Traveling waves and localized structures: An alternative view of nonlinear evolution equations

TL;DR

This paper presents a novel approach to nonlinear evolution equations by extending them into coupled systems that generate localized structures at wave intersections, offering new insights into multi-dimensional wave phenomena.

Contribution

It introduces a method to derive coupled equations from original evolution equations, revealing localized structures near wave intersections in various dimensions.

Findings

Structures are localized near wave intersections in extended systems.

The method applies to known equations in multiple dimensions.

Localized structures can be dynamic and move in space.

Abstract

Given a nonlinear evolution equation in (1+n) dimensions, which has spatially extended traveling wave solutions, it can be extended into a system of two coupled equations, one of which generates the original traveling waves, and the other generates structures that are localized in the vicinity of the intersections of the traveling waves. This is achieved thanks to the observation that, as a direct consequence of the original evolution equation, a functional of its solution exists, which vanishes identically on the single-wave solution. This functional maps any multi-wave solution onto a structure that is confined to the vicinity of wave intersections. In the case of solitons in (1+1) dimensions, the structure is a collection of humps localized in the vicinity of soliton intersections. In higher space dimensions these structures move in space. For example, a two-front system in (1+3) dimensions is mapped onto an infinitely long and laterally bounded rod, which moves in a direction perpendicular to its longitudinal axis. The coupled systems corresponding to several known evolution equations in (1+1), (1+2) and (1+3) dimensions are reviewed.