Researches On The Integrable Systems Of The Soliton Equations And The Exact Solutions Of The Nonlinear Equations

This paper introduce with emphisis the integrable system of the solitonequations include: the formulation and integrability of the nonlinear evolutionhierarchy as well as their expanding integrable systems and the exact solutionsof the nonlinear equations. In the first chapter, historical origin and someresearches of soliton theory together with its research meaning are presented.In the second chapter, firstly, a loop algebra (?)_M with 3M dimensions isconstructed by using of some properties of exterior algebra, which is devotedto establishing many isospectral problems. As its application, a multi-componentsystem BPT hierarchy is obtained. Secondly, a type of (2+1)-dimensionalmulti-component integrable hierarchy is obtained with the help of a(2+1)-dimensional zero-curvature equation and Tu scheme. As an applicationexample, we obtain a generalized (2+1)-dimensional KN hierarchy; at last In termsof the Lie algebra with 3-dimensional, we establishanew loop algebra (?)_M, thenthe extended integrable model of the muti-component KN hierarchy is presentedby use of the generalized isospectral problem and Tu sheme which reduces to ageneralized Burgers equation and generalized coupled kdv equation. In the thirdchapter, firstly, new expanding loop algebras of the loop algebras presented inthe equation systems (2.3.16) of second chapter are constructed, then acorresponding expanding integrable models are engendered by employing properisospectral problems and Tu sheme. Secondly, a 5-dimensional Lie algebra isestablished with the help of the linear combination of a subalgebra, in termsof the Lie algebra, we establish a new loop algebra, then the extended integrablemodel of the muti-component hierarchy (2.4.7) is presented by use of thegeneralized isospectral problem and Tu sheme. The simple approach to generateexpanding integrable models can be used generally. In the fourth chapter, firstintroduced simply the methods of the exact solutions of nonlinear equations, thenintroduced with empnasis the nomogeneous balance principle, finally, thkes the application, has given the exact traveling solutions of the Fisher equation.