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The complex method is systematic and powerful to build various kinds of exact meromorphic solutions for nonlinear partial differential equations on the complex plane C {\mathbb{C}} . By using the complex method, abundant new exact meromorphic solutions to the ( 2 + 1 ) \left(2+1) -dimensional and the ( 3 + 1 ) \left(3+1) -dimensional Boiti-Leon-Manna-Pempinelli equations and the ( 2 + 1 ) \left(2+1) -dimension Kundu-Mukherjee-Naskar equation are investigated. Abundant new elliptic solutions, rational solutions and exponential solutions have been constructed.

Using the traveling wave transformation, the seventh-order KdV equation reduces to a sixth-order complex differential equation (CDE), and we first prove that all meromorphic solutions of the CDE belong to the class W via Nevanlinna's value distribution theory. Then abundant new meromorphic solutions of the sixth-order CDE have been established in the finite complex plane with the aid of an extended complex method and Painlevé analysis, which contains Weierstrass elliptic function solutions and exponential function solutions, some of them are whole new solutions comparing to the opening literature. We give the computer simulations of some elliptic and exponential solutions. At last, we investigate the meromorphic solutions of the nonlinear dispersive Kawahara equation as an application.

The annual exam questions in China National College Entrance Examination (CNCEE, commonly known as Gaokao) have attracted wide attention from the entire society in China. This paper analyzes an important math exam question related to always holding inequalities with an unknown parameter from the 2020 CNCEE and builds different solutions via parameter separation or classified discussion, and then gives a systematic and effective design framework for questions related to problems of always holding inequalities and zeros of functions with an unknown parameter for the CNCEE. The framework reveals the internal mechanism of designing the math exam questions in the CNCEE. According to the design framework, two simulated math exam questions with reference solutions will be proposed.

This paper is concerned with the periodicity of entire functions with finite growth order, and some sufficient conditions are given. Let f is a transcendental entire function with finite growth order, zero is a Picard exceptional value of f , and a given differential monomial Q f of f is periodic, then f is also periodic. We are also interested in finding the following: let f is a transcendental entire function with finite growth order, d is a Picard exceptional value of f and f z Δ c n f z is a periodic function, then f is also a periodic function. These results extend Yang’s conjecture.