Soliton molecules, asymmetric solitons and interactions with T-breathers/M-lumps of the (3+1)-dimensional KDKK equation

Abstract

The (3+1)-dimensional Konopelchenko–Dubrovsky–Kaup–Kupershmidt (KDKK) equation is an important integrable model, which has been widely used in fluid mechanics, plasma physics and ocean dynamics. In this paper, first we obtain soliton molecules, asymmetric solitons of the KDKK equation via using a velocity resonant principle. Next, we construct two types of interaction solutions via different composite methods: one is the interaction solution between a soliton molecule and T-breathers by using the velocity resonant principle and complex conjugate constraints. The other is the interaction solution between a soliton molecule and M-lumps. When $M=1$, the interaction solution is obtained by using the velocity resonant principle and long wave limit. When $M\ge 2$, the interaction solution is obtained by using a new composite method of the velocity resonant principle and partial degeneration of breathers. Dynamical behaviors of the solutions are discussed theoretically and numerically. The method in the paper is very effective that can be employed to construct soliton molecules, asymmetric solitons and interaction solutions of other nonlinear differential equations. The results obtained may be helpful in studying the propagation phenomena of nonlinear localized waves.