Similarity transformations and exact solutions for a family of higher-dimensional generalized Boussinesq equations

In this Letter, we investigate explicitly exact solutions of the higher-dimensional generalized Boussinesq equation. We firstly reduce this equation to one nonlinear ODE and a set of two nonlinear homogeneous PDEs via semi-traveling wave similarity transformation. And then we study solutions of the obtained nonlinear ODE and the set of two nonlinear homogeneous PDEs, respectively. Finally, we can obtain many types of exact solutions of higher-dimensional generalized Boussinesq equation via the semi-traveling wave similarity transformations. These solutions contain an arbitrary function which leads to abundant structures.