A new method, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to nonlinear evolution equation (NEE) is proposed. (1+1)-dimensional Boussinesq equation is used as an example to illustrate the effectiveness of the suggested method. Rational homoclinic wave solution, a new family of two-wave solution, is obtained by inclined periodic homoclinic breather wave solution and is just a rogue wave solution. This result shows that rogue wave originates by the extreme behaviour of homoclinic breather wave in (1+1)-dimensional nonlinear wave fields.

In this paper, a unified formula of a series of rogue wave solutions for the standard (1+1)-dimensional nonlinear Schrödinger equation is obtained through exp-function method. Further, by means of an appropriate transformation and previously obtained solutions, rogue wave solutions of the variable coefficient Schrödinger equation are also obtained. Two free functions of time 𝑡 and several arbitrary parameters are involved to generate a large number of wave structures.