Solitons were first discovered by a Scottish engineer, J. Scott Russell, in 1834 while riding his horse by a water channel when a boat suddenly stopped. A hump of water rolled off the prow of the boat and moved rapidly down the channel for several miles, preserving its shape and speed. The observation was surprising because the hump did not rise and fall, or spread and die out, as ordinary water waves do.

In the 150 years or so since the discovery of Scott Russell, solitons have been discovered in numerous systems besides hydrodynamics. Probably the most important application of these is in the context of optics where they can propagate in optical fibers without distortion: they are being studied for high-data-rate (terabits) communication. Particle physicists have realized that solitons may also exist in their models of fundamental particles, and cosmologists have realized that such humps of energy may be propagating in the far reaches of outer space. There is even speculation that all the fundamental particles (electrons, quarks etc.) may be viewed as solitons owing to their quantum properties, leading to a “dual” description of fundamental matter.

In this book I describe the simplest kinds of solitons, called “kinks” in one spatial dimension and “domain walls” in three dimensions. These are also humps of energy as in Scott Russell's solitons. However, they also have a topological basis that is absent in hydrodynamical solitons. However, they also have a topological basis that is absent in hydrodynamical solitons. This leads to several differences e.g. water solitons cannot stand still and have to propagate with a certain velocity, while domain walls can propagate with any velocity. Another important point in this regard is that strict solitons, such as those encountered in hydrodynamics, preserve their identity after scattering.