Bäcklund transformations and abundant exact explicit solutions of the Sharma–Tasso–Olver equation

By using an extension of the homogeneous balance method and Maple, the Bäcklund transformations for the Sharma–Tasso–Olver equation are derived. The connections between the Sharma–Tasso–Olver equation and some linear partial differential equations are found. With the aid of the transformations given here and the computer program Maple 12, abundant exact explicit special solutions to the Sharma–Tasso–Olver equation are constructed. In addition to all known solutions re-deriving in a systematic way, several entirely new and more general exact explicit solitary wave solutions can also be obtained. These solutions include (a) the algebraic solitary wave solution of rational function, (b) single-soliton solutions, (c) double-soliton solutions, (d) N-soliton solutions, (e) singular traveling solutions, (f) the periodic wave solutions of trigonometric function type, and (g) many non-traveling solutions. By using the Airy’s function and the Bäcklund transformations obtained here, the exact explicit solution of the initial value problem for the STO equation is presented. The variety of the structure of the solutions for the Sharma–Tasso–Olver equation is illustrated.