Vector breathers in the Manakov system

Gelash, Andrey; Расковалов, А. А.

doi:10.1111/sapm.12558

Cited by 17 publications

(10 citation statements)

References 78 publications

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“…We highlight that the complete characterization of the breather's parameters benefits the IST analysis of experimental data on coherent structures in optics, hydrodynamics and other nonlinear waveguides described by nearly integrable models [21,32,34,77,78]. Also, our approach can be generalized to the case of vector breathers of the Manakov system [79][80][81][82][83] and other integrable systems with constant amplitude boundary conditions.…”

Section: Discussionmentioning

confidence: 99%

Numerical direct scattering transform for breathers

Mullyadzhanov,

Gudko,

Mullyadzhanov

et al. 2024

Proc. R. Soc. A.

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We consider the model of the focusing one-dimensional nonlinear Schrödinger equation (fNLSE) in the presence of an unstable constant background, which exhibits coherent solitary wave structures—breathers. Within the inverse scattering transform (IST) method, we study the problem of the scattering data numerical computation for a broad class of breathers localized in space. Such a direct scattering transform (DST) procedure requires a numerical solution of the auxiliary Zakharov–Shabat system with boundary conditions corresponding to the background. To find the solution, we compute the transfer matrix using the second-order Boffetta–Osborne approach and recently developed high-order numerical schemes based on the Magnus expansion. To recover the scattering data of breathers, we derive analytical relations between the scattering coefficients and the transfer matrix elements. Then we construct localized single- and multi-breather solutions and verify the developed numerical approach by recovering the complete set of scattering data with the built-in accuracy providing the information about the amplitude, velocity, phase and position of each breather. To combine the conventional IST approach with the efficient dressing method for multi-breather solutions, we derive the exact relation between the parameters of breathers in these two frameworks.

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Section: Discussionmentioning

confidence: 99%

Numerical direct scattering transform for breathers

Mullyadzhanov,

Gudko,

Mullyadzhanov

et al. 2024

Proc. R. Soc. A.

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“…Finding conditions when this continuity is satisfied is an interesting problem for future studies. We think that our method can be generalized straightforwardly to multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems including vector breathers, [40][41][42][43][44][45][46] making it possible to model the rich dynamics of complex breather interactions 24,62,63,[69][70][71][72] and the behavior of breathers in nearly integrable systems. 73,74 We assume that for such a generalization one will only need to find the solitonic model for the breather background.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…We believe that our method can be applied straightforwardly to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems including vector breathers. [40][41][42][43][44][45][46] The paper is organized as follows. In Section 2, we discuss the DM procedure, the multisoliton and multibreather solutions, and the solitonic model of the plane wave.…”

Section: Introductionmentioning

confidence: 99%

“…At large N , the constructed solutions are practically indistinguishable from breathers in a wide region of space and time. We believe that our method can be applied straightforwardly to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems including vector breathers 40–46 …”

Section: Introductionmentioning

confidence: 99%

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Multisoliton interactions approximating the dynamics of breather solutions

Agafontsev,

Gelash,

Randoux

et al. 2023

Stud Appl Math

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Nowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N‐soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super‐regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies.

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“…Breathers are applied in such physical systems as ocean, plasma, Bose-Einstein condensates, and optical fibers [6][7][8][9][10]. Certain experiments have verified the presence of solitons and breathers in nature, inspiring theorists to predict novel cases for their propagation and interactions [11,12].…”

Section: Introductionmentioning

confidence: 99%

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Qin,

2024

Phys. Scr.

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Utilizing the Riemann-Hilbert approach, we study the inverse scattering transformation, as well as multi-pole solitons and breathers, for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions at infinity. Beginning with the Lax pair, we introduce the uniformization variable to simplify both the direct and inverse problems on the two-sheeted Riemann surface. In the direct scattering problem, we systematically demonstrate the analyticity, asymptotic behaviors and symmetries of the Jost functions and scattering matrix. By solving the corresponding matrix Riemann-Hilbert problem, we work out the multi-pole solutions expressed as determinants for the reflectionless potential. Based on the parameter modulation, the dynamical properties of the simple-, double- and triple-pole solutions are investigated. In the defocusing cases, we show abundant simple-pole solitons including dark solitons, anti-dark-dark solitons, double-hump solitons, as well as double- and triple-pole solitons. In addition, the asymptotic expressions for the double-pole soliton solutions are presented. In the focusing cases, we illustrate the propagations of simple-pole, double-pole, and triple-pole breathers. Furthermore, the multi-pole breather solutions can be reduced to the bright soliton solutions for the focusing nonlocal Lakshmanan-Porsezian-Daniel equation.

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Vector breathers in the Manakov system

Cited by 17 publications

References 78 publications

Numerical direct scattering transform for breathers

Numerical direct scattering transform for breathers

Multisoliton interactions approximating the dynamics of breather solutions

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

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