Variant wave propagation patterns by coupled Bossinesq equations

In this article, the abundant wave solutions of Boussinesq equations are obtained, which are very important to investigate the waves propagation of shallow water. Firstly, we establish the time-fractional form of (1 + 1) dimensional Boussinesq equations to describe the shallow water waves more accurately. Furthermore, by applying the complete discrimination system for polynomial method, all the possible wave solutions are revealed. Finally, in order to analysis the fractionalize effects efficaciously, the numerical simulations for different type of solutions is carried out and the corresponding graphs are obtained. The whole of the solutions presented in this paper consist of all the results in the published literatures, especially the elliptic functions solutions, which are initially shown. In addition, according to the results obtained by advanced fractional method in mathematical physics, the nature of rich wave propagation patterns could be seen clearly.