Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Bao, Shuxin; Chen, Shuangqing

doi:https://doi.org/10.1155/2022/3596620

Advances in Mathematical Physics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3596620 | https://doi.org/10.1155/2022/3596620

Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Shuxin Bao¹and Shuangqing Chen²

Academic Editor: Ming Mei

Received12 Dec 2021

Accepted19 Jul 2022

Published13 Aug 2022

Abstract

The main idea of this paper is to investigate the exact solutions and dynamic properties of a space-time fractional perturbed nonlinear Schrödinger equation involving Kerr law nonlinearity with conformable fractional derivatives. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely, the Hamiltonian, is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the existences of the soliton and periodic solution are established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for the polynomial method.

1. Introduction

For hundred of years, the partial differential equation plays a vital role in many fields of science, and constructing exact solution to it could help us gain a deeper insight into the corresponding phenomena. However, traditional integer-order equation sometimes could not meet the requirement of modeling some special problems such as anomalous diffusion [1]; thus, the fractional calculus is proposed to handle this. Fractional calculus has many definitions, for example, the traditional Riemann–Liouville (RL) definition [2], modified RL definition [3], and conformal definition [4]. However, the classical RL definition is very complex to apply, and the modified RL definition has already been proved wrong [5, 6]; thus, choosing a proper fractional definition is an important and difficult task.

In this paper, we consider the following space-time fractional perturbed nonlinear Schrödinger equation (7).

with conformable fractional derivatives, where is the corresponding fractional order; is the complex valued function defining wave profile in optical fibers; and represent the group velocity dispersion and nonlinear term, respectively; is the intermodal dispersion; is the self-steepeninng perturbation term; and is the nonlinear dispersion coefficient. The nonlinear Schrödinger equation has a broad applications in modeling light waves in nanooptical fibers [7–9]. Especially, when the external electric field exists, this equation can be used to solve nonharmonic motion of electrons bound in molecules [10], so constructing exact solutions to this equation is of great significance. In [11], some optical solitons and singular periodic wave solutions are constructed by the extended Sinh-Gördon equation expansion method, and W-shaped solitons are shown by Al-Ghafri and his colleagues [12]. Other results about soliton theory could be seen in [13–24].

The goal of this paper is to conduct qualitative and quantitative analysis to (1) by the complete discrimination system for polynomial method (CDSPM). The topological structure of this equation is shown, and the existences of soliton and periodic solution are also presented. Moreover, to verify our conclusion explicitly, all exact traveling wave solutions, namely, the classification of traveling wave solutions, are obtained. All conditions of parameters are discussed; thus, this paper contains all results of traveling wave solutions in the existing literatures, and some new solutions are obtained. To the best of our knowledge, this is the first time that the qualitative analysis is conducted to this equation, and we could also see the critical region of the existence of each kind of solution.

The CDSPM is proposed by Liu [25–27] and has been successfully applied to a series of integer-order [28, 29] and fractional-order equations [30–34]. Then, Kai et al. found that this method could also be used to conduct qualitative analysis [35], and combining with the bifurcation method, we can even establish the existence of the soliton and periodic solution [36–38].

The construction of this paper is as follows. The corresponding traveling wave system is given in Section 2, and qualitative analysis is conducted. Moreover, the existences of the soliton and periodic solution are also established in this section. To verify our conclusion explicitly, all single traveling wave solutions are constructed in Section 3, and concrete examples under concrete parameters are also shown to ensure the existence of each solution. In the final part, a brief discussion is given.

2. Dynamic Properties of Equation (1)

By setting

where [33], (1) becomes for the real part, and for the imaginary part. From (3), we have

Substituting (5) into (4) yields

where , and is a constant of integration. By setting , we have

Multiplying (6) with and integrating it once, we have

where , and is a constant of integration. (6) is equivalent to the following dynamic system: and thus, the corresponding Hamiltonian is given by

Now, let us show that the Hamiltonian (12) is a conserved quantity. Taking derivative of right side of (12) with respect to , we have which just proves our conclusion, and we can also conclude that the global phase portrait to the system (11) are just the contour lines of the Hamiltonian (12). In the following, we shall conduct qualitative analysis through this quantity by introducing the complete discrimination system.

From the Hamiltonian (12), we can see that the derivative of the potential energy is given by

where , and . By introducing the following discrimination system we can see that four cases need to be discussed here. This is just the general procedure of the CDSPM—first, rewriting the original equation into the integral form and then introducing the complete discrimination system to discussing the relation between the roots of the corresponding polynomial and the parameters. Then, we can find that all the conditions are discussed, and the results we obtained are integrated.

Case 1. , , we can get is a cuspidal point and is a center for and a saddle point when . For example, when , we have , and corresponding global phase portraits are shown in Figure 1 when .

(a)

(b)

From Figure 1(b), we can see that trajectory I is a closed orbit with a center inside, which just indicates the existence of the periodic solution, and trajectory II is a homoclinic orbit, which indicates the existence of the bell-shaped soliton solution [36].

Case 2. , , we have There is only one equilibrium point here. It is a cuspidal point when and a center when . For example, when , we have , and the corresponding global phase portraits can be seen in Figure 2 when . From Figure 2(b), we can also conclude that the original equation has periodic solution.

Case 3. , we have There is also only one equilibrium point in this case. It is a saddle point when and a center when . So, this case is very similar to Case 2. For example, when and , we have , and the global phase portrait is given in Figure 3 when .

Case 4. , , we have This case is rather interesting due to that there are three equilibrium points , and here. When , , and are two saddle points and is a center, and whereas for , , and are two centers and is a saddle point. Concrete examples of global phase portraits when , and are given in Figure 4.
For Figure 4(a), we can see that trajectory I is a closed orbit with a center inside, which indicates the existence of the periodic solution, and trajectory II and III are two heteroclinic orbits, which indicates the existence of the kink and antikink solitary wave solution, respectively. Figure 4(b) is a “figure eight loop” with trajectories I and II are two closed orbits with a center inside, which indicates the existence of the periodic solution, and trajectories III and IV are two homoclinic orbits, which means the corresponding equation has bright and dark bell-shaped soliton solution.
Now, we have showed the topological structure of system (3) and established the existences of the soliton and the periodic solution. In order to verify the conclusion explicitly, we construct the classification of traveling wave solutions to (3) by the CDSPM.

(a)

(b)

(a)

(b)

(a)

(b)

2.1. Traveling Wave Solutions to (8)

In this section, we construct all traveling wave solutions, namely, the classification of traveling wave solutions to (7). We only focus on the condition of , and could be treated similarly.

By taking the following transformation,

(8) becomes

where and . First, we need to introducing the following complete discrimination system:

The complete discrimination system (27) given in Section 2 is the third-order form, and here is the fourth order. By discussing the relation between the parameters and the coefficients, we shall see that every condition of parameters is discussed; thus, what we have obtained is the classification of traveling wave solutions.

Case 1. and , has a pair of double conjugate complex roots, namely, by substituting (35) into (7), we have the following solution: where is a constant of integration. (36) is a trigonometric function periodic solution. For example if and , then, we have and the solution (36) is given by

Case 2. When . has a real root of multiplicities four, namely, which leads to For example, when , we have

Case 3. When and , has two double distinct real roots, then we have which yields If or , we can get and when , we have

(30) is a solitary wave solution. (22) and (30) have verified the conclusion given in Section 2 that when . (7) has periodic and soliton solution. This shows that the qualitative results obtained are truly correct.

For example, when , and , we have , then solitary wave solution (30) is given by

The corresponding figure of (31) is given in Figure 5. From it, we can see that the main impact of the fractional order is the velocity of convegence, and the position of the soliton is not influenced by it.

Case 4. , then the solution is given by

Thus, the solution in explicit form is given by which is a rational solution. For example, when , and , we have , then

Case 5. , and , we have and then, we can get where Thus, For example, when , and , we have , then the solution is given by

Case 6. . is given by where . Then, we can get the following elliptic function double periodic solutions. When , we have where .
For , similarly we can get where . For instance, when , and , we have , and thus, the solution is given by

Case 7. and . is given by where and . By setting we can get where . For instance, when , , then , and , , the following solution could just be obtained which is also an elliptic function double periodic solution.

Case 8. and , we can get By setting we have where and . For example, when , , we have , , then

Case 9. , and . is given by By setting we have whose solution is given by where and , which is a triangular function periodic solution. For example, when , we have , and , which leads to which is also a periodic solution.

3. Conclusion

In this paper, we consider a space–time fractional perturbed nonlinear Schrödinger equation arising from nanooptical fibers. By taking the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then the corresponding Hamiltonian is constructed, and the qualitative analysis is conducted by introducing the complete discrimination system. The topological structure is given, and the existences of the soliton and periodic solution are established via the bifurcation method. To verify our conclusion explicitly, every kind of traveling wave solutions is constructed by the CDSPM, and some of them are new. In order to analyze the influence of the fractional parameter, a concrete example of the soliton solution is given. From it, we can see to directly that how the fractional order impact the position of the soliton and how these solutions convergent to the same value when the dependent variable tends to infinity. All of the results given in the present paper show the powerfulness of the method adopted in the paper. In the future, we shall further analyze the nonlinear Schrödinger equation with RL definition to give more results to this equation, and we would also like to promote this method to other nonlinear equations like the coupled Boussinesq equation.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflict of interest.

References

B. Zheng, Y. Kai, W. Xu, N. Yang, K. Zhang, and P. M. Thibado, “Exact traveling and non-traveling wave solutions of the time fractional reaction-diffusion equation,” Physica A: Statistical Mechanics and its Applications, vol. 532, article 121780, 2019.
View at: Publisher Site | Google Scholar
V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, no. 8, pp. 2677–2682, 2008.
View at: Publisher Site | Google Scholar
G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers and Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
View at: Publisher Site | Google Scholar
R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014.
View at: Publisher Site | Google Scholar
C.-s. Liu, “Counterexamples on Jumarie's two basic fractional calculus formulae,” Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1-3, pp. 92–94, 2015.
View at: Publisher Site | Google Scholar
C.-s. Liu, “Counterexamples on Jumarie's three basic fractional calculus formulae for non- differentiable continuous functions,” Chaos Solitons and Fractals, vol. 109, pp. 219–222, 2018.
View at: Publisher Site | Google Scholar
A. Yusuf, T. A. Sulaiman, M. Mirzazadeh, and K. Hosseini, “M-truncated optical solitons to a nonlinear Schrödinger equation describing the pulse propagation through a two-mode optical fiber,” Optical and Quantum Electronics, vol. 53, no. 10, pp. 1–17, 2021.
View at: Publisher Site | Google Scholar
M. Yavuz, T. A. Sulaiman, A. Yusuf, and T. Abdeljawad, “The Schrodinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel,” Alexandria Engineering Journal, vol. 60, no. 2, pp. 2715–2724, 2021.
View at: Publisher Site | Google Scholar
A. Yusuf, F. Tchier, and M. Inc, “New interaction and combined multi-wave solutions for the Heisenberg ferromagnetic spin chain equation,” The European Physical Journal Plus, vol. 135, no. 5, pp. 1–8, 2020.
View at: Publisher Site | Google Scholar
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,” Optik, vol. 144, pp. 357–362, 2017.
View at: Publisher Site | Google Scholar
T. A. Sulaiman, H. Bulut, and H. M. Baskonus, “Optical solitons to the fractional perturbed NLSE in nano-fibers,” Discrete and Continuous Dynamical Systems-S, vol. 13, no. 3, pp. 925–936, 2020.
View at: Publisher Site | Google Scholar
K. S. Al-Ghafri, E. V. Krishnan, and A. Biswas, “W-shaped and other solitons in optical nanofibers,” Results in Physics, vol. 23, p. 103973, 2021.
View at: Publisher Site | Google Scholar
H. I. Abdel-Gawad, M. Tantawy, and M. S. Osman, “Dynamic of DNA’s possible impact on its damage,” Mathematical Methods in the Applied Sciences, vol. 39, no. 2, pp. 168–176, 2016.
View at: Publisher Site | Google Scholar
M. S. Osman and H. I. Abdel-Gawad, “Multi-wave solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equations with variable coefficients,” The European Physical Journal Plus, vol. 130, no. 10, pp. 1–11, 2015.
View at: Publisher Site | Google Scholar
H. I. Abdel-Gawad and M. Osman, “On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients,” Journal of Advanced Research, vol. 6, no. 4, pp. 593–599, 2015.
View at: Publisher Site | Google Scholar
S. Kumar, R. Kumar, M. S. Osman, and B. Samet, “A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials,” Numerical Methods for Partial Differential Equations, vol. 37, no. 2, pp. 1250–1268, 2021.
View at: Publisher Site | Google Scholar
K. K. Ali, M. S. Osman, and M. Abdel-Aty, “New optical solitary wave solutions of Fokas-Lenells equation in optical fiber via Sine-Gordon expansion method,” Alexandria Engineering Journal, vol. 59, no. 3, pp. 1191–1196, 2020.
View at: Publisher Site | Google Scholar
K. K. Ali, A. M. Wazwaz, and M. S. Osman, “Optical soliton solutions to the generalized nonautonomous nonlinear Schrodinger equations in optical fibers via the Sine-Gordon expansion method,” Optik, vol. 208, p. 164132, 2020.
View at: Publisher Site | Google Scholar
J. G. Liu, W. H. Zhu, M. S. Osman, and W. X. Ma, “An explicit plethora of different classes of interactive lump solutions for an extension form of 3D-Jimbo-Miwa model,” The European Physical Journal Plus, vol. 135, no. 5, pp. 1–9, 2020.
View at: Publisher Site | Google Scholar
I. Siddique, M. M. M. Jaradat, A. Zafar, K. Bukht Mehdi, and M. S. Osman, “Exact traveling wave solutions for two prolific conformable M-fractional differential equations via three diverse approaches,” Results in Physics, vol. 28, article 104557, 2021.
View at: Publisher Site | Google Scholar
H. F. Ismael, S. S. Atas, H. Bulut, and M. S. Osman, “Analytical solutions to the M-derivative resonant Davey-Stewartson equations,” Modern Physics Letters B, vol. 35, no. 30, article 2150455, 2021.
View at: Publisher Site | Google Scholar
Y. Saliou, S. Abbagari, A. Houwe et al., “W-shape bright and several other solutions to the (3+1)-dimensional nonlinear evolution equations,” Modern Physics Letters B, vol. 35, no. 30, article 2150468, 2021.
View at: Publisher Site | Google Scholar
H. Almusawa, K. K. Ali, A. M. Wazwaz, M. S. Mehanna, D. Baleanu, and M. S. Osman, “Protracted study on a real physical phenomenon generated by media inhomogeneities,” Results in Physics, vol. 31, article 104933, 2021.
View at: Publisher Site | Google Scholar
A. Zafar, M. Raheel, M. Q. Zafar et al., “Dynamics of different nonlinearities to the perturbed nonlinear Schrödinger equation via solitary wave solutions with numerical simulation,” Fractal and Fractional, vol. 5, no. 4, article 213, 2021.
View at: Publisher Site | Google Scholar
L. Cheng-Shi, “Classification of all single travelling wave solutions to Calogero-Degasperis-Focas equation,” Communications in Theoretical Physics, vol. 48, no. 4, pp. 601–604, 2007.
View at: Publisher Site | Google Scholar
L. Cheng-Shi, “All single traveling wave solutions to (3+1)-dimensional Nizhnok-Novikov-Veselov equation,” Communications in Theoretical Physics, vol. 45, no. 6, pp. 991-992, 2006.
View at: Publisher Site | Google Scholar
C.-s. Liu, “The classification of travelling wave solutions and superposition of multi-solutions to Camassa-Holm equation with dispersion,” Chinese Physics, vol. 16, no. 7, pp. 1832–1837, 2007.
View at: Publisher Site | Google Scholar
Y. Kai, “The classification of the single travelling wave solutions to the variant Boussinesq equations,” Pramana, vol. 87, no. 4, pp. 1–5, 2016.
View at: Publisher Site | Google Scholar
Y. Kai, B. Zheng, N. Yang, and W. Xu, “Exact single traveling wave solutions to generalized (2+1)-dimensional Gardner equation with variable coefficients,” Results in Physics, vol. 15, article 102527, 2019.
View at: Publisher Site | Google Scholar
D. Cao, “The classification of the single traveling wave solutions to the time-fraction Gardner equation,” Chinese Journal of Physics, vol. 59, pp. 379–392, 2019.
View at: Publisher Site | Google Scholar
B. Guan, S. Li, S. Chen, L. Zhang, and C. Wang, “The classification of single traveling wave solutions to coupled time- fractional KdV-Drinfel'd-Sokolov-Wilson system,” Results in Physics, vol. 13, article 102291, 2019.
View at: Publisher Site | Google Scholar
S. Chen, Y. Li, Y. Li, B. Guan, and Y. Liu, “Variant wave propagation patterns by coupled Bossinesq equations,” Results in Physics, vol. 24, article 104147, 2021.
View at: Publisher Site | Google Scholar
S. Chen, Y. Li, M. Jiang, B. Guan, Y. Liu, and F. Bu, “Abundant traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line,” Results in Physics, vol. 28, article 104587, 2021.
View at: Publisher Site | Google Scholar
X. Xiao and Z. Yin, “Exact single travelling wave solutions to the fractional perturbed Gerdjikov-Ivanov equation in nonlinear optics,” Modern Physics Letters B, vol. 35, no. 22, article 2150377, 2021.
View at: Google Scholar
Y. Kai, S. Chen, B. Zheng, K. Zhang, N. Yang, and W. Xu, “Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial method,” Chaos, Solitons and Fractals, vol. 141, article 110314, 2020.
View at: Google Scholar
Y. Kai, S. Chen, K. Zhang, and Z. Yin, “A study of the shallow water waves with some Boussinesq-type equations,” Waves in Random and Complex Media, vol. 1, pp. 1–18, 2021.
View at: Publisher Site | Google Scholar
Y. Kai, J. Ji, and Z. Yin, “Exact solutions and dynamic properties of Ito-type coupled nonlinear wave equations,” Physics Letters A, vol. 421, article 127780, 2021.
View at: Google Scholar
Y. Kai, J. Ji, and Z. Yin, “Study of the generalization of regularized long-wave equation,” Nonlinear Dynamics, vol. 107, no. 3, pp. 2745–2752, 2022.
View at: Google Scholar

Copyright

Copyright © 2022 Shuxin Bao and Shuangqing Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

173

Downloads

362

Citations