Ring localized structures in nonlinear shallow water wave dynamics

The nonlinear dynamics of the concentric shallow water waves is described by means of the cylindrical Korteweg-de Vries equation, often referred to as the concentric Korteweg-de Vries equation (cKdVE). By using the mapping that transforms a cKdVE into the standard one-hereafter also referred to as the planar Korteweg-de Vries equation (KdVE)-the spatiotemporal evolution of a cylindrical surface water wave, corresponding to a tilted cylindrical bright soliton, is described. The usual representation of a tilted soliton is 'non-physical'; here the cylindrical coordinate and the retarded time play the role of time-like and space-like variables, respectively. However, we show that, when we express such analytical solution of the cKdVE in the appropriate representation in terms of the two horizontal space coordinates, say X and Y, and the 'true' time, say T, this non-physical character disappears. The analysis is then carried out numerically to consider the surface water wave evolution corresponding to initially localized structures with standard boundary conditions, such as bright soliton, Gaussian and Lorentzian profiles. A comparison among those profiles is finally presented.