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# Mathematical Physics

# Title: Solitons: conservation laws & dressing methods

(Submitted on 31 Dec 2018 (v1), last revised 10 Apr 2020 (this version, v5))

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.

## Submission history

From: Anastasia Doikou [view email]**[v1]**Mon, 31 Dec 2018 17:42:06 GMT (29kb)

**[v2]**Thu, 3 Jan 2019 12:48:16 GMT (29kb)

**[v3]**Thu, 14 Feb 2019 11:30:33 GMT (29kb)

**[v4]**Wed, 20 Mar 2019 12:45:23 GMT (29kb)

**[v5]**Fri, 10 Apr 2020 12:36:40 GMT (30kb)

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