| Solition theory is one of the most important study fields in the nonlinear sciences, it plays a significant role in the sciences and engineers. In this paper, the research object is the integrability and solutions of Variable-Coefficient equation, extended KP equation and two(2+1)-dimensional hierarchies of evolution equations and their Hamiltonian structures.The first chapter is an introduction, we main introduce the development of solition theory and the sumamary of the genetation and its current research status, last is my own works.In the second part, first, we introduce the bilinear derivative forms and bilinear B?cklund transformations of the Variable-Coefficient equation in line with the relationship of Bell polynomials and the bilinear Hirota D-operators. Second, we receive the specific forms of Lax pairs and conjugate Lax pairs for them according to the bilinear B?cklund transformation and the relationship of Bell polynomials and Hopf – Cole transformation v =lnψ. Last, is the infinite conservation laws.In the third chapter, with the help of the Riemann theta function, we obtain the periodic wave solution of the extended KP. Some integrable properties of the equations we obtained.In the fourth chapter, two kinds of appropriate isospectral problems are introduced by using a Lie algebra. With the help of the TAH scheme, we generate two new(2+1)-dimensional hiera of evolution equations, whose Hamiltonian structures are derived from the trace identity proposed by Tu Guizhang.In the fifth chapter, some conclusions are presented. |
