Studies On Two Types Of Generalized Integrable Systems

In the study of integrable system and soliton equations,the integrable extensions or couplings has drawn lots of interests.Soliton equations with self-consistent sources,as an important class of extensions or couplings,has significant applications in mathematics and physics.Squared eigenfunction symmetry is an effective method in the studying of self-consistent sources,which has been applied successfully in many integrable systems.The coupled NLS equation,another important generalization of classical NLS equation,can be reduced in several types of nonlocal NLS equations under specific conditions.Darboux transformations plays an important role in studying those generalized integrable systems.The first part of this thesis concerns about the bilinear identity for extended twodimensional Toda Lattice(2DTL)Hierarchy.With the help of an auxiliary flow which satisfies the squared eigenfunction symmetry,we obtained the bilinear identity for 2DTL hierarchy with squared eigenfunction symmetry.By linearly combining the auxiliary flow and an arbitrary flow of 2DTL hierarchy,we obtained the bilinear identity for extended2 DTL hierarchy.By giving the definition of its ?-function,we furthermore obtained its Hirota bilinear form.From which,two types of Toda lattice equation with self-consistent sources were constructed.The first type is previously known,while the second type seems to be a new result.The second part of this thesis constructed the one-and multi-soliton solutions for reverse-space,reverse-time and reverse-space-time NLS equation.By applying reduction conditions to AKNS system,we obtained the elementary Darboux transformation for the three types of nonlocal NLS equations,and therefore obtaind the explicit form of fundamental Darboux transformations for them by mathematical induction.Using this Darboux transformation,we constructed the one-and N-soliton solutions for the nonlocal NLS equations.Furthermore,we analyzed the behaviors of nonzero boundary-value soliton solutions of nonlocal NLS equations under various values of the parameters.