Several Constructive Techniques In Solving Soliton Equations

This dissertation has mainly done the following three aspects research: First, by means ofsymbolic-numeric computation software, the direct constructive method is introduced to findexact solutions for some nonlinear evolution equations. And the theory of AC= BD is appliedto explain the essence of some well-known classical constructive methods. Next, the bilinearBacldund transformation, the Wronskian technique and the Grammian technique are introducedto find exact solutions for some non-isospectral soliton equations. Finally, the Pfaffianizationprocedure is applied to the non-isospectral KP equation and the non-isospectral mKP equation,and two new integrable systems with Pfaffian solutions are generated.The structure of this paper is as follows:Chapter 1 is concerned with the exposition of the development and the research situationof several subjects which will be discussed, in this paper. The main results of this dissertationare briefly introduced in this chapter.Chapter 2 is devoted to AC=BD theory and its applications in soliton equations. First,based on the idea of AC=BD, the direct constructive method is introduced to find exactsolutions for some soliton equations. This method covers the following conditions: with severalobjective equations; with one objective equations; or with none objective equations. Then, thetheory of AC=BD is applied to explain the essence of some well-known classical constructivemethods, such as the Darboux transformation, the Wronskian technique, and the Lax pairtheory.In Chapter 3, the applications of the Hirota bilinear method and the bilinear Backlundtransformation are briefly introduced. Based on the bilinear Backlund transformation withspectral parameters, the higher-order positons, negatons and complexiton solutions of the KdVequation with loss and non-uniformity terms are generated by chosing appropriate spectralparameters.Chapter 4 is mainly focused on the applications of the Wronskian technique in solitonequations. It is shown that by virtue of some determinantal identities, the Wronskian solutionsare verified by direct substitution into the bilinear form of the soliton equations. In addition, bygeneralizing the equations satisfied by Wronskian entry vectors (which are called the Wronskiancondition equations), the generalized Wronskian solutions of the non-isospectral MKdV equa-tion are obtained, which include the multi-soliton solutions, the positon solutions, the negaton solutions and the complexiton solutions. In the end of this Chapter, the Grammian techniqueis briefly introduced, and the Grammian solutions of the non-isospectral mKP equation areobtained.Chapter 5 first introduces the definition and some important properties of Pfaffian. Thenby applying the Pfaffianization procedure to the non-isospectral KP equation and the non-isospectral mKP equation, two new integrable systems with Pfaffian solutions are generated.