Ring solitons and lump waves of the elliptic cylindrical Kadomtsev-Petviashvili equation in physical coordinates.

The elliptic cylindrical Kadomtsev-Petviashvili (ecKP) equation is derived from the incompressible Euler equations governing surface gravity waves with nearly elliptic fronts. Using Hirota's bilinear method, this paper systematically investigates ring solitons, lump chains, and lump waves of the ecKP-I equation in physical coordinates. It is shown that ring solitons and ring-type lump chains appear only after a critical time; they originate from a centrally symmetric bulge on the water surface that collapses and subsequently propagates outward. The ring soliton is unstable and, under perturbation, gives rise to lump waves or lump chains. In contrast to the KP-I equation, lump waves in the ecKP-I equation exhibit several distinctive features: a dark lump wave with a larger amplitude propagates more slowly and its speed possesses an upper bound; in the standard two-lump solution, the lump waves initially propagate with equal amplitudes along curved trajectories, their collision involves energy transfer, and after the interaction they move along straight lines; degenerate lump waves do not collide, they travel with varying instantaneous velocities while sharing the same asymptotic amplitudes and velocities as their standard counterparts. These findings reveal fundamental properties of asymptotic solutions to the incompressible Euler equations, deepen our understanding of gravity-capillary waves, and may offer new insight into the formation mechanisms of small-scale shallow-water waves.