A Class Of Hierarchy Of Nonlinear Differential Difference Equations And Its Inverse Scattering Transformation

In this dissertation, one of the important branch of nonlinear science, solitontheory, is considered as the research background. Based on the problems ofconstructing exact solutions of nonlinear differential difference equations withvariable coefficients, a class of Toda chain hierarchy with variable coefficients isdeduced through the zero curvature equation which is discrete form, and multiplesoliton solutions of the Toda chain hierarchy are obtained by using inverse scatteringmethod.The main content of this paper is divided into four parts:The first chapter presents the development of the soliton theory, nonlineardifferential-difference equations and several kinds of classic methods for solvingdifferential-difference equations with constant coefficients, the key point isintroducing the inverse scattering transform and its development status.The second chapter summarizes the inverse scattering transformation theory forthe Toda chain hierarchy with constant coefficients, it includes zero curvatureequation, the derivation of Toda chain hierarchy and the issues of solving multiplesoliton solutions of Toda chain hierarchy through the inverse scattering method.In the third chapter, a isospectral variable-coefficient Toda chain hierarchy isderived from the zero curvature equation by introducing an arbitrary function α(t)of time t. Multiple soliton solutions of the derived Toda chain hierarchy areobtained by applying the inverse scattering transform.In the last chapter, a nonisospectral variable-coefficient Toda chain hierarchy isderived from the zero curvature equation by introducing two arbitrary functionsα (t), β (t)of time t and the relationshipλ=()α (t)λ k1λ2t β t (4)spectralparameter satisfies. Multiple soliton solutions of the derived Toda chain hierarchyare obtained by applying the inverse scattering transform.