Periodic-wave and semi-rational solutions for the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation

Abstract This paper investigates the periodic-wave and semi-rational solutions in determinant form for the ( 3 + 1 ) -dimensional Yu-Toda-Sasa-Fukuyama equation via the Kadomtsev–Petviashvili hierarchy reduction. By analyzing the periodic-wave solutions’ properties, we obtain a breather on the x – y plane (or y − z plane) and two kinds of periodic waves on the x − z plane. One of the periodic waves is of growing-decaying amplitude, and the other one is of invariant amplitude. The breather always parallels with the x -direction, and its characteristic lines decide the evolution of the breather. By taking the long-wave limit on the periodic-wave solutions, we derive semi-rational solutions, which generate a lump on the x – y plane and a line rogue wave or moving soliton the x − z plane. Based on the relationship between these two solutions, we conclude that (1) the lump is the limit on the breather; (2) the soliton is the limit of the amplitude-invariant periodic wave; (3) the rogue wave is the limit of the growing-decaying periodic wave.

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