Constructing periodic wave solutions of nonlinear equations by Hirota bilinear method

The investigation of the exact solutions of nonlinear equations plays an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, plasma and elastic media are often modelled by the bell shaped sech solutions and the kink shaped tanh traveling wave solutions. The exact solution, if available, of those nonlinear equations facilitates the verification of numerical solvers and aids in the stability analysis of solutions. In the past decades, there has been significant progression in the development of these methods such as inverse scattering method [1,2], Darboux transformation [3-6], Hirota bilinear method [7-12], algebro-geometric method [13-18] and others. Among them, the algebro-geometric method is an analogue of inverse scattering transformation, which was first developed by Matveev, Its, Novikov, and Dubrovin et al. The method can derive an important class of exact solutions, which is called quasi-periodic or algebro-geometric solution, to many soliton equations such as KdV equation, sin-Gordon equation, and Schrodinger equation. In recent years, such solutions of nonlinear equations have been aroused much interest in the mathematical physics [19-24]. However, this method usually is