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

Jibin Li; Yan Zhou

doi:10.3934/dcdss.2020113

Article Contents

2020, Volume 13, Issue 11: 3083-3097. Doi: 10.3934/dcdss.2020113

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

Jibin Li^1,2, and
Yan Zhou^1,

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

Received: July 31, 2018

Early access: October 2019

Published: July 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 34A05, 34C25-28, 34M55, 35Q51, 35Q53; Secondary: 58F05, 58F14, 58F30.

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Figure 1. The bifurcations of phase portraits of system (8) when $\Delta_1 <0$

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Figure 2. The bifurcations of phase portraits of system (8) when $ k_1\beta\gamma<0 $

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Figure 3. The bifurcations of phase portraits of system (8) when $ \beta<0, k_1<0, \Delta_1>0 $

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Figure 4. The bifurcations of phase portraits of system (8) when $ \beta>0, k_1<0, \Delta_1>0 $

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Figure 5. The level curves of defined by $ H(\psi, y) = 0 $

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Figure 6. The changes of the level curves defined by $ H(\psi, y) = h $ of system (8)

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Figure 7. The bifurcations of phase portraits of system (8) when $ k_1\beta\gamma<0 $

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

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