A degeneration of two-phase solutions of the focusing nonlinear Schrödinger equation via Riemann-Hilbert problems

Bertola, Marco; Giavedoni, Pietro

doi:10.1063/1.4922362

Cited by 8 publications

(9 citation statements)

References 18 publications

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“…On the other hand, even if an explicit formula for a solution of a nonlinear equation is given, the Riemann-Hilbert formalism can be efficiently used to describe certain limiting regimes, in terms of the solution of an appropriately deformed RHP. According to this approach, a representation of two-phase solutions of the focusing NLS equation in terms of a Riemann-Hilbert problem in the plane is used in [4] in order to describe the limiting regime, where some arcs shrink to points. The limiting RH problem has the reduced number of jump arcs, which simplifies the derivation and representation of the limiting solution.…”

Section: Remarkmentioning

confidence: 99%

Planar unimodular Baker-Akhiezer function for the nonlinear Schrödinger equation

Kotlyarov

Shepelsky

2017

Annals of Mathematical Sciences and Applications

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Section: Remarkmentioning

confidence: 99%

Planar unimodular Baker-Akhiezer function for the nonlinear Schrödinger equation

Kotlyarov

Shepelsky

2017

Annals of Mathematical Sciences and Applications

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“…We have some progress in studying of a mixed problem where we come to a necessity of using of the declared matrix BA function. We believe that results of the paper will be useful for further development of the results obtained, for example, in [27,40,42,52,53] and for an investigation of rogue waves (about them see, e.g., [4,5,28,55]) to the Maxwell-Bloch equations.…”

Section: Introductionmentioning

confidence: 80%

A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions

Kotlyarov¹

2018

SIGMA

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The Baker-Akhiezer (BA) function theory was successfully developed in the mid 1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and theory of completely integrable nonlinear equations such as Korteweg-de Vries equation, nonlinear Schrödinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation. Subsequently the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch (MB) system or for the Karpman-Kaup equations, which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency, are unknown. The main goal of the paper is to give a such of extension associated with the Maxwell-Bloch equations. Using different Riemann-Hilbert problems posed on the complex plane with a finite number of cuts we propose such a matrix function that has unit determinant and takes an explicit form through Cauchy integrals, hyperelliptic integrals and theta functions. The matrix BA function solves the AKNS equations (the Lax pair for MB system) and generates a quasi-periodic finite-gap solution to the Maxwell-Bloch equations. The suggested function will be useful in the study of the long time asymptotic behavior of solutions of different initial-boundary value problems for the MB equations using the Deift-Zhou method of steepest descent and for an investigation of rogue waves of the Maxwell-Bloch equations.

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“…(8) with multiplicity of 2 as opposed to nondegenerate points with multiplicity 1) pairs, λ 2 , λ * 2 , λ 3 and λ * 3 . For the case when λ 2 approaches λ 3 , the periodic signal resulting from such an NS can be written as [21]:…”

Section: Signals With Known Nsmentioning

confidence: 99%

Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication

Kamalian¹,

Prilepsky²,

Le³

et al. 2016

Opt. Express

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In this paper we propose the design of communication systems based on using periodic nonlinear Fourier transform (PNFT), following the introduction of the method in the Part I. We show that the famous "eigenvalue communication" idea [A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993)] can also be generalized for the PNFT application: In this case, the main spectrum attributed to the PNFT signal decomposition remains constant with the propagation down the optical fiber link. Therefore, the main PNFT spectrum can be encoded with data in the same way as soliton eigenvalues in the original proposal. The results are presented in terms of the bit-error rate (BER) values for different modulation techniques and different constellation sizes vs. the propagation distance, showing a good potential of the technique.

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A degeneration of two-phase solutions of the focusing nonlinear Schrödinger equation via Riemann-Hilbert problems

Cited by 8 publications

References 18 publications

Planar unimodular Baker-Akhiezer function for the nonlinear Schrödinger equation

Planar unimodular Baker-Akhiezer function for the nonlinear Schrödinger equation

A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions

Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication

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