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Solitons and the Inverse Scattering Transform (Paperback)
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Solitons and the Inverse Scattering Transform (Paperback)
Series: Studies in Applied and Numerical Mathematics, No. 4
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A study, by two of the major contributors to the theory, of the
inverse scattering transform and its application to problems of
nonlinear dispersive waves that arise in fluid dynamics, plasma
physics, nonlinear optics, particle physics, crystal lattice
theory, nonlinear circuit theory and other areas. A soliton is a
localized pulse-like nonlinear wave that possesses remarkable
stability properties. Typically, problems that admit soliton
solutions are in the form of evolution equations that describe how
some variable or set of variables evolve in time from a given
state. The equations may take a variety of forms, for example,
PDEs, differential difference equations, partial difference
equations, and integrodifferential equations, as well as coupled
ODEs of finite order. What is surprising is that, although these
problems are nonlinear, the general solution that evolves from
almost arbitrary initial data may be obtained without
approximation. For such exactly solvable problems, the inverse
scattering transform provides the general solution of their initial
value problems. It is equally surprising that some of these exactly
solvable problems arise naturally as models of physical phenomena.
Simply put, the inverse scattering transform is a nonlinear analog
of the Fourier transform used for linear problems. Its value lies
in the fact that it allows certain nonlinear problems to be treated
by what are essentially linear methods. Chapters 1 and 2 of the
book describe in detail the theory of the inverse scattering
transform. Chapter 3 discusses alternate methods for these exactly
solvable problems and the interconnections among them. Physical
applications are described in Chapter 4, where, for example,
similarities between deep water waves and nonlinear optics become
evident. Because of the fundamental role of linear theory, there is
an extensive appendix that addresses the linear problems and their
solutions.
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