Study On Integrable Properties For Two Kinds Of Variable-coefficient Nonlinear Partial Differential Equations

In almost all fields of natural sciences,nonlinear partial differential equations(NPDEs)can be used to explain a rich variety of practical problems.Therefore,it has become considerably significant to study integrable properties of these equations,including analytical solutions, infinite number of conservation laws and Hamilton structure,which are very useful to understanding various physical phenomena.The number of NPDEs is huge,and more and more equations are springing out in many fields with the development of modern technology,so that it appears more important to study integrable properties of these equations. Meanwhile,soliton solutions are a special kind of solutions for NPDEs and possess certain physical significance when the equations represent some physical processes.Today,soliton,as one significant branch of nonlinear science,has attracted considerable interest both in the theory and experiment.Based on the theory of NPDEs,this paper investigates some integrable properties for variable-coefficient Boussinesq(vcB)equation and variable-coefficient fifth-order Korteweg-de Vries(vcfKdV) equation.The structure of this dissertation is organized as follows:(1)Painlevétest.Through generalizing Painlevétest for ordinary differential equations(ODEs),In 1983,Weiss,Tabor and Carnevale advanced Painlevétest for partial differential equations(PDEs),which provides a necessary condition to judge the integrability of a given equation.This method has been widely used to research the integrablities for NPDEs,including B(?)cklund transformations and Lax pairs.As a "byproduct",the auto-B(?)cklund transformation for PDEs can be obtained through the truncated Painlevéexpansion method.In Chapter 2,by carrying out the Painlevéanalysis for the vcB equation,we will show that this equation possesses Painlevéproperty under certain constraints.At the end of this chapter,the auto-B(?)cklund transformation for this equation is also constructed with the truncated Painlevéexpansion method.(2)In the practical physics and engineering,taking into account the nonuniform boundaries and/or inhomogeneous media,the variable coefficient models can describe various situations more realistically than their constant coefficient counterparts.In Chapter 3,this dissertation introduces an effective method to research variable coefficient models. By the variable transformation relation,the variable coefficient models could be transformed into constant coefficient ones.Then,with the aid of the transformation,many integrable properties of constant coefficient ones are mapped into variable coefficient counterparts.This chapter implements this method on vcB equation and vcfKdV equation and obtains the integrable properties of these equations under certain constraints,respectively.Besides,for the vcB equation,some properties are derived,including the auto-B(?)cklund transformation,nonlinear superposition formula and Lax pairs.(3)The Darboux transformation is useful and powerful for constructing the analytical solutions for integrable NPDEs.After once or several times Darboux transformation,one-soliton or multi-soliton solutions could be gained from a trivial seed solution.The key of this transformation is to construct a gauge transformation to keep the form of Lax pairs invariant.In Chapter 4,on the ground of the theory and results of Chapter 3,the Darboux transformations for vcB equation and vcfKdV equation are obtained.Meanwhile,the Lax pairs,one- and two-soliton solutions for vcfKdV equation are also derived.(4)In the end,with the aid of the auto-B(?)cklund transformation for vcB equation obtained in Chapter 3,new solutions for this equation are gotten by choosing a seed solution.Via Mathematica,Chapter 5 discusses physical interest and possible applications of the solutions for these two equations with the choice of different coefficient functions.