Highly localised horseshoe ripplons and solitons in positive dispersion media

In this study, we systematically review various ripplon solutions to the Kadomtsev–Petviashvili equation with positive dispersion (KP1 equation). We show that there are mappings that allow one to transform the horseshoe solitons and curved lump chains of the KP1 equation into circular solitons of the cylindrical Korteweg–de Vries (cKdV) equation and two-dimensional solitons of the cylindrical Kadomtsev–Petviashvili (cKP) equation. Then, we present analytical solutions that describe new nonlinear highly localised ripplons of a horseshoe shape. Ripplons are two-dimensional waves with an oscillatory structure in space and a decaying character in time; they are similar to lumps but non-stationary. In the limiting case, the horseshoe ripplons reduce to solitons decaying with time and having bent fronts. Such entities can play an important role in the description of strong turbulence in plasma and other media.