High Order Lax Integrable Supersymmetric Systems:Darboux Transformations And Applications

This thesis concerns with supersymmetric integrable systems.The N = 2 ?=-2 supersymmetric KdV equation and the supersymmetric Sawada-Kotera equation,whose Lax matrices are of 4 × 4 and 6×6 respectively,are studied in particular.For those two system-s and some other integrable systems,the corresponding Darboux transformations,Backlund transformations,nonlinear superposition formulas,super-soliton solutions are constructed.In addition,applications of Backlund transformations and discretizations of supersymmetric inte-grable systems are considered.The main results of this thesis are as follows.1.We construct a Darboux transformation and the related Backlund transformation for a N=1 supersymmetric coupled dispersionless integrable system.The associated nonlinear superposition formula is also worked out.By means of Darboux transformation and nonlinear superposition formula,some soliton solutions are obtained for the N = 1 supersymmetric coupled dispersionless integrable system.2.The N = 2?=-2 supersymmetric KdV equation is studied.A Darboux transforma-tion and the corresponding Backlund transformation are constructed for this equation.Also,a nonlinear superposition formula is worked out for the associated Backlund transformation.The Backlund transformation and the related nonlinear superposition formula are used to construct integrable super semi-discrete and full discrete systems.The continuum limits of these discrete systems are also considered.3.A Darboux transformation and the related Backlund transformation for the supersym-metric Sawada-Kotera(SSK)equation are constructed.The associated nonlinear superposition formula is also found.We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation.By the Darboux transformation and nonlinear superposition formula,we construct some soliton solutions.Also,we also present two semi-discrete systems and show that the continuum limit of them go to the SKK equation.4.The study of new integrable defects leads to new type of Backlund transformations named as the type-II Backlund transformations.We show,for the MKdV hierarchy,the type-II Backlund transformation is the compound type-I Backlund transformation.We obtain a Backlund transformation for Tzitzeica equation from an old Tzitzeica's Darboux transforma-tion by eliminating eigenfunction.We study the relation between this BT and all other known Tzitzeica equation's BTs,some of which are called new and type--?" BT.We point out that type-? Backlund transformation of Tzitzeica equation does not exist.