## Topological solitonsIn classical physics, a soliton
is a solution to a
nonlinear
partial
differential equation that behaves like a delocalised (or wavelike)
particle. Often, such equations also admit multisoliton solutions, which can
be interpreted as a superposition of a certain number of solitons, and
in some cases even as a combination of solitons and antisolitons.
The existence of such superposition phenomena is a nontrivial feature
of certain
nonlinear
equations.
The set of rules describing nonlinear
superposition
is particular to a given equation, so there is no systematic way of
studying (multi)solitons. This is in contrast with the principle of linear
superposition, according to which the solutions to any
linear
equation
form an
affine
vector space.
Originally, the word "soliton" was used to describe certain travelling bump solutions of the Korteweg-de Vries (KdV) equation, which models how water waves propagate in one dimension beyond the linear approximation. KdV solitons are very real and were vividly described by the engineer John Scott Russell, who was able to follow one on horseback for over a mile along the Union Canal near Edinburgh in 1834. The stability of such bump waves is due to the existence of a large number of conserved quantities in the KdV dynamical system; this feature is one aspect of the general concept of integrability. The structure of integrable systems can be very
intricate, and it may manifest itself through striking properties of the
solutions. For example, the
speed
of KdV solitons turns out to be determined by their
height and width; in an overtaking process, there is a time
shift effect at superposition, whereby the faster soliton advances and
the slower soliton retards its march.
Topological solitons were so named in analogy with the solutions of the
KdV equation: they are essentially also localised, superpose nonlinearly
and retain individuality in their interactions. They owe their stability to
topology
rather than integrability --- either because they are
maps between
noncontractible
spaces, or because
they must obey nontrivial
boundary conditions.
Quite generally, the topology of a soliton is specified by one
or more
integers
known as
topological
charges; mathematically, these could
specify a
homotopy
class
or a characteristic
class. Topological charges are
conserved under
continuous dynamics, and this means that a party of solitons
must retain the initial overall number as the individuals mingle and move
about. The simplest examples of
topological solitons are known as kinks, and they live on the
real line
just as KdV solitons do. Incidentally, equations for kinks such as
the sine-Gordon
equation are also integrable.
Many examples of topological solitons occur in field theory models in more than one spatial dimension, where the dynamics becomes more interesting. The equations of motion are then no longer integrable, so travelling solutions cannot be constructed explicitly. In theoretical physics, the most interesting examples of solitons occur in gauge field theories, where the equations of motion are invariant under gauge transformations forming an infinite group of internal symmetries; they generalise Maxwell's theory of electromagnetism and are the fundamental building blocks of the standard model of particle physics. A prototypical example of gauge theory solitons are magnetic monopoles, which exist in
three-dimensional space. They can be thought of as sources of
magnetic field
in a theory that looks like Maxwell's from a distance, but has a
smooth and
much richer
(nonabelian) structure
at close range. The topological charge of a magnetic monopole can be
interpreted as a net magnetic charge.
Other examples of gauge theory solitons are vortices in two
dimensions,
which have been important in applications of field theory to
superconductivity,
the
fractional
quantum Hall effect
and other systems in
condensed
matter physics.
Skyrmions
provide yet another example of topological solitons
in a field
theory
with potentially interesting applications;
they
occur as classical solutions in a realistic model for the atomic nucleus,
where the baryon
number
is realised as a topological charge.
Although not much hope can be put on finding exact solutions for solitons in more than one dimension, the models of interest are sometimes "self-dual", and this has allowed some progress on the dynamics of topological solitons to be made. Self-duality roughly means that the static field configurations
of minimal energy satisfy first-order equations, which are usually more
tractable than the time-independent equations of motion.
One striking feature
of such models is that there is a well-behaved space of all solutions to
these first-order equations
(modulo
gauge transformations) with a given
topological charge. This "moduli space" comes equipped with extra
structures such as
riemannian metrics
(which allow one to measure
angles and distances, and to define
geodesics) or
complex
structures,
so we can do
geometry on them.
The geometry of the moduli space encodes physical information about the
system of solitons. For instance, geodesic arcs on the moduli space can be
interpreted as approximating
scattering
processes in the true dynamics at
low speed. As a concrete example, the following process illustrates
slow scattering of topological solitons called lumps, propagating
on an infinite cylinder:
instantons
in
Yang-Mills theory
on
four-manifolds,
on which
Donaldson's
theory is based. The equations for instantons on
euclidean space
are integrable
and, even more surprisingly, most integrable systems known can be obtained
from them by dimensional reduction: the self-duality equation for
magnetic
monopoles and the
sine-Gordon equation
provide two examples.
The geometry of the moduli spaces has been instrumental
in approaching the problem of making sense of topological solitons at the
quantum
level, and in testing
discrete
symmetries known as "dualities" that
one encounters in
quantum field theories.
Further (technical) reading:[1] M. Dunajski: Solitons, Instantons and Twistors, Oxford University Press, 2010 [2] N. Manton, P. Sutcliffe: Topological Solitons, Cambridge University Press, 2004 back to main |