Soliton Solutions for Discrete Hirota Equation. II

Narita, Kazuaki

doi:10.1143/jpsj.60.1497

Cited by 32 publications

(18 citation statements)

References 3 publications

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“…Understanding various properties of dark solitons in discrete lattices is important from physical point of view. [11][12][13] There are analytical results of the IDNLS equation under non-vanishing boundary condition using the perturbative Hirota method by Narita 14 and inverse scattering theory by Vekslerchik and Konotop. 15 However, the detailed anayisis of the solution and understanding the role of a lattice space parameter in the Ndark soliton solutions are missing in their results.…”

Section: Introductionmentioning

confidence: 99%

Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schrödinger Equation

Maruno

Ohta

2006

J. Phys. Soc. Jpn.

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Section: Introductionmentioning

confidence: 99%

Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schrödinger Equation

Maruno

Ohta

2006

J. Phys. Soc. Jpn.

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“…In the case of the focusing Ablowitz -Ladik model [3] iu nt + u n+1 + u n−1 − 2u n + |u n | 2 (u n+1 + u n−1 ) = 0, n ∈ Z, integrable discretization of focusing NLS, the picture is essentially the same. Indeed, using MAEs and in the case of one unstable mode only, it was shown in [33] that a generic periodic initial datum leads to an AW recurrence of Narita solutions (the Narita solution is the discrete analogue of the Akhmediev breather [81,16]). In the case of the PT-symmetric NLS (PT-NLS) equation [4,5,6,7]…”

Section: Remarkmentioning

confidence: 99%

The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

Grinevich¹,

Santini²

2019

Russ. Math. Surv.

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The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number N of unstable modes. We show how the finite gap method adapts to this specific Cauchy problem through three basic simplifications, allowing one to construct the solution, at the leading and relevant order, in terms of elementary functions of the initial data. More precisely, we show that, at the leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals; ii) in each interval I of the partition, only a subset of N (I) ≤ N unstable modes are "visible", and iii) the NLS solution is approximated, for t ∈ I, by the N (I)-soliton solution of Akhmediev type, describing the nonlinear interaction of these visible unstable modes, whose parameters are expressed in terms of the initial data through elementary functions. This result explains the relevance of the m-soliton solutions of Akhmediev type, with m ≤ N , in the generic periodic Cauchy problem of the AWs, in the case of a finite number N of unstable modes. 1 arXiv:1810.09247v2 [math-ph]

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“…23 Its bright and dark soliton solutions can be found using the Hirota bilinear method. 24,25 Discrete rogue waves in the form of rational solutions of IDHE (3) were given in Ref. 26.…”

Section: The Hirota Equation (He)mentioning

confidence: 99%

“…34 It should be remarked that we introduce the term e 2αit in the transformation (5) to get a beautiful bilinear form of ICDHE, which has been previously used to solve the discrete nonlinear Schrödinger equation. 25 Using the perturbation method, we write the power series expansion of f n and g…”

Section: Coupled Bright-bright Soliton Interaction In the Icdhementioning

confidence: 99%

Solitons and dynamic properties of the coupled semidiscrete Hirota equation

Zhao

Zhu

2013

AIP Advances

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In this paper, we obtain the coupled bright-bright and the coupled dark-dark soliton solutions for an integrable coupled semidiscrete Hirota equation (ICDHE) using the bilinear method. These solutions have both singular and nonsingular behaviors depending upon the choice of the parameters. The distinguishing feature of the ICDHE is that the singular solution can possess some nonsingular characteristics in the discrete space at some time. The asymptotic behavior of these coupled bright-bright and the coupled dark-dark solitons is studied in order to explain their collision properties.

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Soliton Solutions for Discrete Hirota Equation. II

Cited by 32 publications

References 3 publications

Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schrödinger Equation

Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schrödinger Equation

The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

Solitons and dynamic properties of the coupled semidiscrete Hirota equation

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