By G. Eilenberger
1.1 Why learn Solitons? The final century of physics, which used to be initiated through Maxwell's of entirety of the idea of electromagnetism, can, with a few justification, be known as the period of linear physi cs. Jith few excepti ons, the tools of theoreti ca 1 phys ics were ruled by means of linear equations (Maxwell, Schrodinger), linear mathematical gadgets (vector areas, specifically Hilbert spaces), and linear tools (Fourier transforms, perturbation conception, linear reaction idea) . certainly the significance of nonlinearity, starting with the Navier-Stokes equations and carrying on with to gravitation concept and the interactions of par ticles in solids, nuclei, and quantized fields, used to be famous. besides the fact that, it was once hardly ever attainable to regard the results of nonlinearity, other than as a according to turbation to the root options of the linearized conception. over the last decade, it has turn into extra well known in lots of parts of "field physics" that nonlinearity can lead to qualitatively new phenom ena which can't be developed through perturbation concept ranging from linear ized equations. via "field physics" we suggest all these parts of theoretical physics for which the outline of actual phenomena leads one to think about box equations, or partial differential equations of the shape (1.1.1) t or tt = F(, x ... ) for one- or many-component "fields" Ht, x, y, ... ) (or their quantum analogs).
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Additional resources for Solitons: Mathematical Methods for Physicists
OR(o) In the more general case, with II 1 0, we consider values of k which are of order t or smaller. For these we have a(k) = o[cosho - te O l 1/2ik + eo(t)] b(k) = o[sinho + te Ol 1/2ik + to(t)] R(o) jumps to the value 1. 13). 16) (l Thus the new soliton appears as if it had entered the system from x = -00. This process of soliton formation can, of course, go the other way, and result in the disappearance of a soliton. We will not pursue this phenomenon further here except to remark that it is clear that the dynamics of the scattering data must be formulated much more carefully in a neighborhood of k = 0, when the "status nascendi" R(o) 1 1 is under consideration.
21) and is uniquely determined by these data. This is the solution to the inverse scattering problem. The result can be stated generally as follows. If the scattering data R, etc. and the Jost solutions f for a potential are known, then one can find the potential ~ = ~ + ~~ which corresponds to ¢ 35 a given set of scattering data R = R + ~,etc. 20). The most important special cases are, on the one hand, ~ = 0 with f ~ exp(ikx), so that M is particularly simple and, on the other hand, ~ - ~ = o~, for which one considers the infinitesimal change in ~ resulting from an infinitesimal change in the scattering data; in this case, we have oK = -OM, since the integral is second order in the infinitesimal quantities.
If one soliton overtakes another, a mutual shift takes place (the faster one to the front, the slower one to the rear). The power of the inverse scattering method is demonstrated by the following simple application: We want to solve the KdV equation with the initial condition ¢(x,o) = - A/cosh 2x . 2]) and are given by 1 Kn = 2 (~ + 1 - 2n) for all n = 1, ... , N such that Kn > O. If A = N(N + 1), the potential is reflectionless, and we have Kn = N + 1 - n, so that the initial state "decays" into N solitons with corresponding vn ; from intuitive consideration of the KdV equation, one would almost certainly never have guessed at such behavior.