Dynamics and Interactions of Truncated, Two-Dimensional Line Solitons in Shallow Water

It is well-known that the Kadomtsev-Petviashvili II (KP-II) equation—an asymptotic model of weakly nonlinear, shallow water gravity waves with weak transverse variation—admits stable, two-dimensional, line soliton solutions characterized by their amplitude and slope. Their dynamics subject to modulation, e.g., truncation and bending, are studied utilizing the soliton limit of the recently derived KP-Whitham modulation equations. A Riemann invariant form for the modulation equations—a system of two hyperbolic equations in two space dimensions and time for the soliton's amplitude and slope—is identified that enables exact solution methods including simple waves and hodograph techniques. This theory is used to describe the evolution of truncated solitons: an isolated line soliton segment and the interaction of two semi-infinite line solitons. The results compare favorably with direct numerical simulation and have application to near-shore nonlinear wave dynamics.