Exact Solutions To The Equation With Non-Uniformity Terms

To find exact solutions to soliton equations is an important aspect in soliton theory. It is of important significance in using Wronskian technique, generated Wronskian and double Wronskian to solve a sort of soliton equations and strive to get much more solutions of these equations. This paper mainly study Wronskian technique, generated Wronskian and double Wronskian to some soliton equations, especially the aspect of finding exact solutions. The dissertation include several aspects: Getting the Hirota's N-soliton solutions of the mKdV equation with non-uniformity terms, through using Hirota's bilinear method; Using Wronskian solution and matrix method to derive the matrix solutions of the mKdV equation with non-uniformity terms, which concluding Positons, Negatons and Complexton solutions; Derivation of a sort of other new exact solutions in terms of double Wronskian solution.In this dissertation, we mainly study from solutions with Hirota form, Wronskian and double Wronskian forms, till to get the exact solutions. It is shown that the bilinear method provides a very powerful tool in searching for exact solutions of soliton equations. The specific work consists as follows:In part one, we first introduce the definition and elementary properties of the bilinear operators. Second, we deduce the bilinear forms for the mKdV equation with non-uniformity terms in detail. Then use the Hirota bilinear method to get Hirota N-soliton solution.In part two, we first introduce Wronskian solution of the soliton equa- tions and the matrix solution of the KdV equation, concluding the elementary of Wronskian solution and some exact solutions derived from matrix solution. Then, we deduce the Wronskian solution of the mKdV equation with non-uniformity terms and its provations. At last, we derive the Positons, Negatons and Complexiton solutions with matrix method.The double Wronskian solution of the mKdV equation with non-uniformity terms is mainly studied in part three, the first step is to introduce double Wronskian solution of the soliton equations. The second step is to get another bilinear forms by means of another transformation, and we also derived its double Wronskian solution of the mKdV equation with non-uniformity terms. The last step is using the double Wronskian solution to get a sort of matrix solutions.