Eur. Phys. J. Plus (2020) 135: 36
https://doi.org/10.1140/epjp/s13360-019-00008-z

Regular Article

Semi-rational and periodic wave solutions for the (3+1)-dimensional Jimbo–Miwa equation

¹ Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100864, China
² Department of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, China

^* e-mail: xiangluo@mail.ustc.edu.cn

Received: 9 April 2019
Accepted: 19 September 2019
Published online: 10 January 2020

Abstract

In this paper, higher-order periodic wave solution in determinant form is investigated via the Kadomtsev–Petviashvili hierarchy reduction for the -dimensional Jimbo–Miwa equation. We obtain the breather and periodic wave via such solution. The breather is periodic in space and localized in time, while amplitude of the periodic wave changes with the value of y. Besides, interaction between the breather and periodic wave shows a different characteristic in which amplitude of the periodic wave changes after the interaction. By taking the long wave limit to the periodic wave solution, we obtain the semi-rational solution, and then derive the rogue wave and kink-like soliton. Furthermore, we conclude that (1) rogue wave can be obtained with the limit on the breather; (2) kink-like soliton can be obtained with the limit on the periodic wave. Via the semi-rational solution, interactions between the rogue wave and breather, between the kink-like soliton and breather, are illustrated.