Painlevé properties and exact solutions for the high-dimensional Schwartz Boussinesq equation

The usual (1+1)-dimensional Schwartz Boussinesq equation is extended to the (1+1)-dimensional space--time symmetric form and the general (n+1)-dimensional space--time symmetric form. These extensions are Painlevé integrable in the sense that they possess the Painlevé property. The single soliton solutions and the periodic travelling wave solutions for arbitrary dimensional space--time symmetric form are obtained by the Painlevé—Bäcklund transformation.