A transformed rational function method and exact solutions to the 3 þ 1 dimensional Jimbo–Miwa equation

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanhfunction type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 þ 1 dimensional Jimbo–Miwa equation is treated, together with a Bäcklund transformation.