Direct integral method, complete discrimination system for polynomial and applications to classifications of all single traveling wave solutions to nonlinear differential equations:a survey

Complete discrimination system for polynomial and direct integral method were discussed systematically. In particularly, we pointed out some mistaken viewpoints. Combining with complete discrimination system for polynomial, direct integral method was developed to become a powerful method and was applied to a lot of nonlinear mathematical physics equations. All single traveling wave solutions to theses equations can be obtained. As examples, we gave all traveling wave solutions to some equations such as mKdV equation, Sine-Gordon equation, Double Sine-Gordon equation, triple Sine-Gordon equation, Fujimoto-Watanabe equation, coupled Harry-Dym equation and coupled KdV equation and so on. At the end, we give the partial answers to Fan’s a problem.