We show that via the complete discrimination system for polynomial method, the bifurcation, critical condition and topological properties of the nonlinear dynamics can be seen very easily. Concrete example of perturbed Gardner’s equation with high order dispersion also verifies our conclusion. The results indicate that the complete discrimination system for polynomial method can not only be used to get quantitative results such as the classification of the traveling wave solutions, but also to conduct qualitative analysis for the nonlinear differential equations.

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多项式完全判别系统对非线性动力学的定性和定量分析

我们表明，通过用于多项式方法的完整判别系统，可以非常容易地看出非线性动力学的分歧，临界条件和拓扑性质。具有高阶色散的摄动Gardner方程的具体例子也验证了我们的结论。结果表明，完备的多项式判别系统不仅可以用来获得行波解的分类等定量结果，而且可以对非线性微分方程进行定性分析。