BIFURCATIONS AND OBTAINED EXACT SOLUTIONS OF THE OPTICAL SOLITON MODEL IN METAMATERIALS DOMINATED BY ANTI-CUBIC NONLINEARITY

Qiuyan Zhang; Yuqian Zhou

doi:10.11948/20220289

2023 Volume 13 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(4): 1931-1971

Next Article Previous Article

Qiuyan Zhang, Yuqian Zhou. BIFURCATIONS AND OBTAINED EXACT SOLUTIONS OF THE OPTICAL SOLITON MODEL IN METAMATERIALS DOMINATED BY ANTI-CUBIC NONLINEARITY[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1931-1971. doi: 10.11948/20220289

Citation:

Qiuyan Zhang, Yuqian Zhou. BIFURCATIONS AND OBTAINED EXACT SOLUTIONS OF THE OPTICAL SOLITON MODEL IN METAMATERIALS DOMINATED BY ANTI-CUBIC NONLINEARITY[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1931-1971. doi: 10.11948/20220289

BIFURCATIONS AND OBTAINED EXACT SOLUTIONS OF THE OPTICAL SOLITON MODEL IN METAMATERIALS DOMINATED BY ANTI-CUBIC NONLINEARITY

Qiuyan Zhang^1,2,,
Yuqian Zhou^1, ,

1.
College of Applied Mathematics, Chengdu University of Information Technology, No.24, Section 1, Xuefu Road, Southwest Airport Economic Development Zone, Chengdu 610225, China
2.
School of Mathematical Sciences, University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, Hi Tech Zone (West District), Chengdu 611731, China

Author Bio: Email: zqy1607@cuit.edu.cn(Q. Zhang)
Corresponding author: Email: cs97zyq@aliyun.com(Y. Zhou)
Fund Project: The authors were supported by National Natural Science Foundation of China (12101090), Sichuan Natural Science Foundation(2023NSFSC0071), Sichuan Science and Technology Program (2021ZYD0009), General Projects of Local Science and Technology Development Funds Guided by the Central Government (2022ZYD0005) and Key Project of Scientific Research and Innovation Team of Chengdu University of Information Technology (KYTD202226)

Abstract

Abstract

In this paper, we study the optical model in metamaterials with nonlinear influence of non-Kerr law and a few Hamiltonian perturbation terms. The nonlinearity of metamaterials is dominated by anti-cubic type. We apply the approach of dynamical system to find the travelling wave solutions of the optical model. Under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, peakon, periodic peakons as well as compacton solutions for this planar dynamical system are given. By using the first integral, exact solutions of can be calculated under different parameter conditions. 83 exact explicit parametric representations are derived.
- Bifurcation /
- exact solution /
- optical soliton /
- anti-cubic nonlinearity
MSC: 34C23, 34C25, 34C37, 74J30

FullText(HTML)

References

[1]	S. S. Afzal, M. Younis and S. T. R. Rizvi, Optical dark and dark-singular solitons with anti-cubic nonlinearity, Optik, 2017, 147, 27–31. doi: 10.1016/j.ijleo.2017.08.067 CrossRef Google Scholar
[2]	K. S. Al-Ghafri, Solitary wave solutions of two KdV-type equations, Open Phys. 2018, 16(1), 311–318. doi: 10.1515/phys-2018-0043 CrossRef Google Scholar
[3]	K. S. Al-Ghafri1 and E. V. Krishnan, Optical solitons in metamaterials dominated by anti-cubic nonlinearity and Hamiltonian perturbations, Int. J. Appl. Comput. Math., 2020, 6(5), 144. doi: 10.1007/s40819-020-00896-1 CrossRef Google Scholar
[4]	K. Al-Ghafri, E. Krishnan, A. Biswas, et al., Optical solitons having anti-cubic nonlinearity with a couple of exotic integration schemes, Optik, 2018, 172, 794–800. doi: 10.1016/j.ijleo.2018.07.101 CrossRef Google Scholar
[5]	L. Alloatti, C. Kieninger, A. Froelich, et al., Second-order nonlinear optical metamaterials: ABC-type nanolaminates, Appl. Phys. Lett., 2015, 107(12), 121903. doi: 10.1063/1.4931492 CrossRef Google Scholar
[6]	A. Arab, Exactly solvable supersymmetric quantum mechanics, J. Math. Anal. Appl., 1991, 158(1), 63–79. doi: 10.1016/0022-247X(91)90267-4 CrossRef Google Scholar
[7]	A. Arab, Exact solutions of multi-component nonlinear Schrdinger and Klein-Gordon equations in two-dimensional space-time, J. Phys. A: Math. Gen., 2001, 34(20), 4281–4288. doi: 10.1088/0305-4470/34/20/302 CrossRef Google Scholar
[8]	A. H. Arnous, M. Ekici, S. P. Moshokoa, et al., Solitons in nonlinear directional couplers with optical metamaterials by trial function scheme, Acta Phys. Polonica A., 2017, 132(4), 1399–1410. doi: 10.12693/APhysPolA.132.1399 CrossRef Google Scholar
[9]	A. Biswas, M. A. Ekici, Sonmezoglu, et al., Solitons in optical metamaterials with anti-cubic nonlinearity, Eur. Phys. J. Plus, 2018, 133, 204. doi: 10.1140/epjp/i2018-12046-6 CrossRef Google Scholar
[10]	A. Biswas, K. R. Khan, M. F. Mahmood, et al., Bright and dark solitons in optical metamaterials, Optik, 2014, 125, 3299–3302 doi: 10.1016/j.ijleo.2013.12.061 CrossRef Google Scholar
[11]	A. Biswas, M. Mirzazadeh, M. Eslami, et al., Solitons in optical metamaterials by functional variable method and first integral approach, Frequenz, 2014, 68(11–12), 525–530. Google Scholar
[12]	P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists, Springer, Berlin, 1971. Google Scholar
[13]	M. Ekici, Exact solitons in optical metamaterials with quadratic-cubic nonlinearity using two integration approaches, Optik, 2018, 156, 351–355. doi: 10.1016/j.ijleo.2017.11.056 CrossRef Google Scholar
[14]	M. Ekici, A. Sonmezoglu, Q. Zhou, et al., Analysis of optical solitons in nonlinear negative-indexed materials with anti-cubic nonlinearity, Opt. Quantum Electron., 2018, 50(2), 75. doi: 10.1007/s11082-018-1341-3 CrossRef Google Scholar
[15]	S. Y. Elnaggar and G. N. Milford, Description and stability analysis of nonlinear transmission line type metamaterials using nonlinear dynamics theory, J. Appl. Phys., 2017, 121(12), 124902. doi: 10.1063/1.4979022 CrossRef Google Scholar
[16]	M. Foroutan, J. Manafian and A. Ranjbaran, Solitons in optical metamaterials with anti-cubic law of nonlinearity by generalized G′/G-expansion method, Optik, 2018, 162, 86–94. doi: 10.1016/j.ijleo.2018.02.087 CrossRef Google Scholar
[17]	M. Foroutan, J. Manafian and I. Zamanpour, Solitonwave solutions in optical metamaterials with anti-cubic law of nonlinearity by ITEM, Optik, 2018, 164, 371–379. doi: 10.1016/j.ijleo.2018.03.025 CrossRef Google Scholar
[18]	Y. Fu and J. Li, Exact stationary-wave solutions in the standard model of the Kerr-nonlinear optical fiber with the bragg grating, J. Appl. Anal, Comput., 2017, 7(3), 1177–1184. Google Scholar
[19]	A. A. Kader, M. A. Latif and Q. Zhou, Exact optical solitons in metamaterials with anti-cubic law of nonlinearity by Lie group method, Opt. Quantum Electron, 2019, 51(1), 30. doi: 10.1007/s11082-019-1748-5 CrossRef Google Scholar
[20]	E. Krishnan, M. Al Gabshi, Q. Zhou, et al., Solitons in optical metamaterials by mapping method, Optoelectron. Adv. Mater. Commun., 2015, 17, 511–516. Google Scholar
[21]	E. Krishnan, A. Biswas, Q. Zhou, et al., Optical solitons with anti-cubic nonlinearity by mapping methods, Optik, 2018, 170, 520–526. doi: 10.1016/j.ijleo.2018.06.010 CrossRef Google Scholar
[22]	M. Lapine, M. Gorkunov and K. H. Ringhofer, Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements, Phys. Rev. E., 2003, 67, 065601. Google Scholar
[23]	J. Li, Geometric properties and exact travelling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation, J. Nonl. Mod. Anal., 2019, 1(1), 1–10. Google Scholar
[24]	J. Li, Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. Google Scholar
[25]	J. Li and G. Chen, Bifurcations of travelling wave solutions for four classes of nonlinear wave equations, Int. J. Bifurcat. Chaos, 2005, 15(12), 3973–3998. doi: 10.1142/S0218127405014416 CrossRef Google Scholar
[26]	J. Li and G. Chen, On a class of singular nonlinear traveling wave equations, Int. J. Bifurcat. Chaos, 2007, 17(11), 4049–4065. doi: 10.1142/S0218127407019858 CrossRef Google Scholar
[27]	J. Li and M. Han, Exact peakon solutions given by the generalized hyperbolic functions for some nonlinear wave equations, J. Appl. Anal, Comput., 2020, 10(4), 1708–1719. Google Scholar
[28]	J. Li, Y. Zhang and X. Zhao, On a class of singular nonlinear traveling wave equations (Ⅱ): an example of GCKdV equations, Int. J. Bifurcat. Chaos, 2011, 19(6), 1955–2007. Google Scholar
[29]	N. N. Potravkin, V. A. Makarov and I. A. Perezhogin, Modeling highly-dispersive transparency in planar nonlinear metamaterials, Opt. Commun., 2017, 385, 177–180. doi: 10.1016/j.optcom.2016.10.056 CrossRef Google Scholar
[30]	E. Prati, Microwave propagation in round guiding structures based on double negative metamaterials, Int. J. Infrared Milli., 2006, 27(9), 1227–1239. Google Scholar
[31]	R. A. Shelby, D. R. Smith and S. Schultz, Experimental Verification of a Negative Index of Refraction, Science, 2001, 292(5514), 77–79. doi: 10.1126/science.1058847 CrossRef Google Scholar
[32]	H. Triki, A. Biswas, Q. Zhou, et al., Solitons in optical metamaterials having parabolic law nonlinearity with detuning effect and Raman scattering, Optik, 2018, 164, 606–609. doi: 10.1016/j.ijleo.2018.03.068 CrossRef Google Scholar
[33]	R. Yang and I. V. Shadrivov, Double-nonlinear metamaterials, Appl. Phys. Lett., 2010, 97(23), 231114. doi: 10.1063/1.3525172 CrossRef Google Scholar
[34]	E. Zayed and K. Alurrfi, New extended auxiliary equation method and its applications to nonlinear Schrödinger-type equations, Optik, 2016, 127, 9131–9151. doi: 10.1016/j.ijleo.2016.05.100 CrossRef Google Scholar
[35]	X. Zeng and X. Yong, A new mapping method and its applications to nonlinear partial differential equations, Phys. Lett. A, 2008, 372(44), 6602–6607. doi: 10.1016/j.physleta.2008.09.025 CrossRef Google Scholar
[36]	Q. Zhang, Bifurcations and exact solutions of the optical soliton model in metamaterials dominated by anti-cubic nonlinearity, J. Appl. Anal. Comput., 2022, 12(4), 1517–1531. Google Scholar
[37]	A. A. Zharov, I. V. Shadrivov and Y. S. Kivshar, Nonlinear Properties of Left-Handed Metamaterials, Phys. Rev. Lett., 2003, 91, 037401. doi: 10.1103/PhysRevLett.91.037401 CrossRef Google Scholar
[38]	Y. Zhou and J. Zhuang, Bifurcations and exact eolutions of the Raman soliton model in nanoscale optical waveguides with metamaterials, J. Nonl. Mod. Anal., 2021, 3(1), 145–165. Google Scholar
[39]	B. Zheng, A new Bernoulli sub-ODE method for constructing traveling wave solutions for two nonlinear equations with any order, U. Politeh. Buch. Ser. A., 2011, 73(3), 85–94. Google Scholar