The new exact solutions of cubic nonlinear schr\"{o}dinger equations

Abstract

Abstract: Constructing exact solutions of cubic nonlinear schr\"{o}dinger equation will be better help us to understand the physical phenomenon of cubic schr\"{o}dinger equation. Generalized exp(-\phi(\xi))-expansion method is given here. Based on this method and the aid of symbolic computation system-Maple，new exact solutions of cubic nonlinear schr\"{o}dinger equation are obtained. These solutions include hyperbolic function solutions， trigonometric function solutions and rational function solutions.As a result, many new exact solutions are obtained. New solutions can be used in optical communications. It has been shown that this method provides a powerful mathematical tool for solving nonlinear partial differential equations in mathematical physics problems.

Key words: nonlinear equations, exact solutions, generalized -expansion method, cubic nonlinear schrödinger equations, optical communications, symbolic computation

CLC Number:

O175.2

MA Zhi-min 1，*， SUN Yu-huai 2. The new exact solutions of cubic nonlinear schr\"{o}dinger equations[J]. Chinese Journal of Quantum Electronics, 2019, 36(4): 416-422.

References

[1] Lai Lianyou, Chen Huibin. Symplectic algorithm simulation of motion and interaction of envelope soliton in Schrödinger equation[J]. Chinese Journal of Quantum Electronics (量子电子学报),2017,34(2): 231-240(in Chinese). [2] Li Yingjie, Zhi Hongyan. Painlevé analysis and solutions of the variable coefficient Schrödinger equation[J]. Chinese Journal of Quantum Electronics (量子电子学报),2017,34(6): 682-690(in Chinese). [3] Lu Dianchen, Seadawy Aly, Arshad M. Applications of extended simple equation method on unstable nonlinear Schrödinger equations[J].Optik, 2017, 140:136–144. [4] Huang Junkun, Zhang Shaowu, Wan Chong.Influence of nonlinear dispersion on Gauss pulse transmission[J].Chinese Journal of Quantum Electronics (量子电子学报),2018,35(5): 603-607(in Chinese). [5] Huang Junkun,Zhang Shaowu.Influence of Kerr dispersion and intrapulse Raman scattering on propagation characteristics of solitons[J]. Chinese Journal of Quantum Electronics (量子电子学报),2018,35(2): 204-208(in Chinese). [6] Yin Dejing. Interaction of self-Fourier optical solutions[J].Chinese Journal of Quantum Electronics (量子电子学报),2012,29(6):741-746(in Chinese). [7] Hou Gansheng，Hou Tong，Wan Chenxing，et al. Calculation method of multi-layer film transmissivity based on Schrödinger equation[J].Chinese Journal of Quantum Electronics (量子电子学报),2018,35(6):660-666(in Chinese). [8] Luo Xiangyi, Liu Xueshen, Ding Peizhu. Dynamic properties and drifting of the solution pattern of cubic nonlinear Schrödinger equation with varying nonlinear parameters[J].Acta Physica Sinica(物理学报), 2007, 56(2): 604-610(in Chinese). [9] Miao Hongli, Wang Jing, Meng Jiwu. Influence of initial chirp on the evolution of spectra [J]. Spectroscopy and Spectral Analysis (光谱学与光谱分析), 2003, 23(1): 38-41 (in Chinese). [10] Melike Kaplan, Ömer Ünsal, Ahmet Bekir. Exact solutions of nonlinear Schrödinger equation by using symbolic computation[J].Mathematical Methods in the Applied Sciences, 2016, 39(8): 2093–2099. [11] Hosseini K, Kumar D, Kaplan M,et al. New exact traveling wave Solutions of the unstable nonlinear Schrödinger equations[J].Communications in Theoretical Physics, 2017, 68(6): 761–767. [12] Abdelhalim Ebaid, Khaled.S.M.New types of exact solutions for nonlinear Schrödinger equation with cubic nonlinearity[J].Journal of Computational and Applied Mathematics, 2011, 235(8): 1984–1992. [13] Taogetusang. The first integral and new infinite sequence solutions of (n+1)-dimensional multiple sine-Gordon equation[J].Chinese Journal of Quantum Electronics (量子电子学报),2017,34(3):316-326.(in Chinese). [14] Sun Yuhui, Ma Zhimin, Li Yan. Explicit solutions for generalized(2+1)-dimensional nonlinear Zakharov—Kuznetsov equation[J].Communications in Theoretical Physics, 2010,54 (3): 397-400. [15] Jiang Ying, Xian Daquan. New non-travelling wave solutions with initial value perturbation of (2+1)-dimensional 　Sawada-Kotera equation[J].Chinese Journal of Quantum Electronics (量子电子学报),2017,34(6):700-704.(in Chinese). [16] Ma Zhimin, Sun Yuhuai, Liu Fusheng. Explicit solutions and bifurcations for a class of generalized boussinesq wave equation[J].Communications in Theoretical Physics, 2013 ,59 (3): 307-310. [17] Taogetusang. New soliton-like solutions of (3+1)-dimensional Burgers equation with variable coefficients[J]. Chinese Journal of Quantum Electronics (量子电子学报),2017,34(5):557-561.(in Chinese). [18] Khan K, Akbar MA. Application of -expansion method to find the exact solutions of modified Benjamin -Bona-Mahony equation[J]. World Applied Sciences Journal,2013, 24(10):1373–1377. [19] Harun-Or Roshid, Md Rashed Kabir, Rajandra Chadra Bhowmik, et al. Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and -expansion method[J].SpringerPlus, 2014, 3: (692),1 -10. [20] Khan Kamruzzaman, Akbar Ali M. Exact traveling wave solutions of Kadomtsev–Petviashvili equation[J].Journal of the Egyptian Mathematical Society,2015,23(2): 278–281. [21] Saima Arsheda, Anjan Biswasb, Mahmoud Abdelatyd,et al. Optical soliton perturbation with Kundu–Eckhaus equation by -expansion scheme and -expansion method[J].Optik,2018, 172: 79–85. [22] Wang Xiaoli, Sirendaoerji. Exact solutions of nonlinear evolution equations with variable coefficients based on expansion method[J].Chinese Journal of Quantum Electronics (量子电子学报),2016,33(6):680-688(in Chinese)