Exact Solutions Of The Soliton Equations Based On The Bilinear Form

The soliton equations usually describe nonlinear phenomena evolving with time.The research objects of soliton equations arise from many fields,for instance applied physics,life sciences,oceanography,ect.Seeking exact solutions of the soliton equations is always an important topic in soliton theory.The research on exact solutions of the soliton equations is not only helpful to understand the essential property and algebraic structure of the equations,but also effectively to explain some natural phenomena properly.In this thesis,based on the Hirota's bilinear method and symbolic computation,various types of exact solutions are derived for soliton equations,which include rational solutions,lump solutions,semi-rational solutions,ect.In addition,properties and figures of the obtained solutions are presented vividly with Mathematica.The thesis is organized as follows: In chapter 1,we briefly introduce the origin and development of soliton theory,the recent research of exact solutions,the structural arrangement and innovation of this thesis.In chapter 2,we mainly introduce the definition of bilinear differential operator,the Pfaff formula and the relevant theorems.In chapter 3,on the basis of the bilinear form,lump solutions and lump-kink solutions of Jimbo-Miwa(JM)equation are obtained by assuming the test function in the form of addition of a quadratic function and an exponential function.Lump-kink solutions reflect the fission and fusion between a lump and kink-type wave,which are nonelastic.In addition,the obtained solutions and their asymptotic behaviors are also analyzed.In chapter 4,by using of the bilinear form and the test function in the form of admixture operation of addition and multiplication of quadratic function,exponential function and trigonometric function,lump solutions and semi-rational solutions of the(2+1)-dimensional Bogoyavlenskii's breaking soliton equation are derived.Lump solutions include 1-lump and 2-lump solution.Semi-rational solutions are composed of lump-bell solutions and rational-sin solutions.Lump-bell solutions reflect the interaction of a lump with a bell-type wave soliton.Rational-sin solutions reflect the interaction of a lump with sinusoidal function.In chapter 5,based on the bilinear form of the potential YTSF equation,on the one hand,the relevant theorems and proofs are given and rational solutions of the potential YTSF equation are derived.And then,the fundamental lump solution,thehigh-order lump solution,the multi-lump solution and their dynamics are studied for some special cases.On the other hand,lump-kink solutions of the potential YTSF equation are obtained via the novel test function.Lump-kink solutions describe the catch-up,head-on and relative static interactions of a lump and kink-type wave.We note that these three interactions are elastic.Summary and discussions are given in chapter 6.