Spatially extended particles hidden in line-soliton dynamics in more than one space dimension

TL;DR

This paper reveals how line soliton interactions in higher dimensions encode dynamics of spatially extended particles, using conservation laws and vertex mappings in solutions of the KP equation.

Contribution

It introduces a novel framework mapping multi-soliton solutions to systems of vertices representing extended particles in higher dimensions.

Findings

Vertices move in space, emulating particles.

Vertices coalesce and split, mimicking particle collisions.

Total momentum conserved, but energy and mass may vary.

Abstract

A dynamics of spatially extended particles, hidden in the dynamics of line solitons in more than one space dimension, is revealed through conservation laws obeyed by the single-soliton solution. These are functions of the solution of a nonlinear evolution equation and its derivatives, which vanish for a single-soliton solution. They map multi-line-soliton solutions into systems of vertices - spatially extended structures, localized around the soliton-collision regions. In more than one space dimension, vertices move in space, emulating the dynamics of spatially extended particles. Examples are provided through the analysis of several soliton solutions of the Kadomtsev-Petviashvili (KP) equation in (1+2)-dimensions. The solution with one collision region is mapped onto a single vertex, which moves at a constant velocity, preserving its spatial structure, thereby emulating a free particle. In solutions with several collision regions, each region is mapped onto a vertex. When vertices are well separated, they also emulate free particles. They move in space, coalesce upon collision, and then split up. The total particle linear momentum is conserved through the collision in all solutions studied. However, depending on the soliton solution, particle masses and the total kinetic energy may change.