Instabilities and oscillations of one- and two-dimensional Kadomtsev-Petviashvili waves and solitons

We investigate the stability and dynamics of nonlinear structures that arise from the Kadomtsev-Petviashvili equation. For initially flat propagating cnoidal waves and solitons, the region of instability in perturbed wavevector K-space is found. This is done by combining information obtained both by approximate and exact calculations. Growth rates are found. Interesting analogies with the deep-water wave problem are pointed out. It might seem strange that this analysis should be necessary when the problem has been 'solved' exactly by Kuznetsov and co-workers. However, as those authors admit, their treatment is extremely difficult to apply to any but a few specific limits. So much so that they miss a stability boundary even in one of those limits. We will comment on this. Here, by combining all our results, we are able to present a reasonably complete picture. Formulae are explicit and simple. Once the instability develops, two-dimensional 'lumps' are produced: as well as a new diminished cnoidal wave. We will concentrate on the lumps and their further fate. The two-dimensional solitons or lumps are then investigated in three dimensions by numerical simulations. A distortion in a plane perpendicular to the motion leads to an oscillation of the lumps. This result is in contradistinction to the result of a perturbation in a plane containing the direction of motion. In that simulation of Senatorski & Infeld, lumps were destroyed completely and three-dimensional solitons were formed. Thus a link was found between two entities that were hitherto considered separately. In all, we have an initially two-dimensional soliton that can either produce a three-dimensional decay product or else exhibit a new oscillation. This will depend on the plane of the perturbation