The "Trigram" Structures And "Trigram" Identities Of The Soliton Equations

Searching for the exact solutions of soliton equations is an important content in the field of nonlinear partial differential equations. Now, there are many methods for obtaining soliton so-lutions, then what is the fundamental structure common to all soliton equations? The viewpoint of Professor Ryogo Hirota is that soliton solutions can be expressed by the Pfaffians and soliton equations (or the bilinear equations) are just equivalent to the "Pfaffian identities". If one expresses the N-soliton solution as a Gram determinant, the bilinear equations arising from the soliton equations become Jacobi identities of determinants. Professor Mikio Sato pointed out that if one expresses the N-soliton solution as a Wronskian determinant, the bilinear equations arising from the soliton equations are equivalent to Plucker relations. We can not only see the structures of the Pfaffian identities clearly, but also complete the proof of the N-soliton solution easily by the Maya diagrams. Based on the above theory and the idea that soliton equations admit the unified structures, we try to explain the structures of the soliton equations by the theory of "Trigrams" in 《The Book of Changes》(YiJing) and the corresponding Maya dia-grams. We obtain a series of integrable equations by constructing the expressions of differential operators for the "Trigrams" and substituting them into "Trigram" identities.This thesis may be divided into five chapters as follows.In chapter1. we review the history and introduce several soliton methods, respectively, of soliton theory briefly such as inverse scattering transform、Hirota’s bilinear method、Wronskian technique and Pfaffian technique.In chapter2, we introduce the preliminary knowledge, including the pseudo-differential operator, Maya diagrams, Pfaffians, Plucker relations and the Jacobi identities of determinants.In chapter3, the main content and idea is AC=BD model and its application. We generalize the model by introducing the pseudo-differential operator. Based on the generalized model, the famous Sato equation is obtained by solving the generalized Lax equation, which proved that the solutions of Dv=0can be expressed by the T function, then the application of AC=BD model is generalized.In chapter4, we try to explain the unified structure of soliton equations by the " Trigrams " theory in 《The Book of Changes》. The main idea, based on the viewpoints of professor Mikio Sato and Ryogo Hirota that soliton equations are equivalent to the Pfaffian identities and the Pfaffian identities can be expressed by the Maya diagrams, is to reduce the Maya diagrams to the corresponding "Trigrams", from the construction of the solutions, we obtain a scries of integrable equations by constructing the expressions of differential operators for the "Trigrams" and substituting them into "Trigram" identities. In chapter5, we obtain the N-soliton solution and the Wronskian solution of a (2+1)-dimensional KdV equation by using Hirota’s bilinear method. At the same time, the corre-sponding pictures arc given.Finally, a short summery of the thesis is given, moreover, some topics of the thesis will be further investigated are also illustrated at the end.