Solitons in Physics

Abstract

Given Bullough’s review of the mathematics of soliton solutions of nonlinear field equations, we see how distinctly different those are from our traditional linear wave experiences. Therein lies the most serious barrier which limits much of physics today in this regard: during this century, generations of physicists have trained in and practiced linear thought, normal mode decomposition or superposition, and nonlinearity introduced only by perturbative corrections was used to varying degrees to correct this limited base. But in no way does an extended wave turn into a soliton by perturbation theory. The other side of the coin, however, is that in strongly nonlinear problems, mathematicians had not generally provided us with a game plan for systematically approaching nonlinearity, in the sense that mode analysis was utilitarian. But that is just what is happening with regard to solitons. It is being discovered that there is a non-negligible class of nonlinear field equations which, if not bearing exact soliton solutions, clearly have both spatially extended and spatially compact solutions, that these solutions are functionally independent to a high degree (if not exactly so), and that they are stable with respect to small perturbations. Now the physicist can do something with these as an augmented set of ingredients in the analysis of nature — in statistical mechanics, quantum liquids, structural phase transitions, quantum field theory, epitaxy in surface physics, polymer science, etc., to name some major areas whose utilization is growing.