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Lectures on nonlinear integrable equations and their solutions

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arxiv 1812.11830 v1 pith:TKOB6ZB5 submitted 2018-12-31 math-ph math.MPnlin.SI

A. Zabrodin This is my paper
classification math-phmath.MPnlin.SI
keywords equationssolutionsdifferentialgiveintegrablelecturesnonlinearoperators
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This is an introductory course on nonlinear integrable partial differential and differential-difference equ\-a\-ti\-ons based on lectures given for students of Moscow Institute of Physics and Technology and Higher School of Economics. The typical examples of Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP) and Toda lattice equations are studied in detail. We give a detailed description of the Lax representation of these equations and their hierarchies in terms of pseudo-differential or pseudo-difference operators and also of different classes of the solutions including famous soliton solutions. The formulation in terms of tau-function and Hirota bilinear differential and difference equations is also discussed. Finally, we give a representation of tau-functions as vacuum expectation values of certain operators composed of free fermions.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On W-algebras and ODE/IM correspondence

    hep-th 2025-08 conditional novelty 6.0 of 10

    The eigenvalues of quantum KdV-type charges in Virasoro, W3, and W4 algebras are computed from Bethe roots via WKB periods of Catalan curves, verified against direct CFT diagonalization.

  2. More on Slavnov Products of Spin Chains and KP Hierarchy Tau Functions

    hep-th 2025-05 reject novelty 5.0 of 10

    Slavnov products of Bethe states in rational spin chains are shown to be KP tau functions, with new Wronskian and Baker-Akhiezer formulas, but the key identification relies on an unverified assumption.

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