Aspects of sine-Gordon solitons, defects and gates

Corrigan, E.; Zambon, C.

doi:10.1088/0305-4470/37/37/l03

Cited by 32 publications

(62 citation statements)

References 17 publications

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“…Therefore, as we see below, it opens up the possibility of creation or annihilation of soliton by the defect point, which was prohibited in earlier studies due to consideration of a frozen BT relation [6]. Apart from these solutions, a single soliton suffering a phase shift, while propagating across the defect point, as found earlier [2,6], seems also to be present. Interestingly, the BT expressed through scalar relations (2.2) can be incorporated more efficiently into the machinery of integrable systems by representing it as a gauge transformation relating the Lax pairs of the DSG:…”

Section: Bridging Condition and Lax Pair For The Sg Model With Defectmentioning

confidence: 87%

“…By taking the diagonal part of this matrix equation one getsD 2 = H W 1 + iaD 1 , which using the relationD 2 …”

Section: Extension To Dsg Modelmentioning

confidence: 99%

“…(1.1) u ± tt − u ± xx + sin u ± = 0, for u + = u(x 0, t), u − = u(x 0, t), exhibiting intriguing properties close to integrable systems was investigated in a series of papers [2,6]. These investigations aimed to find out mainly the additional contribution of the defect point to the conserved Hamiltonian and momentum of the system, using the Lagrangian formalism and the effect of the defect point on the soliton solution using the scattering theory.…”

Section: Introductionmentioning

confidence: 99%

“…Our aim here is to establish the suspected integrability of the above DSG [2], by showing the existence of infinite set of its conserved quantities and finding them explicitly. The idea is to adopt the monodromy matrix approach expressed through matrix Riccati equation [7], a true signature of the integrable systems [8], and couple it with the important concept of extending the domain of defect fields u ± through BT [9].…”

Section: Introductionmentioning

confidence: 99%

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Quantum and classical integrable sine-Gordon model with defect

Habibullin

Kundu

2008

Nuclear Physics B

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Defects which are predominant in a realistic model, usually spoil its integrability or solvability. We on the other hand show the exact integrability of a known sine-Gordon field model with a defect (DSG), at the classical as well as at the quantum level based on the Yang-Baxter equation. We find the associated classical and quantum R-matrices and the underlying q-algebraic structures, analyzing the exact lattice regularized model. We derive algorithmically all higher conserved quantities C n , n = 1, 2, . . . , of this integrable DSG model, focusing explicitly on the contribution of the defect point to each C n . The bridging condition across the defect, defined through the Bäcklund transformation is found to induce creation or annihilation of a soliton by the defect point or its preservation with a phase shift.

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Section: Bridging Condition and Lax Pair For The Sg Model With Defectmentioning

confidence: 87%

“…By taking the diagonal part of this matrix equation one getsD 2 = H W 1 + iaD 1 , which using the relationD 2 …”

Section: Extension To Dsg Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum and classical integrable sine-Gordon model with defect

Habibullin

Kundu

2008

Nuclear Physics B

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“…Bäcklund transformations provide the natural setting for describing integrably such descontinuities since it relates two distinct solutions of the same equation [1]. Examples of such situation can be found for the bosonic [2,3,4,5,8,6,7] and supersymmetric [9], [10] field theories. In a previous note [11] the Grassmanian Thirring model with jump defect was considered by writing its Lagrangian density and the conservation of the modified energy and momentum were established.…”

Section: Introductionmentioning

confidence: 99%

Grassmannian and bosonic Thirring models with jump defects

Aguirre

Gomes

Ymai

et al. 2011

J. High Energ. Phys.

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In this paper we discuss the Lax formulation of the Grassmanian and Bosonic Thirring models in the presence of jump defects. For the Grassmanian case, the defect is described by Bäcklund transformation which is responsible for preserving the integrability of the model. We then propose an extension of the Bäcklund transformation for the Bosonic Thirring model which is verified by some Bäcklund transitions like Vacuum-One soliton, One soliton -One soliton, One soliton -Two solitons and Two solitons -Two solitons. The Lax formulation within the space split by the defect leads to the integrability of Bosonic Thirring model.

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A multisymplectic approach to defects in integrable classical field theory

Caudrelier

Kundu

2015

J. High Energ. Phys.

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We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1 + 1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schrödinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen Bäcklund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions.

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Aspects of sine-Gordon solitons, defects and gates

Cited by 32 publications

References 17 publications

Quantum and classical integrable sine-Gordon model with defect

Quantum and classical integrable sine-Gordon model with defect

Grassmannian and bosonic Thirring models with jump defects

A multisymplectic approach to defects in integrable classical field theory

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