Kadomtsev-Petviashvili II equation: Structure of asymptotic soliton webs

TL;DR

This paper analyzes the structure and dynamics of asymptotic soliton webs in the Kadomtsev-Petviashvili II equation, revealing how they form, expand, and preserve their morphology over time.

Contribution

It provides a detailed description of the structure and propagation of soliton webs, highlighting the dominance of 3- and 4-junctions in their asymptotic configuration.

Findings

Soliton webs expand in time while maintaining their structure.

Junctions propagate at constant velocities determined by wave numbers.

Only 3- and 4-junctions typically form in asymptotic webs.

Abstract

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 par-take in the construction of an asymptotic soliton web.