| In this paper,we study the exact traveling wave solutions of physically significant nonlinear equations such as Benjiamin-BonaMahony equation,Newell equation and Benjiamin Ono equation by using the branching theory of plane dynamical system.With the help of the phase diagram under certain parametric conditions,integrating the elliptic equations,Maple,Mathematica,we analyze the dynamics of these nonlinear equations and obtain the rich accurate traveling wave solutions of the corresponding equations,which are divided into five parts.The first part mainly introduces the generation and development of the soliton theory and an overview of the method for finding traveling wave solutions,and finally expounds the main work of this paper.The second part focuses on the traveling wave solution of the Benjiamin-Bona-Mahony（BBM）equation.First,we row the wave transform to the BBM equation to obtain the first integral and the singular point of the equation.Second,the phase diagram of the planar system is obtained according to the singular point.Finally,its orbits are integrated to obtain the nonlinear wave solutions of the BBM equation at different parameters.The third part mainly studies the traveling wave solution of the Newell equation.First,the traveling wave transformation of the Newell equation is performed to obtain the first integral and singular point of the equation.Secondly,the phase diagram of the plane system is obtained according to different singular point.Finally,the orbital line is integrated.The nonlinear traveling wave solutions of the Newell equation under different parameters are obtained.In the fourth part,we mainly study the traveling wave solution of the Benjiamin Ono equation.First,we row the wave transform to the Benjiamin Ono equation to obtain the first integral and singular point of the equation.Second,the phase diagram of the planar system is obtained according to different singular point.Finally,the elliptic integral formula is used to obtain nonlinear traveling wave solutions with different parameters for the Benjiamin Ono equation.In the fifth part,the work of this paper is summarized and the future research direction is prospected. |
