Research On Integrability And Exact Solutions For A Novel Hierarchy Of Differential Equations

Integrable system is an important research area in the field of nonlinear science.How to find new integrable systems and solve them in view of multiple methods,and discover their potential theoretical and applicable significance further,is an important branch in integrable system.This paper mainly concerns on integrability and exact solutions for a novel hierarchy of nonlinear evolution equations.First,we construct a 33? matrix spectral problem.Then,a novel hierarchy of Boussinesq-type nonlinear evolution equations related to the 33? matrix spectral problem is given by means of the zero curvature equation,which shows that they are integrable in the sense of having Lax pairs.Then,based on the spectral problem and its auxiliary spectral problem,the infinite many conservation laws of the first two nonlinear evolution equations are obtained in view of the associating Riccati type equation.Then it is shown that they are also integrable in the sense of having infinite many conservation laws.Next,we construct the Darboux transformation of the first nontrivial evolution equation by using the guage transformation between the spectral problem.Then two exact solutions of the equation are given by selecting the appropriate seed solution.Next the homoclinic breather wave solution of the equation is obtained by using the extended homoclinic breath test method.The homoclinic breather limit method is used to obtain an rational solution,which is shown to be the rogue wave solution.Finally,the soliton solution of this equation is obtained with the help of the bilinear method.The plots of homoclinic breather wave solution,rogue wave solution,one-soliton solution and two-soliton solution are shown by Maple software.The solutions we given are all checked by inserting them into the equation by the Maple software.