TRAVELING WAVE SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL KUNDU-MUKHERJEE-NASKAR EQUATION

Minrong Ren; Yuqian Zhou; Qian Liu

doi:10.11948/20210192

2021 Volume 11 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(6): 3083-3114

Next Article Previous Article

Minrong Ren, Yuqian Zhou, Qian Liu. TRAVELING WAVE SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL KUNDU-MUKHERJEE-NASKAR EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 3083-3114. doi: 10.11948/20210192

Citation:

Minrong Ren, Yuqian Zhou, Qian Liu. TRAVELING WAVE SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL KUNDU-MUKHERJEE-NASKAR EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 3083-3114. doi: 10.11948/20210192

TRAVELING WAVE SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL KUNDU-MUKHERJEE-NASKAR EQUATION

1.
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, Sichuan, China
2.
School of Computer Science and Technology, Southwest Minzu University, Chengdu 610041, Sichuan, China

Corresponding author: Email address: cs97zyq@aliyun.com(Y. Zhou)
Fund Project: This work is supported by the Natural Science Foundation of China (Nos. 11301043, 11701480), China Postdoctoral Science Foundation (No. 2016M602663) and Sichuan Science and Technology Program (No. 21ZYZYTS0158)

Abstract

Abstract

In this paper, we consider two types of traveling wave systems of the generalized Kundu-Mukherjee-Naskar equation. Firstly, due to the integrity, we obtain their energy functions. Then, the dynamical system method is applied to study bifurcation behaviours of the two types of traveling wave systems to obtain corresponding global phase portraits in different parameter bifurcation sets. According to them, every bounded and unbounded orbits can be identified clearly and investigated carefully which correspond to various traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation exactly. Finally, by integrating along these orbits and calculating some complicated elliptic integral, we obtain all type Ⅰ and type Ⅱ traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation without loss.
- GKMN equation /
- traveling wave solution /
- bifurcation /
- dynamical system
MSC: 35C07, 35Q55, 34C23

FullText(HTML)

References

[1]	A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S. P. Moshokoa and M. Belic, Optical soliton perturbation with quadratic-cubic nonlinearity using a couple of strategic algorithms, Chin. J. Phys., 2018, 56(5), 1990-1998. doi: 10.1016/j.cjph.2018.09.009 CrossRef Google Scholar
[2]	S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer Science & Business Media, 2012. Google Scholar
[3]	M. Ekici, A. Sonmezoglu, A. Biswas and M. R. Belic, Optical solitons in (2+1)-dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme, Chin. J. Phys., 2019, 57, 72-77. doi: 10.1016/j.cjph.2018.12.011 CrossRef Google Scholar
[4]	B. Ghanbari, H. Günerhan, O. A. İlhan and H. M. Baskonus, Some new families of exact solutions to a new extension of nonlinear Schrödinger equation, Phys. Scripta., 2020, 95(7), Article ID: 075208. Google Scholar
[5]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer, New York, 1983. Google Scholar
[6]	A. Jhangeer, A. R. Seadawy, F. Ali and A. Ahmed, New complex waves of perturbed Schrödinger equation with Kerr law nonlinearity and Kundu-Mukherjee-Naskar equation, Results Phys., 2020, 16, Article ID: 102816. Google Scholar
[7]	N. A. Kudryashov, General solution of traveling wave reduction for the Kundu-Mukherjee-Naskar model, Optik., 2019, 186, 22-27. doi: 10.1016/j.ijleo.2019.04.072 CrossRef Google Scholar
[8]	A. Kundu, Novel hierarchies & hidden dimensions in integrable field models: theory & application, Journal of Physics: Conference Series, 2014, 482, Article ID: 012022. Google Scholar
[9]	A. Kundu and A. Mukherjee, Novel integrable higher-dimensional nonlinear Schrödinger equation: properties, solutions, applications, 2013, arXiv: 1305.4023. Google Scholar
[10]	A. Kundu, A. Mukherjee and T. Naskar, Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents, Proc. R. Soc. Lond. Ser. A Math. Phys., 2014, 470(2164), Article ID: 20130576. Google Scholar
[11]	D. Kumar, G. C. Paul, T. Biswas, A. R. Seadawy, R. Baowali, M. Kamal and H. Rezazadeh, Optical solutions to the Kundu-Mukherjee-Naskar equation: mathematical and graphical analysis with oblique wave propagation, Phys. Scripta., 2020, 96(2), Article ID: 025218. Google Scholar
[12]	A. Mukherjee, M. Janaki and A. Kundu, A new (2+1) dimensional integrable evolution equation for an ion acoustic wave in a magnetized plasma, Phys. Plasmas., 2015, 22(7), Article ID: 072302. Google Scholar
[13]	W. Peng, S. Tian and T. Zhang, Optical solitons, complexitons and power series solutions of a (2+1)-dimensional nonlinear Schrödinger equation, Modern Phys. Lett. B., 2018, 32(28), Article ID: 1850336. Google Scholar
[14]	D. Qiu, Y. Zhang and J. He, The rogue wave solutions of a new (2+1)-dimensional equation, Commun. Nonlinear Sci. Numer. Simul., 2016, 30(1-3), 307-315. doi: 10.1016/j.cnsns.2015.06.025 CrossRef Google Scholar
[15]	S. T. R. Rizvi, I. Afzal and K. Ali, Dark and singular optical solitons for Kundu-Mukherjee-Naskar model, Modern Phys. Lett. B., 2020, 34(6), Article ID: 2050074. Google Scholar
[16]	H. Rezazadeh, A. Kurt, A. Tozar, O. Tasbozan and S. Mirhosseini-Alizamini, Wave behaviors of Kundu-Mukherjee-Naskar model arising in optical fiber communication systems with complex structure, Opt. Quant. Electon., 2021. DOI: 10.21203/rs.3.rs-168692/v1. CrossRef Google Scholar
[17]	S. Singh, A. Mukherjee, K. Sakkaravarthi and K. Murugesan, Higher dimensional localized and periodic wave dynamics in an integrable (2+1)-dimensional deep water oceanic wave model, Wave. Random. Complex., 2021, 1-20. DOI: 10.1080/17455030.2021.1874621. CrossRef Google Scholar
[18]	T. A. Sulaiman and H. Bulut, The new extended rational SGEEM for construction of optical solitons to the (2+1)-dimensional Kundu-Mukherjee-Naskar model, Appl. Math. Nonlinear Sci., 2019, 4(2), 513-521. doi: 10.2478/AMNS.2019.2.00048 CrossRef Google Scholar
[19]	R. Talarposhti, P. Jalili, H. Rezazadeh, B. Jalili, D. Ganji, W. Adel and A. Bekir, Optical soliton solutions to the (2+1)-dimensional Kundu-Mukherjee-Naskar equation, Internat. J. Modern Phys. B., 2020, 34(11), Article ID: 2050102. Google Scholar
[20]	X. Wen, Higher-order rational solutions for the (2+1)-dimensional KMN equation, Proc. Rom. Acad. Ser. A Math. Phys., 2017, 18(3), 191-198. Google Scholar
[21]	Y. Yıldırım, Optical soltons to Kundu-Mukherjee-Naskar model with trial equation approach, Optik., 2019, 183, 1061-1065. doi: 10.1016/j.ijleo.2019.02.117 CrossRef Google Scholar
[22]	Y. Yıldırım, Optical soltons to Kundu-Mukherjee-Naskar model with modified simple equation approach, Optik., 2019, 184, 247-252. doi: 10.1016/j.ijleo.2019.02.135 CrossRef Google Scholar
[23]	Y. Yıldırım, Optical soltons to Kundu-Mukherjee-Naskar model in birefringent fibers with trial equation approach, Optik., 2019, 183, 1026-1031. doi: 10.1016/j.ijleo.2019.02.141 CrossRef Google Scholar
[24]	Y. Yıldırım, Optical solitons to Kundu-Mukherjee-Naskar model in birefringent fibers with modified equation approach, Optik., 2019, 184, 121-127. doi: 10.1016/j.ijleo.2019.02.155 CrossRef Google Scholar
[25]	Y. Yıldırım and M. Mirzazadeh, Optical pulses with Kundu-Mukherjee-Naskar model in fiber communication systems, Chinese J. Phys., 2020, 64, 183-193. doi: 10.1016/j.cjph.2019.10.025 CrossRef Google Scholar
[26]	Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, USA, 1992. Google Scholar