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

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.

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 $\Delta_1 <0$

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

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

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

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

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

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

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

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