Energy portrait of rogue waves

Захаров, В. Е.; Shamin, R. V.; Yudin, A. V.

doi:10.1134/s0021364014090136

Cited by 4 publications

(3 citation statements)

References 11 publications

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“…The system is separately considered according to both "conformable" and "M− truncated" derivative operators and the obtained solutions will be compared. The system is a modeling of the spread of the Langmuir waves in the Plasma in ionize plasma [39]. Many theoretical and practical studies are performed on the system.…”

Section: Introductionmentioning

confidence: 99%

A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

Cinar,

Onder,

Secer

et al. 2024

Preprint

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In this paper, the conformable and M− truncated nonlinear complex generalized Zakharov dynamical system are considered. The exact solutions of the dynamical system for different two types of derivatives are derived by using the extended rational sine−cosine and sinh−cosh methods. We make a comparison between the solutions of the Zakharov system with the conformable derivative and the system with M− truncated derivative. The results are depicted in the 2D and 3D graphics. All computations and graphics are fulfilled with the help of Wolfram Mathematica 12. The methods can be successfully applied to get soliton solutions of PDEs (or PDE systems) with the different types of derivatives.

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Section: Introductionmentioning

confidence: 99%

A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

Cinar,

Onder,

Secer

et al. 2024

Preprint

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show abstract

“…We consider the Zakharov system (ZS) in d (d = 1, 2, 3) dimensions for describing the propagation of Langmuir waves in plasma [1][2][3][4],…”

Section: Introductionmentioning

confidence: 99%

“…We consider the Zakharov system (ZS) in

d

(

d&amp;#x0003D;1,2,3

) dimensions for describing the propagation of Langmuir waves in plasma [1–4],

i{\partial}_t{E}&amp;#x0005E;{\varepsilon}\left(\mathbf{x},t\right)&amp;#x0002B;{\nabla}&amp;#x0005E;2{E}&amp;#x0005E;{\varepsilon}\left(\mathbf{x},t\right)-{N}&amp;#x0005E;{\varepsilon}\left(\mathbf{x},t\right){E}&amp;#x0005E;{\varepsilon}\left(\mathbf{x},t\right)&amp;#x0003D;0,\mathbf{x}\in {\mathrm{\mathbb{R}}}&amp;#x0005E;d,t&gt;0,

{\varepsilon}&amp;#x0005E;2{\partial}_{tt}{N}&amp;#x0005E;{\varepsilon}\left(\mathbf{x},t\right)-{\nabla}&amp;#x0005E;2{N}&amp;#x0005E;{\varepsilon}\left(\mathbf{x},t\right)-{\nabla}&amp;#x0005E;2{\left&amp;#x0007C;{E}&amp;#x0005E;{\varepsilon}\Big(\mathbf{x},t\Big)\right&amp;#x0007C;}&amp;#x0005E;2&amp;#x0003D;0,\mathbf{x}\in {\mathrm{\mathbb{R}}}&amp;#x0005E;d,t&gt;0,

E^{ε} false(boldx, 0 false) =

…”

Section: Introductionunclassified

Uniform error bound of a Crank–Nicolson‐type finite difference scheme for Zakharov system in the subsonic limit regime

Wang

Yang

2023

Math Methods in App Sciences

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In this paper, we use the order reduction method to present a Crank–Nicolson‐type finite difference scheme for Zakharov system (ZS) with a dimensionless parameter ε∈false(0,1false]$$ \varepsilon \in \left(0,1\right] $$, which is inversely proportional to the ion acoustic speed. The proposed scheme is proved to perfectly inherit the mass and energy conservation possessed by ZS, while the invariants satisfied by most existing schemes are expressed by two‐level's solution at each time step. In the subsonic limit regime, that is, when 0<ε≪1$$ 0<\varepsilon \ll 1 $$, the solution of ZS propagates rapidly oscillatory initial layers in time, and this brings significant difficulties in designing numerical methods and establishing the error estimates, especially in the subsonic limit regime. After proving the solvability of the proposed scheme, we use the cut‐off function technique and energy method to rigorously analyze two independent error estimates for the well‐prepared, less‐ill‐prepared, ill‐prepared initial data, respectively, which are uniform in both time and space for ε∈false(0,1false]$$ \varepsilon \in \left(0,1\right] $$ and optimal at second order in space. Numerical examples are carried out to verify the theoretical results and show the effectiveness of the proposed scheme.

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Roadmap on optical rogue waves and extreme events

Akhmediev

Kibler

Baronio

et al. 2016

J. Opt.

256

134

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The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.

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Energy portrait of rogue waves

Cited by 4 publications

References 11 publications

A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

Uniform error bound of a Crank–Nicolson‐type finite difference scheme for Zakharov system in the subsonic limit regime

Roadmap on optical rogue waves and extreme events

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