Breather, soliton, multiple-pole, and interaction solutions to the Hirota–Satsuma equation

In this paper, we investigate serval types of localized waves of the Hirota–Satsuma equation by using the Hirota method. By means of two identities, the bilinear form of the Hirota–Satsuma equation is proposed. Then, the one-, two-, and three-soliton solutions are given explicitly and analyzed. The one-soliton solution could present the type of soliton or antisoliton, which depends on the sign of the wave number. The soliton–soliton, soliton–antisoliton, and antisoliton–antisoliton interactions are analyzed through graphics in the two-soliton solution case. The interactions among three solitons/antisolitons are considered in two cases. On basis of the soliton solutions, we give the conditions of obtaining breather solutions and interaction solutions between breathers and solitons. Multiple-pole solutions are derived by taking the limit of wave numbers and appropriate phase parameters. The dynamics of double- and triple-pole solutions are analyzed.