Lump and Rogue Wave Solutions of a Reduced (3 + 1)-Dimensional Shallow Water Equation

Abstract

Considering a reduced (3 + 1)-dimensional shallow water equation, we use Hirota formulation and symbolic calculation to derive positive lump solitons rationally localised in all directions of the $(x, y)$-plane. The interaction of the lump and one stripe solitons is studied. Numerical experiments show that the collision of such solutions is completely inelastic and the lump soliton is swallowed by the stripe one. Exploring the interaction of the lump and a couple of resonance stripe solitons, we note that the lump soliton transforms into a ghost soliton. Most of the time it remains hidden in stripe solitons, but appears at a certain time and fades after that.