doi: 10.3934/dcdss.2020398

Abundant novel solutions of the conformable Lakshmanan-Porsezian-Daniel model

1. 

Department of Mathematics, Faculty of Science, University of Zakho, Zakho, Iraq

2. 

Department of Mathematics and Science Education, Harran University, Sanliurfa, Turkey

3. 

Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey

* Corresponding author: Hajar Farhan Ismael

Received  July 2019 Revised  November 2019 Published  June 2020

In this paper, three images of nonlinearity to the fractional Lakshmanan Porsezian Daniel model in birefringent fibers are investigated. The new bright, periodic wave and singular optical soliton solutions are constructed via the $ \left( m+\frac{G'}{G} \right) $ expansion method, which are applicable to the dynamics within the optical fibers. All solutions are novel compared with solutions obtained via different methods. All solutions verify the conformable Lakshmanan-Porsezian-Daniel model and also, for the existence the constraint conditions are utilized. Moreover, 2D and 3D for all solutions are plotted to more understand its physical characteristics.

Citation: Hajar Farhan Ismael, Haci Mehmet Baskonus, Hasan Bulut. Abundant novel solutions of the conformable Lakshmanan-Porsezian-Daniel model. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020398
References:
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show all references

References:
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K. K. AliH. F. IsmaelB. A. Mahmood and M. A. Yousif, MHD Casson fluid with heat transfer in a liquid film over unsteady stretching plate, Int. J. Adv. Appl. Sci., 4 (2017), 55-58.  doi: 10.21833/ijaas.2017.01.008.  Google Scholar

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A. BiswasM. EkiciA. Sonmezoglu and R. T. Alqahtani, Optical solitons with differential group delay for coupled Fokas–Lenells equation by extended trial function scheme, Optik, 165 (2018), 102-110.  doi: 10.1016/j.ijleo.2018.03.102.  Google Scholar

[12]

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[13]

A. Biswas et al, Optical solitons with Lakshmanan–Porsezian–Daniel model using a couple of integration schemes, Optik, 158 (2018), 705-711.   Google Scholar

[14]

A. BiswasA. H. KaraR. T. AlqahtaniM. Z. UllahH. Triki and M. Belic, Conservation laws for optical solitons of Lakshmanan-Porsezian-Daniel model, Proc. Rom. Acad. Ser. A - Math. Phys. Tech. Sci. Inf. Sci., 19 (2018), 39-44.   Google Scholar

[15]

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[16]

A. BiswasY. YildirimE. YasarQ. ZhouS. P. Moshokoa and M. Belic, Optical solitons for Lakshmanan-Porsezian–Daniel model by modified simple equation method, Optik, 160 (2018), 24-32.  doi: 10.1016/j.ijleo.2018.01.100.  Google Scholar

[17]

C. Cattani, T. A. Sulaiman, H. M. Baskonus and H. Bulut, Solitons in an inhomogeneous Murnaghans rod, Eur. Phys. J. Plus, 133 (2018), 228. Google Scholar

[18]

H. BulutT. A. Sulaiman and H. M. Baskonus, Dark, bright optical and other solitons with conformable space-time fractional second-order spatiotemporal dispersion, Optik, 163 (2018), 1-7.  doi: 10.1016/j.ijleo.2018.02.086.  Google Scholar

[19]

C. Cattani, T. A. Sulaiman, H. M. Baskonus and H. Bulut, On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfeld-Sokolov systems, Opt. Quantum Electron, 50 (2018), 138. Google Scholar

[20]

L. D. MolelekiT. Motsepa and C. M. Khalique, Solutions and conservation laws of a generalized second extended $(3+1)$-dimensional Jimbo-Miwa equation, Appl. Math. Nonlinear Sci., 3 (2018), 459-474.  doi: 10.2478/AMNS.2018.2.00036.  Google Scholar

[21]

M. Dewasurendra and K. Vajravelu, On the method of inverse mapping for solutions of coupled systems of nonlinear differential equations arising in nanofluid flow, heat and mass transfer, Appl. Math. Nonlinear Sci., 3 (2018), 1-14.  doi: 10.21042/AMNS.2018.1.00001.  Google Scholar

[22]

M. Ekici, Optical solitons in birefringent fibers for Lakshmanan–Porsezian–Daniel model by extended Jacobis elliptic function expansion scheme, Optik, 172 (2018), 651-656.   Google Scholar

[23]

M. M. A. El-Sheikh, et al., Optical solitons in birefringent fibers with Lakshmanan–Porsezian–Daniel model by modified simple equation, Optik, 192 (2019), 162899. Google Scholar

[24]

E. İ. EskitąçıoğluM. B. Aktaş and H. M. Baskonus, New complex and hyperbolic forms for Ablowitz–Kaup–Newell–Segur wave equation with fourth order, Appl. Math. Nonlinear Sci., 4 (2019), 105-112.  doi: 10.2478/AMNS.2019.1.00010.  Google Scholar

[25]

E. Fan and J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A, 305 (2002), 383-392.  doi: 10.1016/S0375-9601(02)01516-5.  Google Scholar

[26]

W. Gao and H. F. Ismael, H. Bulut and H. M. Baskonus, Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media, Phys. Scr., 95 (2020), 035207. doi: 10.1088/1402-4896/ab4a50.  Google Scholar

[27]

W. Gao, H. F. Ismael, S. A. Mohammed, H. M. Baskonus and H. Bulut, Complex and real optical soliton properties of the paraxial nonlinear Schrödinger equation in Kerr media with M-fractional, Front. Phys., 7 (2019), 197. Google Scholar

[28]

W. Gao, H. F. Ismael, A. M. Husien, H. Bulut and H. M. Baskonus, Optical soliton solutions of the Cubic-Quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation with the parabolic law, Appl. Sci., 10 (2020), 219. doi: 10.3390/app10010219.  Google Scholar

[29]

Z. Hammouch and T. Mekkaoui, Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives, Journal| MESA, 5 (2014), 489-498.   Google Scholar

[30]

Z. Hammouch, T. Mekkaoui and P. Agarwal, Optical solitons for the Calogero-Bogoyavlenskii-Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative, Eur. Phys. J. Plus, 133 (2018), 248. doi: 10.1140/epjp/i2018-12096-8.  Google Scholar

[31]

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Figure 1.  3D graphic of Eq. (16) when $ \alpha = 0.9,{{A}_{1}} = 3,{{A}_{2}} = 2,a = 0.4,b = 0.1,\delta = 0.2,\nu = 0.2,\kappa = 0.1,\epsilon = 0.4,\beta = 0.9,\lambda = 1,m = 1,\mu = -1 $ and $ t = -2 $ for 2D
Figure 2.  3D figure of Eq. (17), when $ \alpha = 0.8,{{A}_{1}} = 3,{{A}_{2}} = 2,a = 0.4,b = 0.1,\delta = 0.2,\nu = -0.2,\kappa = 0.1,\epsilon = 0.4,\beta = 0.8,\lambda = -1,m = 1,\mu = 1 $ and $ t = 2 $ for 2D
Figure 3.  3D figure of Eq. (18) when $ \alpha = 0.5,{{A}_{1}} = -3,{{A}_{2}} = 0.2,a = 0.4,b = 0.1,\delta = 0.2,\nu = -0.2,\kappa = 0.5,\epsilon = 0.1,\beta = 0.5,\lambda = -2,m = 2,\mu = 1 $ and $ t = 2 $ for 2D
Figure 4.  3D graphic of Eq. (19) when $ \alpha = 0.5,{{A}_{1}} = -3,{{A}_{2}} = 1,a = 0.3,b = -2,\delta = 0.2,\nu = 0.2,\kappa = 0.1,\epsilon = 2,\beta = 0.5,\lambda = 1,m = 1,\mu = -1 $ and $ t = -2 $ for 2D
Figure 5.  3D figure of Eq. (20) when $ \alpha = 0.5,{{A}_{1}} = 3,{{A}_{2}} = 1,a = 0.3,b = 0.2,\delta = 0.2,\nu = -0.2,\kappa = 0.1,\epsilon = 2,\beta = 0.5,\lambda = -1,m = 1,\mu = 1 $ and $ t = 2 $ for 2D
Figure 6.  3D figure of Eq. (21) when $ \alpha = 0.5,{{A}_{1}} = 3,{{A}_{2}} = 1,a = 0.3,b = 0.2,\delta = 0.2,\nu = -0.2,\kappa = 0.1,\epsilon = 2,\beta = 0.5,\lambda = -2,m = 2,\mu = 1 $ and $ t = 2 $ for 2D
Figure 7.  3D figure of Eq. (26) when $ \alpha = 0.9,{{A}_{1}} = 1,{{A}_{2}} = 3,a = 0.3,b = 0.2,\delta = 0.2,\nu = 0.1,\kappa = 2,\epsilon = 0.2,{{a}_{1}} = 2,\beta = 0.9,\lambda = 1,m = 1,\mu = -1 $ and $ t = -2 $ for 2D
Figure 8.  3D figure of Eq. (27) when $ \alpha = 0.9,{{A}_{1}} = 1,{{A}_{2}} = 3,a = 0.3,b = 0.2,\delta = 0.2,\nu = -0.1,\kappa = 2,\epsilon = 0.2,{{a}_{1}} = -2,\beta = 0.9,\lambda = -1,m = 1,\mu = 1 $ and $ t = 2 $ for 2D
Figure 9.  3D figure of Eq. (28) when $ \alpha = 0.5,{{A}_{1}} = 3,{{A}_{2}} = 2,a = 2,b = 0.1,\delta = 0.2,\nu = 0.2,\kappa = 1,\epsilon = 0.2,{{a}_{1}} = 2,\beta = 0.5,\lambda = -2,m = 2,\mu = 1 $ and $ t = 2 $ for 2D
Figure 10.  3D figure of Eq. (29) when $ \alpha = 0.5,{{A}_{1}} = 2,{{A}_{2}} = 3,a = 0.2,b = 0.2,\delta = 0.2,\nu = -0.2,\kappa = 1,\epsilon = 2,{{c}_{1}} = 1,\beta = 0.5,\gamma = -1,\lambda = 1,m = 1,\mu = -1 $ and $ t = 2 $ for 2D
Figure 11.  3D figure of Eq. (30) when $ \alpha = 0.9,{{A}_{1}} = 2,{{A}_{2}} = 3,a = 3,b = -2,\delta = 0.2,\nu = 0.1,\kappa = 2,\epsilon = 0.2,{{c}_{1}} = 1,\beta = 0.9,\gamma = 1,\lambda = -1,m = 1,\mu = 1 $ and $ t = 2 $ for 2D
Figure 12.  3D figure of Eq. (31) when $ \alpha = 0.9,{{A}_{1}} = -3,{{A}_{2}} = 2,a = 2,b = 0.1,\delta = 0.2,\nu = 0.2,\kappa = 1,\epsilon = 0.2,{{c}_{1}} = 1,\beta = 0.9,\gamma = -2,\lambda = -2,m = 2,\mu = 1 $ and $ t = 2 $ for 2D
Figure 13.  3D figure of Eq. (36) when $ \alpha = 1/2,{{A}_{1}} = 0.2,{{A}_{2}} = 0.3,a = -0.2,b = 1,\delta = -0.2,\nu = 0.2,\kappa = 0.1,\epsilon = 0.2,{{c}_{1}} = 2,\gamma = 0.1,{{a}_{0}} = 0.1,{{c}_{3}} = 0.1,\beta = 1/2,\lambda = 1,m = 1,\mu = -1 $ and $ t = -2 $ for 2D
Figure 14.  3D figure of Eq. (37) when $ \alpha = 0.5,{{A}_{1}} = 1,{{A}_{2}} = 3,a = -0.2,b = 2,\delta = -0.2,\nu = 0.2,\kappa = 1,\epsilon = 2,{{c}_{1}} = 1,\gamma = 1,{{a}_{0}} = 5,{{c}_{3}} = 1,\beta = 0.5,\lambda = -1,m = 1,\mu = 1 $ and $ t = -2 $ for 2D
Figure 15.  3D figure of Eq. (38) when $ \alpha = 0.5,{{A}_{1}} = 2,{{A}_{2}} = 0.3,a = -0.2,b = 2,\delta = -0.2,\nu = 0.2,\kappa = 1,\epsilon = 2,{{c}_{1}} = 1,\gamma = 1,{{a}_{0}} = 1,{{c}_{3}} = 0.1,\beta = 0.5,\lambda = -2,m = 2,\mu = 1 $ and $ t = 2 $ for 2D
Figure 16.  3D figure of Eq. (39) when $ \alpha = 0.5,{{A}_{1}} = 0.2,{{A}_{2}} = 0.3,b = -0.2,\delta = 0.2,\nu = 0.2,\kappa = 1,\epsilon = 2,{{c}_{1}} = 0.2,\gamma = 1,{{c}_{3}} = 1,\beta = 0.5,{{a}_{1}} = 1,\omega = 1,a = 0.2,\lambda = 1,m = 1,\mu = -1 $ and $ t = -2 $ for 2D
Figure 17.  3D figure of Eq. (40) when $ \alpha = 0.9,{{A}_{1}} = 0.4,{{A}_{2}} = 2,b = -0.2,\delta = 0.2,\nu = 0.2,\kappa = 1,\epsilon = 2,{{c}_{1}} = 1,\gamma = 1,{{c}_{3}} = 1,\beta = 0.9,{{a}_{1}} = 0.1,\omega = 1,a = 0.2,\lambda = -1,m = 1,\mu = 1 $ and $ t = 2 $ for 2D
Figure 18.  3D figure of Eq. (41) when $ \alpha = 0.9,{{A}_{1}} = 0.4,{{A}_{2}} = 2,b = -0.2,\delta = 0.2,\nu = 0.2,\kappa = 1,\epsilon = 2,{{c}_{1}} = 1,\gamma = 1,{{c}_{3}} = 1,\beta = 0.9,{{a}_{1}} = 0.1,\omega = 1,a = 0.2,\lambda = -2,m = 2,\mu = 1 $ and $ t = 2 $ for 2D
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