Darboux Transformations,Backlund Transformations And Superposition Formula For Three Coupled Systems

In this dissertation,we study several coupled integrable systems.Firstly,we consider the Drinfel'd-Sokolov-Wilson(DSW)equation.DSW equation is an integrable system,and we construct a proper Darboux transformation for it.Rather than considering the six-order Lax operator,we work with a different Lax operator which is a third-order differential operator with a pseudo-differential tail.The corresponding zero-curvature representation or spectral problem is of 4 x 4 matrix type.With the aid of this Lax operator,we calculate the associated Darboux transformation and Backlund transformation.We show that a nonlinear superposition formula may be constructed and demonstrate its usefulness by constructing some solutions for the DSW system.Secondly,we consider the Ito equation.Ito equation is also an integrable system.The corresponding Darboux transformation of Ito equation is presented explicitly.The resulted Backlund transformation is shown to be equivalent to the one found by Hirota.Also,a nonlin-ear superposition formula,which is of differential-algebraic,is obtained.As applications,the results will be used to calculate soliton solutions of the Ito equation.Lastly,we consider the supersymmetric Two-Boson equation.Supersymmetric Two-Boson equation is a supersymmetric integrable system.We use the gauge transformation to the supersymmetric system.We construct two element Darboux transformations and a binary Darboux transformation for this system.And the corresponding nonlinear Backlund trans-formations are constructed.Then,we explore the relationship between our Darboux trans-formations of supersymmetric Two-Boson equation and Darboux transformations of classical Two-Boson equation which are obtained by Li et al.We also give a nonlinear superposition formula of this system.