No abstract

In this paper, we investigate the (2 + 1)-dimensional Hirota–Satsuma–Ito (HSI) shallow water wave model. By introducing a small perturbation parameter ϵ, an extended (2 + 1)-dimensional HSI equation is derived. Further, based on the Hirota bilinear form and the Hermitian quadratic form, we construct the rational localized wave solution and discuss its dynamical properties. It is shown that the oblique and skew characteristics of rational localized wave motion depend closely on the translation parameter ϵ. Finally, we discuss two different interactions between a rational localized wave and a line soliton through theoretic analysis and numerical simulation: one is an absorb-emit interaction, and the other one is an emit-absorb interaction. The results show that the delay effect between the encountering and parting time of two localized waves leads to two different kinds of interactions.

In this paper, we mainly investigate the high-order localized waves in the (2+1)-dimensional Ito equation. By introducing a translation parameter and employing the Hirota derivative operator, we construct and analyze three kinds of high-order localized waves with a translation parameter: high-order line soliton, lump-type localized wave and their hybrid solutions. The obtained results show that nonlinear localized waves with a translation parameter have more plentiful dynamical behaviors. It is shown that the plus and minus resonance phenomena of two line solitons can be controlled by a translation parameter. The direction of propagation and symmetry characteristics of lump-type localized wave can be also governed by this translation parameter. Through analyzing the time delay effect we finally discuss and demonstrate the absorb-emit and emit-absorb interactions between a line soliton and a lump-type localized wave.