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, doi: 10.3934/dcdss.2020054
References:
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M. ArshadA. 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.

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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.

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H. M. BaskonusH. 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.

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H. BulutT. 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.

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H. BulutT. A. SulaimanH. 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.

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C. CattaniT. A. SulaimanH. 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.

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show all references

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.

[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.

[3]

K. K. AliR. 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.

[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.

[5]

A. H. ArnousM. Z. UllahM. AsmaS. P. MoshokoaQ. ZhouM. MirzazadehA. 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.

[6]

M. ArshadA. 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.

[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.

[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.

[9]

H. M. BaskonusT. A. SulaimanH. 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.

[10]

H. M. BaskonusH. 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.

[11]

H. M. BaskonusT. 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.

[12]

H. M. BaskonusT. 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.

[13]

I. BendahmaneH. TrikiA. BiswasA. S. AlshomraniQ. ZhouS. 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.

[14]

A. H. BhrawyA. A. AlshaeryE. M. HilalZ. 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.

[15]

A. BiswasM. EkiciA. SonmezogluQ. ZhouS. 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.

[16]

A. BiswasQ. ZhouS. P. MoshokoaH. TrikiM. 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.

[17]

A. BiswasQ. ZhouM. Z. UllahH. TrikiS. 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.

[18]

A. BiswasA. H. KaraM. Z. UllahQ. ZhouH. 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.

[19]

A. BiswasH. TrikiQ. ZhouS. P. MoshokoaM. 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.

[20]

H. BulutT. 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.

[21]

H. BulutT. A. SulaimanH. 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.

[22]

H. BulutT. A. SulaimanH. M. BaskonusH. RezazadehM. 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.

[23]

H. BulutT. 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.

[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.

[25]

C. CattaniT. A. SulaimanH. 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.

[26]

C. CattaniT. A. SulaimanH. 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.

[27]

M. T. DarvishiS. AhmadianS. 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.

[28]

M. EkiciA. SonmezogluQ. ZhouS. P. MoshokoaM. Z. UllahA. H. ArnousA. 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.

[29]

M. EkiciM. MirzazadehA. SonmezogluQ. ZhouS. P. MoshokoaA. 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.

[30]

M. EkiciM. MirzazadehM. EslamiQ. ZhouS. P. MoshokoaA. 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.

[31]

A. EsenT. A. SulaimanH. 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.

[32]

M. EslamiM. MirzazadehB. 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.

[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.

[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.

[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.

[36]

M. EslamiH. RezazadehM. 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.

[37]

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Figure 1.  The 3D and 2D surfaces of Eq. (19) at α = β = 0.7
Figure 2.  The 3D and 2D surfaces of Eq. (19) at α = β = 0.8
Figure 3.  The 3D and 2D surfaces of Eq. (21) at α = β = 0.7
Figure 4.  The 3D and 2D surfaces of Eq. (21) at α = β = 0.8
Figure 5.  The 3D and 2D surfaces of Eq. (22) at α = β = 0.7
Figure 6.  The 3D and 2D surfaces of Eq. (22) at α = β = 0.8
Figure 7.  The 3D and 2D surfaces of Eq. (29) at α = β = 0.7
Figure 8.  The 3D and 2D surfaces of Eq. (29) at α = β = 0.8
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