Integrable semi-discretization of the massive Thirring system in laboratory coordinates

Joshi, Nalini; Pelinovsky, Dmitry E.

doi:10.1088/1751-8121/aaf2c2

Cited by 6 publications

(4 citation statements)

References 50 publications

(185 reference statements)

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“…These semi-discretizations are not relevant for the time-evolution problem related to the MTM in laboratory coordinates. It was only recently [11] when the integrable semi-discretization of the MTM in laboratory coordinates was derived. The corresponding semi-discrete MTM is written as the following system of three coupled equations:…”

Section: Introductionmentioning

confidence: 99%

“…It is shown in [11] that the semi-discrete MTM system (1) is the compatibility condition (4) d dt N n (λ) = P n+1 (λ)N n (λ) − N n (λ)P n (λ), of the following Lax pair of two linear equations:…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, since the semi-discrete MTM system (1) has the Lax pair of linear equations (5), it is integrable by the inverse scattering transform method which implies existence of infinitely many conserved quantities, exact solutions, transformations between different solutions, and reductions to other integrable equations [10]. These properties of integrable systems were not explored for the semi-discrete MTM system (1) in the previous work [11].…”

Section: Introductionmentioning

confidence: 99%

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Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

Pelinovsky

2019

Physics Letters A

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

Pelinovsky

2019

Physics Letters A

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“…37,38 Most recently, Pelinovsky and Joshi proposed a semidiscrete integrable MT model in laboratory coordinates and studied its soliton solution via Darboux transformation. 39,40 Recently, by combining the Hirota's bilinear method 41 and the Kadomtsev-Petviashvili (KP) hierarchy reduction method, 42 we have constructed general soliton solutions to many soliton equations such as the vector nonlinear Schrödiner equation, 43 the complex short pulse equation, 44,45 and the derivative Yajima-Oikawa system 46 for both the vanishing boundary condition (VBC) and nonvanishing boundary condition (NVBC). Therefore, the motivation of the present work is to bilinearize the MT model and find its soliton solutions under VBC and NVBC.…”

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confidence: 99%

Tau‐function formulation for bright, dark soliton and breather solutions to the massive Thirring model

Chen

Feng

2022

Stud Appl Math

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In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two‐component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single‐component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one‐ and two‐soliton for bright, dark soliton and breather solutions are analyzed in details.

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The massive Thirring system in the quarter plane

Xia¹

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The unified transform method (UTM) for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM in general is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model posed in the quarter plane. We show for this integrable model, the UTM is as effective as the IST method: the Riemann-Hilbert (RH) problems we formulated for such a problem have explicit (x, t)dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.Keywords: massive Thirring system, initial-boundary value problem, unified transform method, inverse scattering transform. Transformations of the Lax pairFollowing [17], we transform the Lax pair (1.2) with the spectral parameter λ to two equivalent forms: one, with the new spectral parameter k, is suitable to study the behaviour for λ near origin, and the other one, with the new spectral parameter z, is suitable to study the behaviour for λ near infinity. More precisely, we introduce the following two transformationsand we introduce the new spectral parameterswhere the jump matrix J (x, t, z) and the oriented contour L = L 1 ∪ L 2 ∪ L 3 ∪ L 4 are defined as follows 47)Using the global relation (3.52) we can verify directly that {Ω (j) (t, k)} 4 1 satisfy J 1 (0, t, k)Ω (2) (t, k) = Ω (1) (t, k)J (t) (t, k), k ∈ L 1 , J 2 (0, t, k)Ω (2) (t, k) = Ω (3) (t, k)J (t) (t, k), k ∈ L 2 , J 3 (0, t, k)Ω (4) (t, k) = Ω (3) (t, k)J (t) (t, k), k ∈ L 3 , J 4 (0, t, k)Ω (4) (t, k) = Ω (1) (t, k)J (t) (t, k), k ∈ L 4 .(4.27)

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Integrable semi-discretization of the massive Thirring system in laboratory coordinates

Cited by 6 publications

References 50 publications

Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

Tau‐function formulation for bright, dark soliton and breather solutions to the massive Thirring model

The massive Thirring system in the quarter plane

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