Solitons

1.1 Why Study Solitons? The last century of physics which was initiated by Maxwells completion of the theory of electromagnetism can with some justification be called the era of linear physi cs. Jith few excepti ons the methods of theoreti ca 1 phys- ics 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 par- ticles in solids nuclei and quantized fields was recognized. However it was hardly possible to treat the effects of nonlinearity except as a per- turbation 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 phenom- ena which cannot be constructed via perturbation theory starting from linear- ized 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).