Painlevé analysis and inelastic interactions of the lumps for a generalized (2+1)-dimensional Korteweg-de Vries system for the shallow-water waves

Abstract

Water waves, one of the common natural phenomena, are recognized as complex and often turbulent. A generalized (2+1)-dimensional Korteweg-de Vries system for the shallow-water waves is conducted in this paper. We perform the Painlevé analysis and find that the system is Painlevé integrable. We study the inelastic interactions of the lumps for the system. We find that two lumps, which propagate along the curves with the equal amplitude, are symmetric about the x axis before the interaction, where x is a scaled spatial variable. After the interaction, amplitudes of the two lumps are different, but in the process of moving, the lower lump gradually increases, while the higher lump gradually decreases, and the velocities of two lumps at the infinity are equal. We observe two different inelastic interactions of the three lumps: (1) the three lumps are symmetric in time and space, and they slowly contract (before the interaction) and swell (after the interaction); (2) the three lumps slowly fuse and after the interaction they form a straight line forward, and their amplitudes are gradually equal when t → ∞.