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.

中文翻译：

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具有四阶色散和双幂律非线性的非线性薛定ding方程的分叉和精确行波解

对于具有四阶色散和双重幂律非线性的非线性薛定ding（NLS）方程，采用动力学系统方法，研究了分叉和精确的行波解。因为所获得的行波系统是具有奇异直线的可积分奇异行波系统，并且相平面中的原点是高阶平衡点。我们需要使用奇异系统理论来分析系统解的动力学和分叉行为。对于$ m> 1 $和$ 0 <m = \ frac1n <\ frac12 $，对应于$ H（\ psi，y）= 0 $给出的电平曲线，可以给出精确的显式有界行波解。对于$ m = 1 $，对应于所有有界相位轨道，并取决于系统参数的变化，