| In the past decades, nonlinear evolution equations (NLEEs) have been widely used to describe complex phenomena in various sciences, such as fluid dynamics, plasma physics, field theory and condensed matter physics. Currently, more and more attention has been paid to the higher-dimensional NLEEs. On the other hand, a vast variety of methods have been developed including scattering method, Painleve test, Hirota bilinear method and so on. Among them, Hirota bilinear method has been providing a powerful and effective tool for finding exact solutions of various NLEEs. Based on the theory of Hirota bilinear method, this paper investigates the periodic wave solutions of the (2+1)-dimensional Boussinesq equation.The structure of this dissertation is organized as follows:Chapter one is devoted to summarize some properties of soliton solutions, including the history and development of the soliton theory and several corresponding research methods. In chapter two, Hirota bilinear method is generalized and investigated. The properties of the Hirota bilinear operator is introduced firstly. Then, we list three transformations, which are used to transform NLEEs into the corresponding bilinear forms. Finally, we generalized the methods about how to derive the above transformations.Chapter three is about the bilinear Backlund transformation and the bilinear nonlinear superposition formulae.Chapter four is mainly focused on the construction of explicit solutions for NLEEs on the basis of bilinear formalism. First, we summarize two methods of obtaining soliton solutions of the (2+1)-dimensional Boussinesq equation. Then, the one- and two-periodic wave solutions for the (2+1)-dimensional Boussinesq equation are obtained by the Hirota bilinear method in terms of Riemann theta functions. Moreover, it is shown that the above periodic wave solutions can be reduced to the corresponding soliton solutions under certain limits. |
