: 10.1140/epjp/i2017-11778-y.  Google Scholar [11] H. M. Baskonus, T. A. Sulaiman and H. Bulut, Dark, bright and other optical solitons to the decoupled nonlinear Schr$\bf{\ddot o}$dinger equation arising in dual-core optical fibers, Opt Quant Electron, 50 (2018), 165.   Google Scholar [12] H. M. Baskonus, T. A. Sulaiman and H. Bulut, Bright, dark optical and other solitons to the generalized higher-order NLSE in optical fibers, Opt Quant Electron, 50 (2018), 253.  doi: 10.1007/s11082-018-1522-0.  Google Scholar [13] I. Bendahmane, H. Triki, A. Biswas, A. S. Alshomrani, Q. Zhou, S. P. Moshokoa and M. Belic, Bright, dark and W-shaped solitons with extended nonlinear Schr$\bf{\ddot o}$dinger's equation for odd and even higher-order terms, Superlattices Microstruct., 114 (2018), 53-61.   Google Scholar [14] A. H. Bhrawy, A. A. Alshaery, E. M. Hilal, Z. Jovanoski and A. Biswas, Bright and dark solitons in a cascaded system, Optik, 125 (2014), 6162-6165.  doi: 10.1016/j.ijleo.2014.06.118.  Google Scholar [15] A. Biswas, M. Ekici, A. Sonmezoglu, Q. Zhou, S. P. Moshokoa and M. Belic, Optical soliton perturbation with full nonlinearity for Kundu-Eckhaus equation by extended trial function scheme, Optik, 160 (2018), 17-23.  doi: 10.1016/j.ijleo.2018.01.111.  Google Scholar [16] A. Biswas, Q. Zhou, S. P. Moshokoa, H. Triki, M. Belic and R. T. Alqahtani, Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations, Optik, 145 (2017), 14-17.  doi: 10.1016/j.ijleo.2017.07.036.  Google Scholar [17] A. Biswas, Q. Zhou, M. Z. Ullah, H. Triki, S. P. Moshokoa and M. Belic, Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle, Optik, 143 (2017), 131-134.  doi: 10.1016/j.ijleo.2017.06.087.  Google Scholar [18] A. Biswas, A. H. Kara, M. Z. Ullah, Q. Zhou, H. Triki and M. Belic, Conservation laws for cubic-quartic optical solitons in Kerr and power law media, Optik, 145 (2017), 650-654.  doi: 10.1016/j.ijleo.2017.08.047.  Google Scholar [19] A. Biswas, H. Triki, Q. Zhou, S. P. Moshokoa, M. Z. Ullah and M. Belic, Cubic-quartic optical solitons in Kerr and power law media, Commun. Theor. Phys., 144 (2017), 357-362.  doi: 10.1016/j.ijleo.2017.07.008.  Google Scholar [20] H. Bulut, T. A. Sulaiman and B. Demirdag, Dynamics of soliton solutions in the chiral nonlinear Schr$\bf{\ddot o}$dinger equations, Nonlinear Dynamics, 91 (2018), 1985-1991.   Google Scholar [21] H. Bulut, T. A. Sulaiman, H. M. Baskonus and T. Akturk, Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media, Opt Quant Electron, 50 (2018), 19.  doi: 10.1007/s11082-017-1286-y.  Google Scholar [22] H. Bulut, T. A. Sulaiman, H. M. Baskonus, H. Rezazadeh, M. Eslami and M. Mirzazadeh, Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation, Optik, 127 (2018), 20-27.  doi: 10.1016/j.ijleo.2018.06.108.  Google Scholar [23] H. Bulut, T. A. Sulaiman and H. M. Baskonus, On the new soliton and optical wave structures to some nonlinear evolution equation, The European Physical Journal Plus, 132 (2017), 459.  doi: 10.1140/epjp/i2017-11738-7.  Google Scholar [24] C. Cattani, Harmonic wavelet solutions of the Schrodinger equation, International Journal of Fluid Mechanics Research, 30 (2003), 463-472.  doi: 10.1615/InterJFluidMechRes.v30.i5.10.  Google Scholar [25] C. Cattani, T. A. Sulaiman, H. M. Baskonus and H. Bulut, Solitons in an inhomogeneous Murnaghan's rod, Eur. Phys. J. Plus, 133 (2018), 228.  doi: 10.1140/epjp/i2018-12085-y.  Google Scholar [26] C. Cattani, T. A. Sulaiman, H. M. Baskonus and H. Bulut, On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel'd-Sokolov systems, Opt Quant Electron, 50 (2018), 138.  doi: 10.1007/s11082-018-1406-3.  Google Scholar [27] M. T. Darvishi, S. Ahmadian, S. B. Arbabi and M. Najafi, Optical solitons for a family of nonlinear (1+1)-dimensional time-space fractional Schr$\bf{\ddot o}$dinger models, Optical and Quantum Electronics, 50 (2018), 32.   Google Scholar [28] M. Ekici, A. Sonmezoglu, Q. Zhou, S. P. Moshokoa, M. Z. Ullah, A. H. Arnous, A. Biswas and M. Belic, Analysis of optical solitons in nonlinear negative-indexed materials with anti-cubic nonlinearity, Opt. Quant. Electron., 50 (2018), 75.  doi: 10.1007/s11082-018-1341-3.  Google Scholar [29] M. Ekici, M. Mirzazadeh, A. Sonmezoglu, Q. Zhou, S. P. Moshokoa, A. Biswas and M. Belic, Dark and singular optical solitons with Kundu-Eckhaus equation by extended trial equation method and extended $G'/G$–expansion scheme, Optik, 127 (2016), 10490-10497.  doi: 10.1016/j.ijleo.2016.08.074.  Google Scholar [30] M. Ekici, M. Mirzazadeh, M. Eslami, Q. Zhou, S. P. Moshokoa, A. Biswas and M. Belic, Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives, Optik, 127 (2016), 10659-10669.  doi: 10.1016/j.ijleo.2016.08.076.  Google Scholar [31] A. Esen, T. A. Sulaiman, H. Bulut and H. M. Baskonus, Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schr$\bf{\ddot o}$dinger equation, Optik, 167 (2018), 150-156.   Google Scholar [32] M. Eslami, M. Mirzazadeh, B. F. Vajargah and A. Biswas, Optical solitons for the resonant nonlinear Schr$\bf{\ddot o}$dinger's equation with time-dependent coefficients by the first integral method, Optik, 125 (2014), 3107-3116.   Google Scholar [33] M. Eslami and M. Mirzazadeh, Optical solitons with Biswas-Milovic equation for power law and dual-power law nonlinearities, Nonlinear Dyn., 83 (2016), 731-738.  doi: 10.1007/s11071-015-2361-1.  Google Scholar [34] M. Eslami, Soliton-like solutions for the coupled Schrodinger-Boussinesq equation, Optik, 126 (2016), 3987-3991.  doi: 10.1016/j.ijleo.2015.07.197.  Google Scholar [35] M. Eslami, Trial solution technique to chiral nonlinear Schr$\bf{\ddot o}$dinger equation in (1+2)-dimensions, Nonlinear Dyn., 85 (2016), 813-816.  doi: 10.1007/s11071-016-2724-2.  Google Scholar [36] M. Eslami, H. Rezazadeh, M. Rezazadeh and S. S. Mosavi, Exact solutions to the space-time fractional Schr$\bf{\ddot o}$dinger-Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation, Optical and Quantum Electronics, 49 (2017), 279.   Google Scholar [37] M. Eslami and H. Rezazadeh, The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo, 53 (2016), 475-485.  doi: 10.1007/s10092-015-0158-8.  Google Scholar [38] O. A. Ilhan, H. Bulut, T. A. Sulaiman and H. M. Baskonus, Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd-Bullough-Mikhailov equation, Indian Journal of Physics, 92 (2018), 999-1007.  doi: 10.1007/s12648-018-1187-3.  Google Scholar [39] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar [40] J. Manafian, Optical soliton solutions for Schrdinger type nonlinear evolution equations by the $tan(\varphi/2)$-expansion method, Optik, 127 (2016), 4222-4245.   Google Scholar [41] J. Manafian and M. F. Aghdaei, Abundant soliton solutions for the coupled Schr$\bf{\ddot o}$dinger-Boussinesq system via an analytical method, The European Physical Journal Plus, 131 (2016), 97.   Google Scholar [42] M. Mirzazadeh, M. Ekici, A. Sonmezoglu, M. Eslami, Q. Zhou, E. Zerrad, A. Biswas and M. Belic, Optical Solitons in Nano-Fibers with Fractional Temporal Evolution, Journal of Computational and Theoretical Nanoscience, 13 (2016), 5361-5374.  doi: 10.1166/jctn.2016.5425.  Google Scholar [43] K. M. Owolabi and A. Atangana, Numerical solution of fractional-in-space nonlinear Schr$\bf{\ddot o}$dinger equation with the Riesz fractional derivative, The European Physical Journal Plus, 131 (2016), 335.   Google Scholar [44] I. Podlubny, Fractional Differential Equations, 1$^{st}$ edition, Academic Press, an Diego, 1999.  doi: 10.1007/978-1-4612-0873-0.  Google Scholar [45] A. Sardar, K. Ali, S. T.R. Rizvi, M. Younis, Q. Zhou, E. Zerrad, A. Biswas and A. Bhrawy, Dispersive optical solitons in nanofibers with Schr$\bf{\ddot o}$dinger-Hirota equation, Journal of Nanoelectronics and Optoelectronics, 11 (2016), 382-387.   Google Scholar [46] M. Savescu, A. H. Bhrawy, E. M. Hilal, A. A. Alshaery and A. Biswas, Optical solitons in birefringent fibers with four-wave mixing for Kerr law nonlinearity, Rom. J. Phys., 59 (2014), 582-589.   Google Scholar [47] A. R. Seadawy, Modulation instability analysis for the generalized derivative higher order nonlinear Shr$\bf{\ddot o}$dinger equation and its the bright and dark soliton solutions, Journal of Electromagnetic Waves and Applications, 31 (2017), 1353-1362.   Google Scholar [48] A. R. Seadawy and D. Lu, Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Shr$\bf{\ddot o}$dinger equation and its stability, Results Phys., 7 (2017), 43-48.   Google Scholar [49] T. A. Sulaiman, T. Akturk, H. Bulut and H. M. Baskonus, Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation, Journal of Electromagnetic Waves and Applications, 32 (2018), 1093-1105.  doi: 10.1080/09205071.2017.1417919.  Google Scholar [50] H. Triki and A. M. Wazwaz, New solitons and periodic wave solutions for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation, Journal of Electromagnetic Waves and Applications, 30 (2016), 788-794.  doi: 10.1080/09205071.2016.1153986.  Google Scholar [51] X. Xian-Lin and T. Jia-Shi, Travelling wave solutions for Konopelchenko-Dubrovsky equation using an extended sinh-Gordon equation expansion method, Commun. Theor. Phys., 50 (2008), 1047-1051.  doi: 10.1088/0253-6102/50/5/06.  Google Scholar [52] X. J. Yang, F. Gao and H. M. Srivastava, Exact Travelling Wave solutions for the Local Fractional Two-Dimensional Burgers-Type Equations, Computers and Mathematics with Applications, 73 (2017), 203-210.  doi: 10.1016/j.camwa.2016.11.012.  Google Scholar [53] H. C. Yaslan, New analytic solutions of the conformable space-time fractional Kawahara equation, Optik, 140 (2017), 123-126.  doi: 10.1016/j.ijleo.2017.04.015.  Google Scholar [54] R. Yilmaza and E. Bas, Explicit Solutions of Fractional Schr$\bf{\ddot o}$dinger Equation via Fractional Calculus Operators, Int. J. Open Problems Compt. Math., 5 (2012), 133-141.   Google Scholar [55] A. Yokus, H. M. Baskonus, T. A. Sulaiman and H. Bulut, Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numer Methods Partial Differential Eq., 34 (2018), 211-227.  doi: 10.1002/num.22192.  Google Scholar [56] M. Younis, N. Cheemaa, S. A. Mahmood and S. T. R. Rizvi, On optical solitons: The chiral nonlinear Schr$\bf{\ddot o}$dinger equation with perturbation and Bohm potential, Opt Quant Electron, 48 (2016), 542.   Google Scholar [57] Q. Zhou, Optical solitons for Biswas-Milovic model with Kerr law and parabolic law nonlinearities, Nonlinear Dynamics, 84 (2016), 677-681.  doi: 10.1007/s11071-015-2516-0.  Google Scholar [58] Q. Zhou and A. Biswas, Optical solitons in parity-time-symmetric mixed linear and nonlinear lattice with non-Kerr law nonlinearity, Superlattices Microstruct., 109 (2017), 588-598.  doi: 10.1016/j.spmi.2017.05.049.  Google Scholar [59] Q. Zhou, A. Sonmezoglu, M. Ekici and M. Mirzazadeh, Optical solitons of some fractional differential equations in nonlinear optics, J. Mod. Opt., 64 (2017), 2345-2349.  doi: 10.1080/09500340.2017.1357856.  Google Scholar [60] Q. Zhou, Analytical study on optical solitons in s kerr-law medium with an imprinted parity-time-symmetric mixed linear-nonliear lattice, Proc. Rom. Acad. Ser. A, 18 (2017), 223-230.   Google Scholar [61] Q. Zhou, C. Wei, H. Zhang, J. Lu, H. Yu, P. Yao and Q. Zhu, Exact solutions to the resonant nonlinear schr$\bf{\ddot o}$dinger equation with both spatio-temporal and inter-modal dispersions, Proc. Rom. Acad. Ser. A, 17 (2016), 307-313.   Google Scholar

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##### References:
 [1] A. Abdon and B. Dumitru, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.   Google Scholar [2] M. A. Akinlar and M. Kurulay, A novel method for analytical solutions of fractional partial differential equations, Mathematical Problems in Engineering, 2013 (2013), Art. ID 195708, 4 pp. doi: 10.1155/2013/195708.  Google Scholar [3] K. K. Ali, R. I. Nuruddeen and K. R. Raslan, New structures for the space-time fractional simplified MCH and SRLW equations, Chaos, Solitons and Fractals, 106 (2018), 304-309.  doi: 10.1016/j.chaos.2017.11.038.  Google Scholar [4] S. Arbabi and M. Najafi, Exact solitary wave solutions of the complex nonlinear Schr$\bf{\ddot o}$dinger equations, Optik, 127 (2016), 4682-4688.   Google Scholar [5] A. H. Arnous, M. Z. Ullah, M. Asma, S. P. Moshokoa, Q. Zhou, M. Mirzazadeh, A. Biswas and M. Belic, Dark and singular dispersive optical solitons of Schr$\bf{\ddot o}$dinger-Hirota equation by modified simple equation method, Optik, 136 (2017), 445-450.   Google Scholar [6] M. Arshad, A. R. Seadawy and D. Lu, Elliptic function and solitary wave solutions of the higher-order nonlinear Schr$\bf{\ddot o}$dinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability, The European Physical Journal Plus, 132 (2017), 371.   Google Scholar [7] A. Atangana and D. Baleanu, Application of Fixed Point Theorem for Stability Analysis of a Nonlinear Schrodinger with Caputo-Liouville Derivatives, Filomat, 31 (2017), 2243-2248.  doi: 10.2298/FIL1708243A.  Google Scholar [8] E. Bas, R. Yilmaza and E. Panakhov, Fractional solutions of bessel equation with $N$-method, The Scientific World Journal, 2013 (2013), Article ID 685695, 8 pages. doi: 10.1155/2013/685695.  Google Scholar [9] H. M. Baskonus, T. A. Sulaiman, H. Bulut and T. Akturk, Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schr$\bf{\ddot o}$dinger equation with $\delta$-potential, Superlattices and Microstructures, 115 (2016), 19-29.   Google Scholar [10] H. M. Baskonus, H. Bulut and T. A. Sulaiman, Investigation of various travelling wave solutions to the extended (2+1)-dimensional quantum ZK equation, The European Physical Journal Plus, 132 (2017), 482.  doi: 10.1140/epjp/i2017-11778-y.  Google Scholar [11] H. M. Baskonus, T. A. Sulaiman and H. Bulut, Dark, bright and other optical solitons to the decoupled nonlinear Schr$\bf{\ddot o}$dinger equation arising in dual-core optical fibers, Opt Quant Electron, 50 (2018), 165.   Google Scholar [12] H. M. Baskonus, T. A. Sulaiman and H. Bulut, Bright, dark optical and other solitons to the generalized higher-order NLSE in optical fibers, Opt Quant Electron, 50 (2018), 253.  doi: 10.1007/s11082-018-1522-0.  Google Scholar [13] I. Bendahmane, H. Triki, A. Biswas, A. S. Alshomrani, Q. Zhou, S. P. Moshokoa and M. Belic, Bright, dark and W-shaped solitons with extended nonlinear Schr$\bf{\ddot o}$dinger's equation for odd and even higher-order terms, Superlattices Microstruct., 114 (2018), 53-61.   Google Scholar [14] A. H. Bhrawy, A. A. Alshaery, E. M. Hilal, Z. Jovanoski and A. Biswas, Bright and dark solitons in a cascaded system, Optik, 125 (2014), 6162-6165.  doi: 10.1016/j.ijleo.2014.06.118.  Google Scholar [15] A. Biswas, M. Ekici, A. Sonmezoglu, Q. Zhou, S. P. Moshokoa and M. Belic, Optical soliton perturbation with full nonlinearity for Kundu-Eckhaus equation by extended trial function scheme, Optik, 160 (2018), 17-23.  doi: 10.1016/j.ijleo.2018.01.111.  Google Scholar [16] A. Biswas, Q. Zhou, S. P. Moshokoa, H. Triki, M. Belic and R. T. Alqahtani, Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations, Optik, 145 (2017), 14-17.  doi: 10.1016/j.ijleo.2017.07.036.  Google Scholar [17] A. Biswas, Q. Zhou, M. Z. Ullah, H. Triki, S. P. Moshokoa and M. Belic, Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle, Optik, 143 (2017), 131-134.  doi: 10.1016/j.ijleo.2017.06.087.  Google Scholar [18] A. Biswas, A. H. Kara, M. Z. Ullah, Q. Zhou, H. Triki and M. Belic, Conservation laws for cubic-quartic optical solitons in Kerr and power law media, Optik, 145 (2017), 650-654.  doi: 10.1016/j.ijleo.2017.08.047.  Google Scholar [19] A. Biswas, H. Triki, Q. Zhou, S. P. Moshokoa, M. Z. Ullah and M. Belic, Cubic-quartic optical solitons in Kerr and power law media, Commun. Theor. Phys., 144 (2017), 357-362.  doi: 10.1016/j.ijleo.2017.07.008.  Google Scholar [20] H. Bulut, T. A. Sulaiman and B. Demirdag, Dynamics of soliton solutions in the chiral nonlinear Schr$\bf{\ddot o}$dinger equations, Nonlinear Dynamics, 91 (2018), 1985-1991.   Google Scholar [21] H. Bulut, T. A. Sulaiman, H. M. Baskonus and T. Akturk, Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media, Opt Quant Electron, 50 (2018), 19.  doi: 10.1007/s11082-017-1286-y.  Google Scholar [22] H. Bulut, T. A. Sulaiman, H. M. Baskonus, H. Rezazadeh, M. Eslami and M. Mirzazadeh, Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation, Optik, 127 (2018), 20-27.  doi: 10.1016/j.ijleo.2018.06.108.  Google Scholar [23] H. Bulut, T. A. Sulaiman and H. M. Baskonus, On the new soliton and optical wave structures to some nonlinear evolution equation, The European Physical Journal Plus, 132 (2017), 459.  doi: 10.1140/epjp/i2017-11738-7.  Google Scholar [24] C. Cattani, Harmonic wavelet solutions of the Schrodinger equation, International Journal of Fluid Mechanics Research, 30 (2003), 463-472.  doi: 10.1615/InterJFluidMechRes.v30.i5.10.  Google Scholar [25] C. Cattani, T. A. Sulaiman, H. M. Baskonus and H. Bulut, Solitons in an inhomogeneous Murnaghan's rod, Eur. Phys. J. Plus, 133 (2018), 228.  doi: 10.1140/epjp/i2018-12085-y.  Google Scholar [26] C. Cattani, T. A. Sulaiman, H. M. Baskonus and H. Bulut, On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel'd-Sokolov systems, Opt Quant Electron, 50 (2018), 138.  doi: 10.1007/s11082-018-1406-3.  Google Scholar [27] M. T. Darvishi, S. Ahmadian, S. B. Arbabi and M. Najafi, Optical solitons for a family of nonlinear (1+1)-dimensional time-space fractional Schr$\bf{\ddot o}$dinger models, Optical and Quantum Electronics, 50 (2018), 32.   Google Scholar [28] M. Ekici, A. Sonmezoglu, Q. Zhou, S. P. Moshokoa, M. Z. Ullah, A. H. Arnous, A. Biswas and M. Belic, Analysis of optical solitons in nonlinear negative-indexed materials with anti-cubic nonlinearity, Opt. Quant. Electron., 50 (2018), 75.  doi: 10.1007/s11082-018-1341-3.  Google Scholar [29] M. Ekici, M. Mirzazadeh, A. Sonmezoglu, Q. Zhou, S. P. Moshokoa, A. Biswas and M. Belic, Dark and singular optical solitons with Kundu-Eckhaus equation by extended trial equation method and extended $G'/G$–expansion scheme, Optik, 127 (2016), 10490-10497.  doi: 10.1016/j.ijleo.2016.08.074.  Google Scholar [30] M. Ekici, M. Mirzazadeh, M. Eslami, Q. Zhou, S. P. Moshokoa, A. Biswas and M. Belic, Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives, Optik, 127 (2016), 10659-10669.  doi: 10.1016/j.ijleo.2016.08.076.  Google Scholar [31] A. Esen, T. A. Sulaiman, H. Bulut and H. M. Baskonus, Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schr$\bf{\ddot o}$dinger equation, Optik, 167 (2018), 150-156.   Google Scholar [32] M. Eslami, M. Mirzazadeh, B. F. Vajargah and A. Biswas, Optical solitons for the resonant nonlinear Schr$\bf{\ddot o}$dinger's equation with time-dependent coefficients by the first integral method, Optik, 125 (2014), 3107-3116.   Google Scholar [33] M. Eslami and M. Mirzazadeh, Optical solitons with Biswas-Milovic equation for power law and dual-power law nonlinearities, Nonlinear Dyn., 83 (2016), 731-738.  doi: 10.1007/s11071-015-2361-1.  Google Scholar [34] M. Eslami, Soliton-like solutions for the coupled Schrodinger-Boussinesq equation, Optik, 126 (2016), 3987-3991.  doi: 10.1016/j.ijleo.2015.07.197.  Google Scholar [35] M. Eslami, Trial solution technique to chiral nonlinear Schr$\bf{\ddot o}$dinger equation in (1+2)-dimensions, Nonlinear Dyn., 85 (2016), 813-816.  doi: 10.1007/s11071-016-2724-2.  Google Scholar [36] M. Eslami, H. Rezazadeh, M. Rezazadeh and S. S. Mosavi, Exact solutions to the space-time fractional Schr$\bf{\ddot o}$dinger-Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation, Optical and Quantum Electronics, 49 (2017), 279.   Google Scholar [37] M. Eslami and H. Rezazadeh, The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo, 53 (2016), 475-485.  doi: 10.1007/s10092-015-0158-8.  Google Scholar [38] O. A. Ilhan, H. Bulut, T. A. Sulaiman and H. M. Baskonus, Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd-Bullough-Mikhailov equation, Indian Journal of Physics, 92 (2018), 999-1007.  doi: 10.1007/s12648-018-1187-3.  Google Scholar [39] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar [40] J. Manafian, Optical soliton solutions for Schrdinger type nonlinear evolution equations by the $tan(\varphi/2)$-expansion method, Optik, 127 (2016), 4222-4245.   Google Scholar [41] J. Manafian and M. F. Aghdaei, Abundant soliton solutions for the coupled Schr$\bf{\ddot o}$dinger-Boussinesq system via an analytical method, The European Physical Journal Plus, 131 (2016), 97.   Google Scholar [42] M. Mirzazadeh, M. Ekici, A. Sonmezoglu, M. Eslami, Q. Zhou, E. Zerrad, A. Biswas and M. Belic, Optical Solitons in Nano-Fibers with Fractional Temporal Evolution, Journal of Computational and Theoretical Nanoscience, 13 (2016), 5361-5374.  doi: 10.1166/jctn.2016.5425.  Google Scholar [43] K. M. Owolabi and A. Atangana, Numerical solution of fractional-in-space nonlinear Schr$\bf{\ddot o}$dinger equation with the Riesz fractional derivative, The European Physical Journal Plus, 131 (2016), 335.   Google Scholar [44] I. Podlubny, Fractional Differential Equations, 1$^{st}$ edition, Academic Press, an Diego, 1999.  doi: 10.1007/978-1-4612-0873-0.  Google Scholar [45] A. Sardar, K. Ali, S. T.R. Rizvi, M. Younis, Q. Zhou, E. Zerrad, A. Biswas and A. Bhrawy, Dispersive optical solitons in nanofibers with Schr$\bf{\ddot o}$dinger-Hirota equation, Journal of Nanoelectronics and Optoelectronics, 11 (2016), 382-387.   Google Scholar [46] M. Savescu, A. H. Bhrawy, E. M. Hilal, A. A. Alshaery and A. Biswas, Optical solitons in birefringent fibers with four-wave mixing for Kerr law nonlinearity, Rom. J. Phys., 59 (2014), 582-589.   Google Scholar [47] A. R. Seadawy, Modulation instability analysis for the generalized derivative higher order nonlinear Shr$\bf{\ddot o}$dinger equation and its the bright and dark soliton solutions, Journal of Electromagnetic Waves and Applications, 31 (2017), 1353-1362.   Google Scholar [48] A. R. Seadawy and D. Lu, Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Shr$\bf{\ddot o}$dinger equation and its stability, Results Phys., 7 (2017), 43-48.   Google Scholar [49] T. A. Sulaiman, T. Akturk, H. Bulut and H. M. Baskonus, Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation, Journal of Electromagnetic Waves and Applications, 32 (2018), 1093-1105.  doi: 10.1080/09205071.2017.1417919.  Google Scholar [50] H. Triki and A. M. Wazwaz, New solitons and periodic wave solutions for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation, Journal of Electromagnetic Waves and Applications, 30 (2016), 788-794.  doi: 10.1080/09205071.2016.1153986.  Google Scholar [51] X. Xian-Lin and T. Jia-Shi, Travelling wave solutions for Konopelchenko-Dubrovsky equation using an extended sinh-Gordon equation expansion method, Commun. Theor. Phys., 50 (2008), 1047-1051.  doi: 10.1088/0253-6102/50/5/06.  Google Scholar [52] X. J. Yang, F. Gao and H. M. Srivastava, Exact Travelling Wave solutions for the Local Fractional Two-Dimensional Burgers-Type Equations, Computers and Mathematics with Applications, 73 (2017), 203-210.  doi: 10.1016/j.camwa.2016.11.012.  Google Scholar [53] H. C. Yaslan, New analytic solutions of the conformable space-time fractional Kawahara equation, Optik, 140 (2017), 123-126.  doi: 10.1016/j.ijleo.2017.04.015.  Google Scholar [54] R. Yilmaza and E. Bas, Explicit Solutions of Fractional Schr$\bf{\ddot o}$dinger Equation via Fractional Calculus Operators, Int. J. Open Problems Compt. Math., 5 (2012), 133-141.   Google Scholar [55] A. Yokus, H. M. Baskonus, T. A. Sulaiman and H. Bulut, Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numer Methods Partial Differential Eq., 34 (2018), 211-227.  doi: 10.1002/num.22192.  Google Scholar [56] M. Younis, N. Cheemaa, S. A. Mahmood and S. T. R. Rizvi, On optical solitons: The chiral nonlinear Schr$\bf{\ddot o}$dinger equation with perturbation and Bohm potential, Opt Quant Electron, 48 (2016), 542.   Google Scholar [57] Q. Zhou, Optical solitons for Biswas-Milovic model with Kerr law and parabolic law nonlinearities, Nonlinear Dynamics, 84 (2016), 677-681.  doi: 10.1007/s11071-015-2516-0.  Google Scholar [58] Q. Zhou and A. Biswas, Optical solitons in parity-time-symmetric mixed linear and nonlinear lattice with non-Kerr law nonlinearity, Superlattices Microstruct., 109 (2017), 588-598.  doi: 10.1016/j.spmi.2017.05.049.  Google Scholar [59] Q. Zhou, A. Sonmezoglu, M. Ekici and M. Mirzazadeh, Optical solitons of some fractional differential equations in nonlinear optics, J. Mod. Opt., 64 (2017), 2345-2349.  doi: 10.1080/09500340.2017.1357856.  Google Scholar [60] Q. Zhou, Analytical study on optical solitons in s kerr-law medium with an imprinted parity-time-symmetric mixed linear-nonliear lattice, Proc. Rom. Acad. Ser. A, 18 (2017), 223-230.   Google Scholar [61] Q. Zhou, C. Wei, H. Zhang, J. Lu, H. Yu, P. Yao and Q. Zhu, Exact solutions to the resonant nonlinear schr$\bf{\ddot o}$dinger equation with both spatio-temporal and inter-modal dispersions, Proc. Rom. Acad. Ser. A, 17 (2016), 307-313.   Google Scholar
The 3D and 2D surfaces of Eq. (19) at α = β = 0.7
The 3D and 2D surfaces of Eq. (19) at α = β = 0.8
The 3D and 2D surfaces of Eq. (21) at α = β = 0.7
The 3D and 2D surfaces of Eq. (21) at α = β = 0.8
The 3D and 2D surfaces of Eq. (22) at α = β = 0.7
The 3D and 2D surfaces of Eq. (22) at α = β = 0.8
The 3D and 2D surfaces of Eq. (29) at α = β = 0.7
The 3D and 2D surfaces of Eq. (29) at α = β = 0.8
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