American Institute of Mathematical Sciences

Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity

 1 School of Mathematcal Science, Huaqiao University, Quanzhou, Fujian 362021, China 2 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Received  August 2018 Published  October 2019

Fund Project: The authors are supported by the National Natural Science Foundation of China (11871231, 11162020).

For the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity, by using the method of dynamical systems, we investigate the bifurcations and exact traveling wave solutions. Because obtained traveling wave system is an integrable singular traveling wave system having a singular straight line and the origin in the phase plane is a high-order equilibrium point. We need to use the theory of singular systems to analyze the dynamics and bifurcation behavior of solutions of system. For $m>1$ and $0<m = \frac1n<\frac12$, corresponding to the level curves given by $H(\psi, y) = 0$, the exact explicit bounded traveling wave solutions can be given. For $m = 1$, corresponding all bounded phase orbits and depending on the changes of system's parameters, all exact traveling wave solutions of the equation can be obtain.

Citation: Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020113
References:
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References:
 [1] P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists, Second edition, revised. Die Grundlehren der mathematischen Wissenschaften, Band 67 Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar [2] J. Li, J. Wu and H. Zhu, Travelling waves for an integrable higher order KdV type wave equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2235-2260.  doi: 10.1142/S0218127406016033.  Google Scholar [3] J. Li and G. Chen, On a class of singular nonlinear traveling wave equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4049-4065.  doi: 10.1142/S0218127407019858.  Google Scholar [4] J. Li and H. Dai, On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical Approach, Science Press, Beijing, 2007.   Google Scholar [5] J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact solutions, Science Press, Beijing, 2013.   Google Scholar [6] J. Li, W. Zhou and G. Chen, Understanding peakons, periodic peakons and compactons via a shallow water wave equation,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650207, 27 pp. doi: 10.1142/S0218127416502072.  Google Scholar [7] A. M. Shahoot, K. A. E. Alurrfi, I. M. Hassan and A. M. Almsri, Solitons and other exact solutions for two nonlinear PDEs in mathematical physics using the generalized projective riccati equations method,, Adv. Math. Phys., 2018 (2018), Art. ID 6870310, 11 pp. doi: 10.1155/2018/6870310.  Google Scholar [8] N. K. Vitanov, Z. D. Dimitrova and T. I. Ivanova, On solitary wave solutions of a class of nonlinear partial differential equations based on the function $\frac{1}{\cosh^n(\alpha x+\beta t)}$,, Appl. Math. Compu., 315 (2017), 372-380.  doi: 10.1016/j.amc.2017.07.064.  Google Scholar [9] G. Q. Xu, New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation, Appl. Math. Comput., 217 (2011), 5967-5971.  doi: 10.1016/j.amc.2010.12.008.  Google Scholar [10] E. M. Zayed, A. G. Al-Nowehy and M. E. Elshater, Solitons and other solutions to nonlinear schrödinger equation with fourth-order dispersion and dual power law nonlinearity, Ric. Mat., 66 (2017), 531-552.   Google Scholar
The bifurcations of phase portraits of system (8) when $\Delta_1 <0$
The bifurcations of phase portraits of system (8) when $k_1\beta\gamma<0$
The bifurcations of phase portraits of system (8) when $\beta<0, k_1<0, \Delta_1>0$

Parameters: (a) $\gamma_0>\gamma>4k_1\beta>0.$ (b) $\gamma = \gamma_0 = \frac{4(m+1)^2k_1\beta}{2m+1}.$ (c) $\gamma>\gamma_0.$

The bifurcations of phase portraits of system (8) when $\beta>0, k_1<0, \Delta_1>0$

Parameters: (a) $\gamma_0<\gamma<4k_1\beta.$ (b) $\gamma = \gamma_0.$ (c) $\gamma<\gamma_0<0.$

The level curves of defined by $H(\psi, y) = 0$

Parameters: (a) $\beta<0, \Delta_1<0.$ (b) $k_1<0, \beta<0, \Delta_1>0, \gamma=\gamma_0>0, h_2=0.$ (c) $k_1<0, \beta<0, \Delta_1>0, \gamma>\gamma_0>0.$ (d) $k_1<0, \beta>0, \Delta_1>0, \gamma_0<\gamma<0.$ (e) $\beta\gamma k_1<0, \beta<0$ or $k_1=0, \beta<0, \gamma>0.$ (f) $\beta\gamma k_1<0, \beta>0.$

The changes of the level curves defined by $H(\psi, y) = h$ of system (8)
The bifurcations of phase portraits of system (8) when $k_1\beta\gamma<0$

(a) $h_1<0<h_2.$ (b) $h_1<h_2 = 0.$ (c) $h_1<h_2<0.$ (d) $h_1 = h_2<0.$

The bifurcations of phase portraits of system (8) for $m = \frac1n, \beta<0$

(a) $\gamma<\gamma_0<0, h_2<0<h_1; \gamma = \gamma_0, 0 = h_2<h_1; \gamma_0<\gamma<4k_1\beta, 0<h_2<h_1.$ (b) $\gamma = 4k_1\beta, 0<h_1 = h_2.$

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