Localized excitations in (2+1)-dimensional systems

By means of a special variable separation approach, a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various (2+1)-dimensional models such as the Davey-Stewartson (DS) model, the Nizhnik-Novikov-Veselov (NNV) system, asymmetric NNV equation, asymmetric DS equation, dispersive long wave equation, Broer-Kaup-Kupershmidt system, long wave-short wave interaction model, Maccari system, and a general (N+M)-component Ablowitz-Kaup-Newell-Segur (AKNS) system. Selecting the arbitrary functions appropriately, one may obtain abundant stable localized interesting excitations such as the multidromions, lumps, ring soliton solutions, breathers, instantons, etc. It is shown that some types of lower dimensional chaotic patterns such as the chaotic-chaotic patterns, periodic-chaotic patterns, chaotic line soliton patterns, chaotic dromion patterns, fractal lump patterns, and fractal dromion patterns may be found in higher dimensional soliton systems. The interactions between the traveling ring type soliton solutions are completely elastic. The traveling ring solitons pass through each other and preserve their shapes, velocities, and phases. Some types of localized weak solutions, peakons, are also discussed. Especially, the interactions between two peakons are not completely elastic. After the interactions, the traveling peakons also pass through each other and preserve their velocities and phases, however, they completely exchange their shapes.