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

Yair Zarmi Jacob Blaustein Institutes for Desert Research Ben-Gurion University of the Negev Midreshet Ben-Gurion, 8499000, Israel

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) di- mensions, 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.

Keywords: nonlinear evolution equations; spatially extended wave solutions; traveling wave inter- action regions; localized structures

PACS: 02.30.Ik, 03.65.Pm, 05.45.Yv, 02.30.Ik E-mail: zarmi@bgu.ac.il

1. Introduction The interest in nonlinear systems that admit solutions in the form of spatially localized structures in (1+n) dimensions has been growing rapidly over the years. The usual approach focuses on find- ing finite-amplitude, spatially localized solutions of a single evolution equation [1-25]. Many works have considered the generation of localized structures in systems of coupled equations that are obtained as possible descriptions for a variety of physical systems (see. e.g., Refs. [26-43]).

This paper also focuses on the generation of localized structures in systems of coupled equations.
The novelty is that the systems are simple extensions of nonlinear evolution equations in (1+n) dimensions, which have spatially extended traveling wave solutions such as solitons or fronts. (In this context, a soliton in (1+1) dimensions, while localized along the x-axis, is viewed as spatially extended in the x-t plane.) The algorithm exploited leads to the extension of an evolution equation into a system of two coupled equations, one of which generates the traveling wave solutions of the original equation, whereas the second equation generates localized structures.

The starting point is a direct consequence of the evolution equation: Given the solution, u, of that equation, a functional, R[u], exists, which vanishes identically when u is a single-wave solution.
The functional, R[u], maps multi-wave solutions onto structure that are confined to the vicinity of wave intersections. In the case of solitons in (1+1) dimensions these structures are humps in the vicinity of soliton intersections. In the case of fronts in (1+1) dimensions of the Burgers equation these structures are KdV-like solitons on a half line. In more than one space dimension, the local- ized structures move in space. For example, the multi-soliton solutions of the Kadomtsev- Petviashvili (KP) II equation are mapped onto collections of humps that move in the x-y plane, and a two-front solution of the Sine-Gordon equation in (1+3) dimensions is mapped onto an infinitely long, but laterally bounded, rod that moves in a direction perpendicular to its longitudinal axis.

A physically interesting outcome of this approach is that the emergence of solutions with more than one traveling wave (i.e., one soliton, or one front) may be viewed as the splitting of a single wave into several waves under the effect of the accompanying localized structure. Concurrently, generation of the localized structures may be viewed as a consequence of the interaction amongst spatially extended traveling waves.

The cases of several known evolution equations in (1+1), (1+2) and (1+3) dimensions are re- viewed. The generation of localized structures from the multi-soliton solutions of evolution equa- tions in (1+1) dimensions is reviewed in Section 2. In these cases, R[u] describes humps in the x-t plane. Section 3 is devoted to the Burgers equation. The images of its front solutions under R[u] are KdV-like solitons on a half line. The generation of moving spatially localized structures in (1+2) dimensions in the case of the KP II equation is reviewed in Section 4. Section 5 reviews the generation of localized structures from multi-front solutions of the Sine-Gordon equation. In (1+1) dimensions, these are humps in the

…(Full text truncated)…