Kadomtsev-Petviashvili II equation: Structure of asymptotic soliton webs

A wealth of observations, recently supported by rigorous analysis, indicate that, asymptotically in time, most multi-soliton solutions of the Kadomtsev-Petviashvili II equation self-organize in webs comprised of solitons and soliton-junctions. Junctions are connected in pairs, each pair - by a single soliton. The webs expand in time. As distances between junctions grow, the memory of the structure of junctions in a connected pair ceases to affect the structure of either junction. As a result, every junction propagates at a constant velocity, which is determined by the wave numbers that go into its construction. One immediate consequence of this characteristic is that asymptotic webs preserve their morphology as they expand in time. Another consequence, based on simple geometric considerations, explains why, except in special cases, only 3-junctions ("Y-shaped", involving three wave numbers) and 4-junctions ("X-shaped", involving four wave numbers) can partake in the construction of an asymptotic soliton web.