Formation of rogue waves from a locally perturbed condensate

Gelash, Andrey

doi:10.1103/physreve.97.022208

Cited by 60 publications

(67 citation statements)

References 37 publications

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“…The spectral curve Γ 0 corresponding to the background (2) is rational, and a point γ ∈ Γ 0 is a pair of complex numbers γ = (λ, µ) satisfying the quadratic equation µ 2 = λ 2 + |a| 2 . The corresponding monodromy matrix: tr T 0 (λ) = 2 cos(µL) (37) defines the branch points (λ ± 0 , µ 0 ) = (±i|a|, 0) and the resonant (double) points (λ ± n , µ n ) = (± (nπ/L) 2 − |a| 2 , nπ/L), n ∈ Z, n = 0. Near the resonant points:…”

Section: Finite-gap Approximation Of the Aw Cauchy Problemmentioning

confidence: 99%

“…Concerning the NLS Cauchy problems in which the initial condition consists of a perturbation of the exact background (2), what we call the Cauchy problem of the AWs, if such a perturbation is localized, then slowly modulated periodic oscillations described by the elliptic solution of (1) play a relevant role in the longtime regime [16,17]. The relevance of the Kuznetsov -Kawata -Inoue -Ma solitons and of the superregular solitons (constructed by Zakharov and Gelash [93], see also [94], [95]) in this problem was investigated in [37].…”

Section: Introductionmentioning

confidence: 99%

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Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

2020

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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. Using the finite gap method, two of us (PGG and PMS) have recently solved, to leading order and in terms of elementary functions of the initial data, 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 of unstable modes. In this paper, concentrating on the simplest case of a single unstable mode, we study the periodic Cauchy problem of the AWs for the NLS equation perturbed by a linear loss or gain term. Using the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic model describing quantitatively how the solution evolves, after a suitable transient, into slowly varying lower dimensional patterns (attractors) in the (x, t) plane, characterized by ∆X = L/2 in the case of loss, and by ∆X = 0 in the case of gain, where ∆X is the x-shift of the position of the AW during the recurrence, and L is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces O(1) effects on the periodic AW dynamics, generating the slowly varying loss/gain attractors analytically described in this paper, we expect that these attractors, together with their generalizations corresponding to more unstable modes, will play a basic role in the theory of periodic AWs in nature.

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Section: Finite-gap Approximation Of the Aw Cauchy Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

2020

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“…These explicit expressions come from the exact SR breather solution (see Appendix A), where V grj is extracted from the hyperbolic function cosh Θ j , while V phj is extracted from the trigonometric function cos Φ j . If the higher-order terms are absent, α n>2 = 0, V grj , V phj reduce to their values for the simplest NLSE [18][19][20][21]. Figure 1 shows the evolution of V grj , V phj with q.…”

Section: Sr Breather Propertymentioning

confidence: 99%

Growth rate of modulation instability driven by superregular breathers

Liu

Yang

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schrödinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.

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“…Just as solitons, breathers may collide with each other creating complex interference patterns. They reveal high amplitude peaks when the breathers are synchronised [26,35,36], similar to the case of the soliton synchronisation [37,38].…”

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confidence: 66%

Chessboard-like spatio-temporal interference patterns and their excitation

Liu

Yang

et al. 2019

J. Opt. Soc. Am. B

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We discover new type of interference patterns generated in the focusing nonlinear Schrödinger equation (NLSE) with localised periodic initial conditions. At special conditions, found in the present work, these patterns exhibit novel chess-board-like spatio-temporal structures which can be observed as the outcome of collision of two breathers. The infinitely extended chess-board-like patterns correspond to the continuous spectrum bands of the NLSE theory. More complicated patterns can be observed when the initial condition contains several localised periodic swells. These patterns can be observed in a variety of physical situations ranging from optics and hydrodynamics to Bose-Einstein condensates and plasma.

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Formation of rogue waves from a locally perturbed condensate

Cited by 60 publications

References 37 publications

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

Growth rate of modulation instability driven by superregular breathers

Chessboard-like spatio-temporal interference patterns and their excitation

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