Application Of Bilinear Methods To Several Exact Solutions Of Two Classes Of Partial Differential Equations

The Hirota bilinear method is one of the most important issues in the soliton theory of nonlinear partial differential equations and mathematical physics. The paper discussed the power of the Hirota bilinear method and its generalization to establish exact solutions for nonlinear wave equations and nonlinear heat equation. We pointed out a class of nonlinear heat equations which have no Hirota bilinear counterparts. Then, from the generalized bilinear method, we developed a new technology to obtain multi-wave solutions for these equations. Our results illustrate that the generalized bilinear method is an essential extension to the Hirota bilinear method. Moreover, the proposed technology can help us to discover more nontrivial equation.The main work of this dissertation is as follows:First, by using the Hirota bilinear method and appropriate exchange formulas, we obtained the bilinear B(?)ckluand transform for a class of seventh-order KdV equations. Further, by using gauge transformations and symbolic computation technology, we got the other three bilinear B(?)cklund transformations. By applying these transformations and the truncation method, we can get some new N-soliton solutions of the original equation.Secondly, by using the Hirota bilinear method and Wronskian technique, this paper presented a sufficient condition which provides the nonlinear wave equation with a double Wronskian solution. Then, by classifying eigenvalues of a matrix, we established a series of soliton solutions, rational solutions, Matveev solutions and complexiton solutions for the corresponding equations. The above solutions present the diversity of solutions for the studied nonlinear equations. Classification methods and ideas used herein can help researchers to establish more double Wronskian solution of nonlinear partial differential equations.Third, based on the Bell polynomial theory and an auxiliary argument, we established a suitable transformation and obtained Hirota bilinear forms for a seventh-order KdV equation, and calculated their multi-soliton solutions. As we all know, it is easy for the low-order nonlinear partial differential equations to establish corresponding Hirota bilinear form. Because of nonlinear terms in the high-order nonlinear partial differential equations, it is often complex to create a corresponding Hirota bilinear form for them. This method shows that the Bell polynomial theory can be used to construct Hirota bilinear forms for high-order equations. We further used the Riemann 0 function to establish periodic wave solutions for some of the preceding equations.Fourthly, we studied a class of heat conduction equations. We used above methods and read literature to find that:These equations have not been transformed into any Hirota bilinear forms. However, such a heat equation is able to have a generalized bilinear form. To calculate the corresponding multi-wave solutions of these equations, this paper developed a new technology, and established new multi-wave solutions for the above equations.