Applications Of Squared Eigenfunction Symmetries

We mainly,in this dissertation,investigate application of eigenfunctions in integrable systems.Squared eigenfunctions can not only explore the relationship between different integrable systems,but also act as additional symmetries to obtain new integrable system,e.g.integrable system with self-consistent sources(SCS).The dissertation is organized as the following:First,we investigate application of eigenfunctions in fully discrete integrable systems.After recalling those 4-dimensionally consistent octahedral equations,we give squared eigenfunction symmetries of the lattice potential KP(lp KP)equation.Then we develop the Cauchy matrix approach to derive octahedral equations combined with squared eigenfunc-tions,which can be considered as octahedral equations with SCS.Next,applications of eigenfunctions in semi-discrete KP systems are investigated.We recall the D~2?KP system(with two continuous and one discrete variables)and its squared eigenfunction symmetry(also known as additional symmetry).Then,we extend the Cauchy matrix approach to obtain the D~2?KP system with SCS.Solutions of the D~2?KP system with SCS are also obtained.We also derive the D?~2KP system from two quasi-differential operators with difference information and obtain its squared eigenfunction symmetry.In ad-dition,by respectively introducing continuous and discrete arbitrary functions in dispersion relation,we derive two types of D?~2KP systems with SCS.Finally,we investigate connections between some classical physical models and inte-grable systems with SCS.Firstly,aftering recall the derivation of several Maxwell-Bloch type equations,we present their connection with the reduced AKNS+SCS system.Then,utilizing the Cauchy matrix approach,we obtain solutions of the AKNS+SCS system.New dynamics of the M-B equations are analyzed.Next,by means of the Cauchy matrix ap-proach,we derive a lower-order Kadomtsev–Petviashvili(KP)equation with SCS,and then we reduce it to the multi-component Yajima–Oikawa(Y-O)system.New solutions of the Y-O system are obtained,in which there is more freedom in dispersion relations and am-plitudes.In addition,relations of several continuous integrable systems with self-consistent sources are given by reduction.These results not only enable us to obtain multiple pole solutions of integrable systems with SCS,but also help us understand applications of eigenfunctions in both continuous and discrete integrable systems.