The Generation and Evolution of Lump Solitary Waves in Surface-Tension-Dominated Flows

Three-dimensional solitary waves or lump solitons are known to be solutions to the Kadomtsev–Petviashvili I equation, which models small-amplitude shallow-water waves when the Bond number is greater than 1 3 . Recently, Pego and Quintero presented a proof of the existence of such waves for the Benney–Luke equation with surface tension. Here we establish an explicit connection between the lump solitons of these two equations and numerically compute the Benney– Luke lump solitons and their speed-amplitude relation. Furthermore, we numerically collide two Benney–Luke lump solitons to illustrate their soliton wave character. Finally, we study the flow over an obstacle near the linear shallow-water speed and show that three-dimensional lump solitons are periodically generated.