Applications Of Constructive Methods In Several Nonlinear Problems

The present Ph.D. dissertation is concerned with the applications of constructive methods in some nonlinear problems. We mill use both the method of integrable systems and the method of reduction with symmetries. These methods have played important roles in a large number of problems arose from physical mathematics and differential geometry.The structure of this paper is as follows.Chapter I: Preface. This chapter is devoted to the exposition of the development and the reseach situation of these problems which will be discussed in this paper. The main results of this PhD. dissertation are briefly introduced in this chapter.Chapter : We construct Darboux transformation for reduced Maxwell-Bloch equations in this chapter. Darboux transformation will be constructed from axbitary solution of the above equations. Moreover, this construction could be realized in an algebraic and inductive way. Hence multi-soliton solutions could be obtained. This genrelized the work in[l].Chapter : Affine spheres via Loop group method. In this capter we construct affine spheres by using Loop group methods. First we prove an Iwasawa type decomposition theorem for the Loop group, and then use it to obtain the Weierstrass type representation for definite affine spheres. We consider finite type construction in terms of Weierstrass deta. The indefinite case win also be disscussed.Chapter : We construct Hamiltoniaa minimal submanifolds with SO(n)- symmetry in complex hyperbolic space Bn. We prove that these examples are complete. Moreover, these examples are not minimal submanifblds in the commom sense when the parameter...