TY - UNPB

T1 - Traveling waves and localized structures: An alternative view of nonlinear evolution equations

AU - Zarmi, Yair

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

KW - Nonlinear Sciences - Exactly Solvable and Integrable Systems

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

ER -