DEGENERATION OF LUMP-TYPE LOCALIZED WAVES IN THE (2+1)-DIMENSIONAL ITO EQUATION

The degeneration of lump-type localized waves in the (2+1)-dimensional Ito equation is investigated through the parallel relationship of wave numbers. These lump-type localized waves can degenerate into three different kinds of localized wave solutions: singular lump-type localized wave, periodic variable amplitude localized wave, rogue wave. In the process of propagation, the lump-type localized waves keep the same waveform structure and amplitude. However, the periodic variable amplitude localized wave demonstrates three different kinds of waveform structures, which presents an interesting emit-absorb interaction phenomenon. By an emitting and absorbing interaction, the localized wave realizes the energy exchange from one localized wave to another, and keeps the original waveform structure. Rogue wave is a rational growing-and-decaying localized wave which is localized in both space and time.