Solitons: conservation laws & dressing methods

Abstract

We review some of the fundamental notions associated to the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux-Backlund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov-Shabat dressing scheme and the Gelfand-Levitan-Marchenko equation are also introduced. Via this method generic solutions are produced, and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete non-linear Schrodinger model, the Korteweg-de Vries and non-linear Schrodinger equations as well as the sine-Gordon and Liouville models.

arXiv comment39 pages, LaTex. A few typos corrected

References in corpus (6)

• Multicomponent integrable wave equations I. Darboux-Dressing Transformation
• Multicomponent integrable wave equations II: Soliton solutions
• A Simple Lattice Version of the Nonlinear Schrodinger Equation and its Deformation with Exact Quantum Solution
• Systematic classical continuum limits of integrable spin chains and emerging novel dualities
• Darboux-Backlund transformations, dressing & impurities in multi-component NLS
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• Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics
• The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory
• General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations