Resonant interactions of the rational waves and breathers in the third-type of Davey–Stewartson equation for shallow water

In this work, we investigate the resonant interactions of the rational waves and breathers in the third-type of Davey–Stewartson equation. The structures of the nonlinear waves generated by resonant interactions are characterized by a general semi-rational solution, constructed using the Kadomtsev–Petviashvili hierarchy reduction method combined with Hirota’s bilinear method. The complete resonant interactions studied involve the propagation of rational waves in the background of parallel breathers. It is worth noting that rational waves only appear for a short period of time and are localized in the middle of parallel breathers. As time approaches infinity, only parallel breathers exist. Furthermore, depending on the constraints imposed on the phase parameters $p_{\alpha }^{\left[i\right]}$, the propagating rational waves exhibit two distinct types of localized structures: lump-type rogue waves and line-segment rogue waves.