Bifurcation Analysis And The Exact Solutions For Two Classes Of Nonlinear Partial Differential Equations

The qualitative theory of differential equation and the bifurca-tion method of dynamical systems have great.significance for the structure and properties of the solutions of nonlinear partial differential equations.By using the qualitative theory of differential equation,the bifurcation method and symbolic computation method,the exact traveling wave solutions,the bifurcation phase portraits and the relations of the traveling wave solutions for two classes of par-tial differential equation have been investigated.These two classes equations are the integral order of(2+1)-dimensional Bogoyavlenskii equation,the extended(2+1)-dimensional Bogoyavlenskii-Schieff equation and the fractional order of(2+1)-dimensional Nizhnik-Novikov-Veslov equation.Firstly,by using the trav-eling wave transformation,the nonlinear equation can be turned into a planar dynamical system;secondly,by using the bifurcation method of dynamical sys-tems,the bifurcation phase portraits of the system and the exact traveling wave solutions have been obtained according to the phase diagram;finally,the relations between the solutions have been discussed.The approximation process between different types of solutions have been given by the Mathematical software.These solutions include periodic blow up wave solution,periodic wave solution,torsion-al wave solution,unbounded wave solution,blasting wave solution and solitary wave solution.