| In the last few decades,non-linear science has a great development as a new subject.The non-linear factors can be met in all natural sciences and even in social science,and the non-linear science mainly investigates non-linear interaction between a variety of factors and scale changes as well as the resulted complex phenomena. The non-linear science has become an important symbol of modern science,whose related studies have turned to be the basic research of various branches of natural science.Solitons as an important branch of the non-linear science,have been well studied and widely applied in fluid mechanics,biology,mathematics,plasma,optics, communications and other fields of natural science,which is of important physics significance.It has become one of the most important topics to find the exact solutions which are helpful for better understanding of the physical systems described by nonlinear partial differential equations(NPDEs).Various powerful methods have been proposed,such as the inverse scattering method,the B(a|¨)cklund transformation method,the Darboux transformation method,the Hirota direct method and the Wronskian method.The Painlev(?) analysis method has been proved to be one of the most successful and widely applied tools in studying the integrability of NPDEs, from which further study can be continued,such as the Hirota bilinear form, associated B(a|¨)cklund transformation,Lax pair and other integrable properties. Based on the theory of NPDEs,this paper investigates some integrable properties and several kinds of significant methods for WBK equation,perturbed variable-coefficient KdV equation,(2+1)-dimensional variable-coefficient KdV equation and(3+1)-dimensional variable-coefficient KP equation using symbolic computation.The structure of the present paper is organized as follows:In chapter 1,we first introduce the history and development of the solitons,and then by means of several examples we explain three methods—traveling wave method,general form of B(a|¨)cklund transformation method and non-linear superposition method.Chapter 2 gives a review of the well-known Painlev(?) analysis.In the 1980s, Weiss,Tabor and Carnevale advanced Painlev(?) test for partial differential equations (PDEs) via the generalizing Painlev(?) test for ordinary differential equations(ODEs), which provides a necessary condition to judge the integrability of a given equation. We first introduce the related concepts,ideas and steps.Then,for illustration,we apply this technique to the WBK equation and(2+1)-dimensional variable-coefficient KdV equation.In chapter 3,we study the Hirota direct method and bilinear B(a|¨)cklund transformation method.The Hirota direct method is an important tool to deal with NPDEs and soliton problems,and it can be used to effectively construct the N-soliton solution in the form of an N-th-order polynomial in N exponentials for a large class of NPDEs.The main advantage of the bilinear B(a|¨)cklund transformation method is which establishs a relationship between a seed solution and new one,theoretically, we can obtain a rich family of new solutions using iteration.In this chapter,we introduce the D-operator,the special properties of the D-operator,the dependent variable transformation which can transform a NLEE into its bilinear form and the applications of truncated Painlev(?) expansion.To illustrate,we construct the bilinear form,soliton solutions and bilinear B(a|¨)cklund transformation for the perturbed variable-coefficient KdV equation,WBK equation and(2+1)-dimensional variable-coefficient KdV equation.In chapter 4,we study the Wronskian technique.The Wronskian technique provides a simple and straightforward way of verifying the validity of the exact solutions by virtue of properties of the Wronskian determinant.For the perturbed variable-coefficient KdV equation,(2+1)-dimensional variable-coefficient KdV equation and(3+1)-dimensional variable-coefficient KP equation,the exact solutions in the Wronskian form are constructed and proved by direct substitution into the bilinear equations.
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