Dynamics of soliton interaction solutions of the Davey-Stewartson I equation.

Abstract

In this paper, we first modify the binary Darboux transformation to derive three types of soliton interaction solutions of the Davey-Stewartson I equation, namely the higher-order lumps, the localized rogue wave on a solitonic background, and the line rogue wave on a solitonic background. The uniform expressions of these solutions contain an arbitrary complex constant, which plays a key role in obtaining diverse interaction scenarios. The second-order dark-lump solution contains two hollows that undergo anomalous scattering after a head-on collision, and the minimum values of the two hollows evolve in time and reach the same asymptotic constant value 0 as t→±∞. The localized rogue wave on a solitonic background describes the occurrence of a waveform from the solitonic background, quickly evolving to a doubly localized wave, and finally retreating to the solitonic background. The line rogue wave on the solitonic background does not create an extreme wave at any instant of time, unlike the one on a constant background, which has a large amplitude at the intermediate time of evolution. For large t, the solitonic background has multiple parallel solitons possessing the same asymptotic velocities and heights. The obtained results improve our understanding of the generation mechanisms of rogue waves.