Spectral Transform and Solitons: How to Solve and Investigate Nonlinear Evolution Equations

Calogero, F.; Degasperis, A.

doi:10.1007/978-1-4899-2058-4_3

Cited by 202 publications

(337 citation statements)

References 248 publications

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“…In fact, another way to obtain this expansion could be to substitute directly the regular expansion (2.19) in (2.22), but the recursive relations (2.29) (with the initial conditions (2.26)) will provide a contact with the use of the recursive operator [15] in the theory of integrable hierarchies. Now, the action (2.28) of the symmetry generators on the bosonic field is given in terms of a X n (x) by making use of (2.29)…”

Section: Regular Expansion Of the Transfer Matrix And The Dressingmentioning

confidence: 99%

“…The linear differential operators R a , R b are called recursive operators [15] and they generate the integrable flows of an hierarchy (next Section). We have proven here that the proper dressing transformation (2.30),(2.31) can be thought of as generated by the inverse power of the recursive operators, i.e., in a compact notation,…”

Section: Regular Expansion Of the Transfer Matrix And The Dressingmentioning

confidence: 99%

“…The arbitrariness in the initial condition for a 2k+1 and b 2k will be fixed in the following using the geometrical interpretation of the resolvent (equation (2.74)). The recursive operators R a , R b [15] generate the integrable flows of the hierarchy as implied by (2.67,2.68):…”

Section: The Integrable Hierarchy and The Asymptotic Dressingmentioning

confidence: 99%

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Hidden local, quasi-local and non-local symmetries in integrable systems

Fioravanti¹,

Stanishkov²

2000

Nuclear Physics B

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The knowledge of non usual and sometimes hidden symmetries of (classical) integrable systems provides a very powerful setting-out of solutions of these models. Primarily, the understanding and possibly the quantisation of intriguing symmetries could give rise to deeper insight into the nature of field spectrum and correlation functions in quantum integrable models. With this perspective in mind we will propose a general framework for discovery and investigation of local, quasi-local and non-local symmetries in classical integrable systems. We will pay particular attention to the structure of symmetry algebra and to the rôle of conserved quantities. We will also stress a nice unifying point of view about KdV hierarchies and Toda field theories with the result of obtaining a Virasoro algebra as exact symmetry of Sine-Gordon Model. *

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Section: Regular Expansion Of the Transfer Matrix And The Dressingmentioning

confidence: 99%

Section: Regular Expansion Of the Transfer Matrix And The Dressingmentioning

confidence: 99%

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Hidden local, quasi-local and non-local symmetries in integrable systems

Fioravanti¹,

Stanishkov²

2000

Nuclear Physics B

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“…Finally, between the different excellent existing books on solitons let us suggest to the reader [57] and [58], as the most accurate and complete for the nonlinear evolution equations associated to the Schrödinger and to the Zakharov-Shabat spectral equation in 1+1 dimensions, respectively, and [59] as the most updated and comprehensive. In particular for those who want to have a general overlooking on the subject and a rich bibliography to pick over this book is particularly recommended.…”

Section: Guidelines For Additional Readingmentioning

confidence: 99%

Multidimensional localized solitons

Boiti

Martina

Pempinelli

1995

Chaos, Solitons & Fractals

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Recently it has been discovered that some nonlinear evolution equations in 2 + 1 dimensions, which are integrable by the use of the Spectral Transform, admit localized (in the space) soliton solutions. This article briefly reviews some of the main results obtained in the last five years thanks to the renewed interest in soliton theory due to this discovery. The theoretical tools needed to understand the unexpected richness of behaviour of multidimensional localized solitons during their mutual scattering are furnished. Analogies and especially discrepancies with the unidimensional case are stressed.

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“…It is well known that one of the methods of examining such problems for nonlinear evolution equations is the inverse scattering transform [1,7]. Another approach has been proposed by Naumkin and Shishmarev [18], who have studied nonlocal nonlinear equations of the first order in time and with small initial data.…”

mentioning

confidence: 99%