By A. Kundu

Overlaying either classical and quantum types, nonlinear integrable structures are of substantial theoretical and sensible curiosity, with purposes over a variety of themes, together with water waves, pin versions, nonlinear optics, correlated electron platforms, plasma physics, and reaction-diffusion strategies. Comprising one half on classical theories and functions and one other on quantum points, Classical and Quantum Nonlinear Integrable structures: concept and alertness stories the advances made in nonlinear integrable platforms, with emphasis at the underlying ideas instead of technical information. It varieties a great introductory textbook in addition to an invaluable reference for experts.

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Translated and revised from the 1986 Russian variation.

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**Additional info for Classical and Quantum Nonlinear Integrable Systems: Theory and Application (Series in Mathematical and Computational Physics)**

**Example text**

6, with appropriate scale change and redefinition of parameters. 76b) as N F (x + y, t) = Cn2 (t) e−κn (x+y) = n=1 = N Cn e−κn x Cn e−κn y n=1 gn (x, t)gn (y, t) gn (x, t) = Cn (t) e−κn x . 90) Then defining N K(x, y) = ωn (x)gn (y) n=1 Copyright © 2003 IOP Publishing Ltd. 76), we obtain ∞ N ωm (x) + gm (x) + ωn (x) gm (z)gn (z) dz = 0. 93a) x ω(x) = (ω1 (x), ω2 (x), . . , ωN (x))T g(x) = (g1 (x), g2 (x), . . 93) can be rewritten as the matrix equation P(x)ω(x) = −g(x). 91), we have K(x, x) = gT (x)ω(x) = −gT (x)P−1 (x)g(x).

Also how does the linearization property discussed in the previous section help in this regard? 45) through a three-step process. The procedure was originally developed by Gardner, Greene, Kruskal and Miura. This method, now called the inverse scattering transform (IST) method, may be considered as a nonlinear Fourier transform method. It will be now described as applicable to the KdV equation. 1 The IST method for the KdV equation The analysis proceeds in three steps similar to the case of the Fourier transform method applicable for linear dispersive systems: (i) direct scattering transform analysis, (ii) analysis of time evolution of scattering data and (iii) IST analysis.

The linear one of R Fuchs [10] as an isomonodromy condition. 10) (A, B, C, D denote constants and a, b parameters). The requirement that the monodromy matrix (which transforms two independent solutions ψ1 , ψ2 when t goes around a singularity) be independent of the non-apparent singularity x results in the condition that u, as a function of x, satisfies P6. A useful by-product of this search for new functions is the construction of several exhaustive lists (classifications) of second [11–15], third [1, 4, 6], fourth [4,5] or higher-order [16] ODEs, whose general solution is explicitly given because they have the PP.