Symmetry Constraints Of Semi-discrete Integrable Systems

In this dissertation we mainly investigate the differential-difference Kadomtsev-Pet-viashvili(D2?KP)hierarchy,the differential-difference modified Kadomtsev-Petviashvili(D1?mKP)hierarchy,their squared eigenfunction symmetries,and their links to(1+1)?dimensional semi-discrete integrable systems,the Ragnisco-Tu(RT),Volterra and rela-tivistic Toda lattice(RTL)hierarchies.We first introduce the RT spectral problem which is gauge equivalent to the Ablowitz-Ladik(AL)spectral problem and is a Darboux transformation(DT)as well as a bidi-rectional discretization of the Ablowitz-Kaup-Newell-Segur(AKNS)spectral problem.Isospectral and nonisospectral flows derived from the spectral problem compose a Lie algebra.Infinitely many symmetries of the nonisospectral hierarchy are obtained.By considering infinite dimensional subalgebras of the algebra and continuum limit of recur-sion operator,three semi-discrete AKNS hierarchies are constructed.For the scalar D2?KP hierarchy,we prove its squared eigenfunction symmetry.Through a constraint of the symmetry,Lax triad of the D2? KP hierarchy together with adjoined forms of temporal parts in the Lax triad are reduced to the RT spectral problem and the related semi-discrete AKNS hierarchy.For the D2?mKP hierarchy,we consider two ways to derive it,which are related to two different forms of quasi-difference operators.In the first case the Lax triad leads to the Volterra spectral problem and Volterra lattice hierarchy,while in the second case the Lax triad gives rise to the RTL spectral problem and its hierarchy.The RTL(+)and RTL(-)hierarchies are equivalent to each others under a transformation.These results are helpful to understand connections between semi-discrete(2 + 1)-dimensional and(1 + 1)-dimensional integrable systems.