Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial method

Abstract

• We show that the complete discrimination system for polynomial method can be used to conduct qualitative analysis. • As a concrete example, the critical region for perturbed Gardner’s equation is given. • The exact solutions for perturbed Gardner’s equation are presented. 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 .