Darboux transformation, positons and general superposition formula for the sine-Gordon equation

Andreev, V.; Brezhnev, Yu. V.

doi:10.1016/0375-9601(95)00663-n

Cited by 20 publications

(11 citation statements)

References 15 publications

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“…A second 1-Darboux transformation solution s 2 [1] is considered using the constraint a 0 = 0 on solution (31) and (32), which leads to the particular solution…”

Section: Explicit Solutions Used For the Bosonic Supersymmetric Sym-tmentioning

confidence: 99%

On integrability aspects of the supersymmetric sine-Gordon equation

Bertrand

2017

J. Phys. A: Math. Theor.

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Abstract. In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zerocurvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-Bäcklund transformation of the supersymmetric sine-Gordon equation.In addition, a detailed description of the associated Darboux transformation is presented and non-trivial super multisoliton solutions are constructed. These integrability properties allow us to provide new explicit geometric characterizations of the bosonic supersymmetric version of the Sym-Tafel formula for the immersion of surfaces in a Lie superalgebra. These characterizations are expressed only in terms of the independent bosonic and fermionic variables.

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“…A second 1-Darboux transformation solution s 2 [1] is considered using the constraint a 0 = 0 on solution (31) and (32), which leads to the particular solution…”

Section: Explicit Solutions Used For the Bosonic Supersymmetric Sym-tmentioning

confidence: 99%

On integrability aspects of the supersymmetric sine-Gordon equation

Bertrand

2017

J. Phys. A: Math. Theor.

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“…In this chapter, we will combine the general auxiliary linear problem and coordinate reconstruction formula (2.46)-(2.50) with the method of Darboux and Crum transformations [39][40][41][42], in order to produce new string solutions corresponding to multikink configurations of the elliptic (Euclidean) sinh-Gordon, or sinh-Laplace, equation.…”

Section: Spectral Parameter and A Convenient Gaugementioning

confidence: 99%

“…Returning to our problem at hand, we may use the Lax pair equation (2.47) for∂ in order to eliminate all derivatives from the spinor components in (3.22), and in fact it turns out that inside the Wronskian, we can simply replace 19 [40]…”

Section: 22)mentioning

confidence: 99%

“…In light of the potential application open strings may have in the study of the two aforementioned topics, in this note we complement Pohlmeyer reduction in Euclidean worldsheet AdS 3 with a method for constructing new solutions based on Darboux and Crum transformations [39][40][41][42]. Given a known solution of the reduced Lax pair equations, the Darboux transformation maps it algebraically to a new solution, very similarly to how the dressing method [43][44][45] works directly at the level of the sigma model.…”

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confidence: 99%

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Pohlmeyer reduction and Darboux transformations in Euclidean worldsheet AdS 3

Παπαθανασίου

2012

J. High Energ. Phys.

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Pohlmeyer reduction has been instrumental both in the program for computing gluon scattering amplitudes at strong coupling, and more recently in the progress towards semiclassical three-point correlators of heavy operators in AdS/CF T . After a detailed review of the method, we combine it with Darboux and Crum transformations in order to obtain a class of string solutions corresponding to an arbitrary number of kinks and breathers of the elliptic sinh-Gordon equation. We also use our construction in order to identify the previously found dressed giant gluon with the single breather solution.

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“…[1][2][3][4][5] The socalled positon is a fundamental solution of NLEEs on zero background, which was first proposed in the KdV equation. [6] Further, it was extended to other equations such as the Sine-Gordon equation, [7,8] the modified KdV equation, [9,10] the NLS-MB equations. [11] Positon can be viewed as a long-range analog of multi-soliton and slowly decay, [12] which are derived from the multi-soliton by the limits λ 2 j−1 → λ 1 and λ 2 j → λ 2 (λ j is the eigenvalue of Lax pair).…”

Section: Introductionmentioning

confidence: 99%

Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations

Yuan

Ghanbari

2023

Chinese Phys. B

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Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method has been used to derive several positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg-de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied which shows that multi-positons of (2+1)-dimensional equations are decomposed into multi-solitons as well as the (1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.

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Darboux transformation, positons and general superposition formula for the sine-Gordon equation

Cited by 20 publications

References 15 publications

On integrability aspects of the supersymmetric sine-Gordon equation

On integrability aspects of the supersymmetric sine-Gordon equation

Pohlmeyer reduction and Darboux transformations in Euclidean worldsheet AdS 3

Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations

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