Exact Solutions Of Some Variable-coefficient Hierarchies Of Nonlinear Evolution Equations

Based on the introduction of the origin and the research status of soliton, thisdissertation aim to construct the exact solutions of nonlinear evolution equationshierarchy which is variable-coefficient, extended the application of Hirota’s bilinearmethod to variable-coefficient KdV hierarchy, MKdV hierarchy and AKNS hierarchy.As a result, the bilinear forms of these hierarchies are obtained, then multiple solitonsolutions of these hierarchies are finally given. The concrete content of this dissertationcan be divided into four parts as followed:The first part mainly introduces the origin and the development status of solitontheory, and some methods for solving nonlinear evolution equations are introduced,including the Hirota’s bilinear method, Wronskian determinant, inverse scatteringmethod, Backlund transformation, well as the Exp-function method.The second part summarizes the basic theory of Hirota’s bilinear method and itssome properties, key point is the introduction of the transformations of the bilinearmethod commonly used logarithmic transformation, double logarithmic transformation,rational transformation.The third part gives the new formal KdV hierarchy, MKdV hierarchy and AKNShierarchy by introducing some coefficient functions. Then the bilinear forms of thevariable-coefficient KdV hierarchy, MKdV hierarchy and AKNS hierarchy are derivedby the use of Hirota’s bilinear method.With the aid of the bilinear forms of the variable-coefficient KdV hierarchy, MKdVhierarchy and AKNS hierarchy derived in third part, the forth part obtains multisolitonsolutions of the variable-coefficient KdV hierarchy, MKdV hierarchy and AKNShierarchy by using the perturbation and truncation technique.