Research On Realted Properties Of A Generalized Integrable Hierarchy

The main contents of this paper are as follows: the first one is to construct the integrable couplings system of the Levi integrable hierarchy and find the bi-Hamilton structure and make a preliminary study on the multi-component KWI integrable hierarchy and the multi-component KN integrable hierarchy.The other one is to seek the exact solutions of several kinds of nonlinear evolution equations.At first,taking the classical Levi integrable hierarchy as an example,we briefly introduce the general steps of Tu scheme: the first step is to construct an appropriate spectral problem for spatial variables;the second step is to obtain a recursive relationship by solving the stationary zero curvature equation;the third step is to construct an appropriate spectral problem for temporal variables and generate the integrable hierarchy by the zero curvature equations;the fourth step is to construct bi-Hamilton structure by using the trace identity.Next,based on a non-semi-simple Lie algebra,the classical Levi integrable hierarchy are used to construct the corresponding integrable coupled system and the bi-Hamilton structure is given by the multi-component trace identities.Then two new Levi integrable hierarchies obtained with the similar researchs and then the integrable couplings and bi-Hamilton structures of these two new Levi integrable hierarchies are obtained by Tu scheme.Next,we construct the multi-component KN hierarchy and the multi-component WKI hierarchy,and study their Hamilton structures and the corresponding non-isospectral flows.Finally,we study the exact solutions of several kinds of nonlinear evolution equations.For the(1+1)dimensional Drinfel’d-Sokolov-Wilson(DSW)equations,the exact solutions of the DSW equations are expressed in different forms f.Then two sets of interaction solutions between lump and solitons are introduced,and the corresponding lumpoff behavior(lump-one soliton type solution)and rogue wave(lump-twin soliton type solution)are discussed respectively.For(3+1)dimensional generalized Burgers equation,starting from a Backlund transformation,abundant localized solutions are provided with the help of different options for the seed solution and a multi-linear variable separation ansatz.For the Ito equation,we seek the exact solution of the multi-cosh function type,and by the relationship of the cosh function and the cos function,we can get the exact solution of the multi-cos function type.For the delay reaction-diffusion systems,we use the functional constraints method to reduce and solve the systems.