The Wronskian Representation Of Exact Solutions For Soliton Equations

Searching for exact solutions is one of the important topics in the study of nonlin-ear evolution equations. Among the existing solving methods, Wronskian technique, together with the Hirota bilinear method, is considered as one of the most efficient and direct approaches in deriving soliton solutions for nonlinear evolution equations possessing bilinear forms.In1971, Hirota first proposed the formal perturbation tech-nique, named the Hirota method, to obtain N-soliton solution of the KdV equation. At the same time a more compact form of N-soliton solution was presented through Wronskian technique. Satsuma gave the Wronskian representation of the multisoli-ton solution to the KdV equation.Then the Wronskian technique was developed by Freeman and Nimmo.A meaningful generalization came from Siriaunpiboon and co-workers and more solutions, such as positons, negatons, rational solutions and mixed solutions, were expressed in terms of Wronskian. Recently, Ma and You considered the coefficient matrix in the Wronskian condition equations to be its canonical form, i.e.,a diagonal or Jordan form and gave a systematic analysis on how to solving these Wronskian condition equations. All these works had promoted the development of Wronskian technique.Based on the existing Wronskian technique, this essay states its further application and new development in Soliton equation.The third chapter uses Wronskian technique to construct the most general Wron-skian conditions for the (3+1)-dimensional Jimbo-Miwa equation. As a result, the gen- eralized Wronskian solutions including rational solutions, positons, negatons, solitons and interaction solutions are obtained. Moreover, some new solutions, together with some known solutions given by other solving methods, are derived from these Wron-skian solutions.Chapter four is about the improved Wronskian technique and its application for solving equations. A group of identities, named Wronskian identities of bilinear KP hi-erarchy here, are discovered as well as two useful properties of D-operator. This makes it possible to easily search for some new Wronskian solutions for PDE which owns bilinear form, and to simplify the process of the proof. As an application, We propose the Wronskian condition for both the first and the second equations of BKP hierarchy for the first time by using this new method and obtain the Wronskian representation of its Soliton solutions,which has denied the statement that the Soliton solutions to BKP-type equations are expressed in terms of Pfaffian rather than Wronskian determinant. In addition, we also applied this method to the high-dimensional equations.As an ex-ample, we discuss two existed generalized BKP equations and obtain their Wronskian conditions.Chapter five gives the conclusion and expectation about this research.