By A. Kundu

Masking either classical and quantum types, nonlinear integrable structures are of substantial theoretical and sensible curiosity, with functions over quite a lot of subject matters, together with water waves, pin versions, nonlinear optics, correlated electron structures, plasma physics, and reaction-diffusion approaches. Comprising one half on classical theories and purposes and one other on quantum facets, Classical and Quantum Nonlinear Integrable platforms: concept and alertness studies the advances made in nonlinear integrable platforms, with emphasis at the underlying recommendations instead of technical information. It types an excellent introductory textbook in addition to an invaluable reference for experts.

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**Sample text**

83), we have K(x, y, t) = −C02 e8κ 3 t −κ(x+y) [1 + (C02 /2κ) e8κ 3 t −2κx ] . 77) as d K(x, x + 0, t) dx = −2∂K(x, y, t)/∂x|y=x − 2∂K(x, y, t)/∂y|y=x u(x, t) = −2 = −2κ 2 e−2κ(x−4κ 2 t )−2δ [1 + e−2κ(x−4κ t )−2δ ]2 2 = −2κ sech2 [κ(x − 4κ 2t) + δ]. 32) obtained by the Hirota method. 2 Two-soliton solution Again, let us consider a reflectionless potential such that R(k, t) = 0 but now with two bound states (example: u(x) = −6 sech2 x has two bound states with λ1 = −4, λ2 = −1), specified by the discrete values κ1 and κ2 and the corresponding Copyright © 2003 IOP Publishing Ltd.

Examples of these various situations √ are, respectively: the PDE ux ut + uuxt = 0 with the general solution u = f (x) + g(t), which presents movable critical singularities and can be transformed into the d’Alembert equation; the Burgers PDE ut + uxx + 2uux = 0, linearizable into the heat equation ψt + ψxx = 0; the KdV PDE ut + uxxx − 6uux = 0, which is integrable by the inverse spectral transform (IST) [19]; and the Liouville PDE uxt + eu = 0, which possesses a BT to the d’Alembert equation ψxt = 0.

Solving them, one obtains h1 (x, t) and h2 (x, t). 89a) where γi = κi x − 4κi3t − δi δi = 1 log 2 2 Ci0 (κ2 − κ1 ) 2κi (κ2 + κ1 ) i = 1, 2. 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), .