Solitons Mathematical Methods for Physicists - Eilenberger, Gert (Springer Berlin Heidelberg, 1981)
1.1 Why Study Solitons? The last century of physics, which was initiated by Maxwell's completion of the theory of electromagnetism, can, with some justification, be called the era of linear physics. With few exceptions, the methods of theoretical physics have been dominated by linear equations (Maxwell, Schrodinger), linear mathematical objects (vector spaces, in particular Hilbert spaces), and linear methods (Fourier transforms, perturbation theory, linear response theory) . Naturally the importance of nonlinearity, beginning with the Navier-Stokes equations and continuing to gravitation theory and the interactions of particles in solids, nuclei, and quantized fields, was recognized. However, it was hardly possible to treat the effects of nonlinearity, except as a perturbation to the basis solutions of the linearized theory. During the last decade, it has become more widely recognized in many areas of "field physics" that nonlinearity can result in qualitatively new phenomena which cannot be constructed via perturbation theory starting from linearized equations. By "field physics" we mean all those areas of theoretical physics for which the description of physical phenomena leads one to consider field equations, or partial differential equations of the form (1.1.1) ~t or ~tt = F(~, ~x ... ) for one- or many-component "fields" Ht,x,y, ... ) (or their quantum analogs)