Integrable Couplings Of Nonisospectral Integrable Hierarchies,Hamiltonian Structure As Well As Symmetry

Soliton theory is a power tool in expanding and describing the nonlinear phenomena in the fields of plasma physics,magnetic fluid,nonlinear optices,and so on.Searching for new integrable systems is an interesting research topic.Among them,Searching for new the isospectral and nonisospectral soliton equations and integrable couplings systems are new and important research direction.Therefore,it is meaningful and indispensable to derive the isospectral and non-isospectral soliton equations hierarchies and their integrable couplings systems and then to study their properties.In this paper,we mainly study the generation of several kinds of non-isospectral integrable equations,the construction of integrable couplings systems,and the derivation of new multi-component integrable systems,and the derivation of the bi-Hamiltonian structures of the isospectral and non-isospectral soliton hierarchies and their integrable couplings systems using the improved Tu scheme and ZNε-trace identity.The main contributions are as follows:First,we consider two spectral problems associated with so(3,R)under the case of(?),Solving the zero-curvature equation,we derive the AKNS type and KN type soliton equation hierarchies,and reduce some nonlinear equations with physical significance.Based on the Tu scheme,we obtain the bi-Hamiltonian structures of the non-isospectral AKNS type and KN type soliton hierarchies.Secondly,we consider the modified Jaulent-Miodek linear spectrum problem and derive the nonisospectral modified Jaulent-Miodek soliton hierarchy and its bi-Hamiltonian structure in this case of(?)using the nonisospectral zero curvature equation and the improved Tu scheme.The bi-Hamiltonian structure is obtained to prove the Integrability of the soliton hierarchy in the sense of Liouville.The bi-Hamiltonian is obtained to prove the integrability of nonisospectral AKNS type soliton hierarchy,nonisospectral KN type soliton hierarchy and nonisospectral mJM soliton hierarchy in the sense of Liouville.Second,by introducing a two-dimensional non-semisimple Lie algebra g,and applying it to the spectral problem associated with matrix loop algebra so(3,R)and the semi-discrete spatial spectral problem,respectively.The integrable coupling system of non-isospectral soliton hierarchy of AKNS type and the integrable coupling system of generalized Toda lattice hierarchy are derived using continuous zero curvature equations and discrete zero curvature equations,respectively.The generalized Toda lattice equations,relativistic Toda lattice equations,and some interesting nonlinear systems are derived from them.Then the bi-Hamiltonian structure of the AKNS integrable coupling hierarchy is obtained by using the Z2ε trace identity.Finally,a new high-dimensional Lie algebra g and the corresponding loop algebra g are introduced,and a new linear spectral problem is constructed.By considering the variation of the spectral parameters with time,a integrable coupling system of nonisospectral modified Jaulent-Miodek equation hierarchy is derived from the zero curvature equation.The bi-Hamiltonian structure is derived using quadratic-form identity.Third,based on the non-semisimple Lie algebra g,the spectral matrix of modified Jaulent-Miodek spatial spectral problem is extended to the N-dimensional case and a multi-component nonisospectral modified Jaulent-Miodek integrable hierarchy is generated starting from the zero curvature equation.the zero curvature equation.The Hamiltonian structure is derived based on the ZNε-trace identity.