Bosonization Of Supersymmetric KdV Equation And Studies Of Its Integrable Properties

A new field of supersymmetric integrable systems started with the N=1 supersymmetric extensions of the Korteweg-de Vries equation and the discussions of them twenty years ago. Now, they hold an important position in many study fields with a young figure and great influence. The far-reaching significance lies not only in mathematics, but also in applications in various areas of modern physics. Therefore, the supersymmetric integrable systems are attracting grow-ing attention. And investigating their properties and searching for their exact solutions are of great importance. However, it is difficult to study any supersym-metric integrable system including the supersymmetric KdV equation due to the restriction fermionic fields. In this dissertation, taking the N= 1 supersymmet-ric KdV equation as an object of study we propose a bosonization approach to effectively avoid difficulties caused by intractable fermionic fields which are an-ticommuting. It greatly enrich our understanding of supersymmetric integrable systems and open up a new effective path of studying in this area.In this dissertation, three topics are discussed. Firstly, the bosonization procedure of the supersymmetric systems is applied to the sKdV equation with N fermionic parameters to get the bosonized equation, and then the bosonized equation is solved by the Lie symmetry method. Secondly, we extend the object of study to the general N=1 supersymmetric KdV equation, and study the exact solutions and integrability of it. Lastly, the bosonization approach of N=1 supersymmetric KdV equation is generalized, then singularity analysis and the exact solutions are concerned.Chapter 1 is an introduction which is devoted to reviewing the mathematical and physical backgrounds of the KdV equation and its supersymmetric extensions in this dissertation. The significance and development of the nonlinear science, including the mathematical tools involved and supersymmetric nonlinear equa-tions, are reviewed, too. The basic knowledge related to the supersymmetry and the bosonization approach are introduced in this chapter, and we also briefly report the main works of this dissertation.In Chapter 2, the bosonization approach is applied to a specific supersym-metric integrable system, say the sKdV equation for the first time in detail. Firstly, the sKdV equation is bosonized with two and three ferniionic parameters and some systems of differential equations are obtained. Then the bosonized sKdV equation is solved by the travelling wave reduction method, and various new exact solutions are obtained and discussed. On this basis, some special types of exact supersymmetric extensions of any solutions of the uaual KdV equation is constructed straightforwardly through the exact solutions of KdV equation and the related symmetries, which include the famous single soliton solution and multiple soliton solution. And finally, the general bosonized equa-tions and the general travelling wave solutions of sKdV equation are constructed with N fermionic parameters. We can conclude that the bosonization approach can be applicable to not only the supersymmetric integrable systems but also allthe models with fermionic fields, such as super integrable systems and pure integrable fermionic systems.In Chapter 3, bosonization approch with two fermionic parameters is applied in solving the general N=1 supersymmetric KdV equation with an arbitrary parameter a (sKdV-a equation). Then the Lie point symmetries of the coupled bosonic equations are considered and several types of similarity reductions and solutions of it are conducted for all values of a. Taking a= 2 for example, a soliton solution of sKdV-2 system is constructed by using an exact solution of KdV equation. At last, a certain type of exact solution of the sKdV-a system is discussed, which is independent of the parameter a. This type of solution can satisfy all possible N=1 supersymmetric extensions of KdV equation. Especially, some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously for a≠3 and a≠0. In other words, any kind of solutions of the usual KdV equation such as the N-soliton solutions, τ-function solutions can be extended to those of the sKdV-a quation for all a.In Chapter 4, the bosonization approach with N fermionic parameters is extended to an important subject. The bosonic fields in the bosonized sKdV (B-sKdV) systems are defined on even Grassmann algebra and are not just restricted to c-number algebra. Then it will enriches the solutions of sKdV equation. With the help of the singularity analysis, the Painleve property of the BsKdV system is proved and a Backlund transformation (BT) is found. The BT related nonlocal symmetry, we call it as residual symmetry, is used to find symmetry reduction solutions of the BsKdV system. Hinted from the symmetry reduction solutions, a more generalized but much simpler method is established to find exact solutions of the BSKdV and then the SKdV systems, which actually can be applied to any fermionic systems. Using the residual symmetry with a free spectrum parameter, the infinitely many nonlocal symmetries are obtained.The last chapter concerns the summary and discussion for the whole disser-tation, and the prospect for the further works are also discussed in this chapter.