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

Xu, Tao; Pelinovsky, Dmitry E.

doi:10.1016/j.physleta.2019.125948

Cited by 22 publications

(5 citation statements)

References 30 publications

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“…We admit that the resolution of numerical data is poor near the origin on the right panels of Figure 5 because resolution of the Lax spectrum is poor near the points ±1 on the left panels. It is likely that sensitivity of numerical detected eigenvalues is related to evaluating the square roots in (35) and (39) near 𝑧 = ±2. To obtain the rogue waves, as we do in Section 6 by using the analytical theory, we will consider here eigenfunctions of the linear system ( 21)-( 22) for the eigenvalue 𝜆 = 𝜆 1 given by a root of the polynomial 𝑃(𝜆) in (29).…”

Section: Cnoidal Wavementioning

confidence: 99%

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

Chen,

Pelinovsky

2023

Stud Appl Math

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We study the standing periodic waves in the semidiscrete integrable system modeled by the Ablowitz–Ladik (AL) equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a nonstandard linear Lax system, which is different from the standard Lax representation of the AL equation.

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Section: Cnoidal Wavementioning

confidence: 99%

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

Chen,

Pelinovsky

2023

Stud Appl Math

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“…Therefore, our first task is to extend the 1-fold DT to the eigenfunctions of the Lax system (2.1). We achieve the task with direct computations similarly to computations in [37] and [11].…”

Section: Construction Of Rogue Wavesmentioning

confidence: 99%

Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation

Chen

Pelinovsky

2021

Phys. Rev. E

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We study the standing periodic waves in the semi-discrete integrable system modelled by the Ablowitz-Ladik equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a non-standard linear Lax system, which is different from the standard Lax representation of the Ablowitz-Ladik equation.This article is dedicated to Athanassios S. Fokas on the occasion of his 70th anniversary for his many contributions to studies of integrable nonlinear PDEs and boundary-value problems.

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“…Regarding the integrable discretization, Nijhoff et al gave the integrable discretization of the MT model in light cone coordinates [36,37] in 1980s. Most recently, Pelinovsky et al proposed a semi-discrete integrable MT model in laboratory coordinates [38] and studied its solution solution via Darboux transformation [39]. Surprisingly, as far as we are aware, the bilinear formulation is missing in the literature.…”

Section: Introductionmentioning

confidence: 99%

General bright and dark soliton solutions to the massive Thirring model via KP hierarchy reductions

Chen¹,

Feng²

2021

Preprint

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In the present paper, we are concerned with the tau function and its connection with the Kadomtsev-Petviashvili (KP) theory for the massive Thirring (MT) model. First, we bilinearize the massive Thirring model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP-Toda hierarchy, we derive the multi-bright solution to the MT model by the KP hierarchy reductions. Then, we show that the discrete KP equation can generate a set of bilinear equations of a deformed KP-Toda hierarchy through Miwa transformation. By imposing constraints on the parameters of the tau function, the general dark soliton solution to the MT model is constructed from the tau function of the discrete KP equation. Finally, the dynamics and properties of oneand two-soliton for both the bright and dark cases are analyzed in details.

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

Cited by 22 publications

References 30 publications

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation

General bright and dark soliton solutions to the massive Thirring model via KP hierarchy reductions

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