Construction Of Exact Traveling Wave Solutions Of The Kaup-Newell Equation And Exact Solutions Of The Two-Component DGH System

With the advancement of nonlinear scientific research,nonlinear partial differential equations have gradually become a common mathematical model in fields such as fluid mechanics and engineering technology.In many complex problems,the analysis of exact solutions of these nonlinear partial differential equations with a strong physical background can explain abstract natural phenomena more clearly,and on this basis,reasonably predict the changes of the phenomena.In this paper,the problem of exact traveling wave solutions of the Kaup-Newell equation in the field of optics and the two-component Dullin-Gottwald-Holm system in the field of shallow water waves is investigated using the direct integral method and the complete discrimination system for polynomial method.For the Kaup-Newell equation,exact solutions of the equation,which include solitary wave solutions,trigonometric function type solutions and Jacobi elliptic function type solutions,are obtained using the complete discrimination system for polynomial method,and a complete classification of chirp envelope modes is given.In addition,three propagation mode dynamics are plotted under the model,given specific values for the parameters of the exact solution of the equation.The images provide a clearer understanding of the transmission dynamics of femtosecond optical solitons in non-linear materials and allow information on the dynamics of light propagation to be obtained in the model.The two-component Dullin-Gottwald-Holm system is generalised from a class of 1+1-dimensional productable DGH equations that more accurately describe the motion of shallow water waves with non-zero spin.The system can be reduced to the form of an ordinary differential equation by a traveling wave transformation,on the basis of which two methods,the direct integral method and the complete discrimination system for polynomial method,are used to construct exact traveling wave solutions of the system in the richly classified case,and to show images of the relevant solutions by appropriately selecting the values of the exact solution parameters of the system.