Prolongation approach to Bäcklund transformation of Zhiber–Mikhailov–Shabat equation

Yang, Hangsheng; Li, Youquan

doi:10.1063/1.531576

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…However, that fact is at least consistent with the apparent absence of defects in most of these models, at least of the kind previously considered [9]. On the other hand, there are several types of Bäcklund transformation available in the literature for the Tzitzéica model [12,13,14,15] c and, therefore, one might suppose there should be a generalisation of the defect, at least for this model, and possibly for others. The purpose of this article is to propose a generalisation by allowing a defect to have its own degree of freedom in a certain well-defined manner, which is just general enough to encompass the c Note: the model introduced by Tzitzéica is the a (2) 2 member of the affine Toda collection of field theories and is also known as the Bullough-Dodd or Zhiber-Mikhailov-Shabat equation.…”

Section: Introductionsupporting

confidence: 74%

“…(5.12) Inevitably, this depends on the fields u and v. The Bäcklund transformation (5.8)-(5.11) seems not to have been reported elsewhere in the literature [14][15][16].…”

Section: Defects and Bäcklund Transformationsmentioning

confidence: 97%

“…Setting u 0 ≡ u, v 0 ≡ v, λ 2 ≡ −λ and sending σ → 1/( √ 2 σ) the two sets of defect conditions lead to Inevitably, this depends on the fields u and v. The Bäcklund transformation (5.8)-(5.11) seems not to have been reported elsewhere in the literature [13,14,15].…”

Section: Defects and Bäcklund Transformationsmentioning

confidence: 99%

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A new class of integrable defects

Corrigan¹,

Zambon²

2009

J. Phys. A: Math. Theor.

158

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An alternative Lagrangian definition of an integrable defect is provided and analyzed. The new approach is sufficiently broad to allow a description of defects within the Tzitzéica model, which was not possible in previous approaches, and may be generalizable. New, two-parameter, sine-Gordon defects are also described, which have characteristics resembling a pair of 'fused' defects of a previously considered type. The relationship between these defects and Bäcklund transformations is described and a Hamiltonian description of integrable defects is proposed.

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Section: Introductionsupporting

confidence: 74%

“…(5.12) Inevitably, this depends on the fields u and v. The Bäcklund transformation (5.8)-(5.11) seems not to have been reported elsewhere in the literature [14][15][16].…”

Section: Defects and Bäcklund Transformationsmentioning

confidence: 97%

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A new class of integrable defects

Corrigan¹,

Zambon²

2009

J. Phys. A: Math. Theor.

158

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“…Taking γ= 1, we could study the realizations of the vector fields e i , f i , Z , A in an infinite‐dimensional “q‐space” with coordinates following the procedure in Ref. [6].…”

Section: Classification Of the Algebramentioning

confidence: 99%

Stuart Vortices Extended to a Sphere Admits Infinite‐Dimensional Generalized Symmetries

Ifidon

2010

Stud Appl Math

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To generalize Stuart vortices to the surface of a sphere, Crowdy recently obtained a nonlinear partial differential equation (involving a parameter γ ) that has no known solution except for γ = 0 for which he gives a solution [1]. γ = 0 is therefore, assumed to be a kind of "solvability" condition for the equation. In this paper, to examine the integrability of this equation, we apply a generalized form of the Wahlquist-Estabrook prolongation procedure given in [2] to the equation for all nonzero values of γ . We see that the generalized symmetries of the Stuart vortices when extended to the surface of a sphere are infinite dimensional for all nonzero values of γ and are isomorphic to A + 1 which becomes the minimal prolongation algebra of the equation.

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