&lt;title&gt;Rotational multipetal waves in optical system with 2D feedback&lt;/title&gt;

Razgulin, A. V.

doi:10.1117/12.164778

Cited by 31 publications

(12 citation statements)

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“…(6), replace z im z exp( ( ) ) × ® h , and then equate the forms of the second and third degrees in z, z. Assume m = 0 and n = 0 and use (8), then obtain the following equation in s 2 : …”

Section: Approximate Stationary Solutionsmentioning

confidence: 99%

“…Structurization in these equations has been studied in the mathematical literature since the 1990s. The autowave phenomena were substantiated, for a fixed-angle rotation in a circle or a ring, using the Andronov-Hopf bifurcation theory in [6][7][8], where multipetal rotational waves were described and analyzed for stability. Methods of constructing periodic solutions for an arbitrary domain and nondegenerate smooth transformation were developed in [9][10][11].…”

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confidence: 99%

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Optical buffering in stationary structures

Belan

2008

Cybern Syst Anal

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517.9The local dynamics of a nonlinear parabolic equation on a circle with a shifted spatial argument and a small diffusion is studied. It is proved that the interaction of stationary structures satisfies the 1:3 principle. The number of stable stationary structures increases as the diffusion coefficient tends to zero. INTRODUCTIONRecently, there have been many studies on nonlinear optics, which is due to the wide use of optical systems in information technologies [1]. The key advantages of optical methods for data storage and conversion are parallel signal processing and high performance. Owing to their natural benefits, optical systems are used to create elements of content-addressable memory, pattern recognition systems, and learning analog computers (see [1][2][3], etc.).One of the most popular nonlinear optical systems is a system consisting of a thin layer of nonlinear Kerr-type medium and a two-dimensional feedback loop that can be organized in different ways [1,3]. The fundamental feature of such systems is that the external feedback loop can be used to directly influence the nonlinear dynamics of the system by using prisms, lenses, dynamic holograms, and other devices for controlled transformation of spatial variables. Experiments show that even elementary transformations (rotation, rotation with contraction, reflection) allow the self-organization of a light field to be implemented in a great variety of ways: multipetal rotational waves, optical spirals, stationary structures, switching waves, etc. (see [1,4] and references therein). Note that field stochastization (optical turbulence) was observed in the systems under certain conditions [5, 1].Parabolic functional differential equations with transformed arguments of the unknown function, used to model optical systems with two-dimensional feedback, are a new class of equations for studying the structurization phenomenon. Structurization in these equations has been studied in the mathematical literature since the 1990s. The autowave phenomena were substantiated, for a fixed-angle rotation in a circle or a ring, using the Andronov-Hopf bifurcation theory in [6][7][8], where multipetal rotational waves were described and analyzed for stability. Methods of constructing periodic solutions for an arbitrary domain and nondegenerate smooth transformation were developed in [9][10][11]. The central manifold method was used in [12][13][14][15] to study the bifurcations of rotating structures in a ring and a circle for rotation, and in a circle for rotation together with radial contraction. As shown in [16], a method based on approximate periodic solutions [17] can be used to study bifurcation rotating structures. The bifurcations of stationary structures were studied in [12,18].Parabolic equations with transformed argument and small diffusion are of interest for studying optical turbulence. The method of quasinormal forms for these equations with rotation was applied in [19,20] to describe the dynamics of traveling waves and slowly varying structures. The optical ...

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Section: Approximate Stationary Solutionsmentioning

confidence: 99%

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confidence: 99%

Optical buffering in stationary structures

Belan

2008

Cybern Syst Anal

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“…Such equations arise, for example, in models of nonlinear optics [1][2][3] and also have a purely theoretical value as equations with nonclassical lower-order terms (e.g., see [4][5][6][7][8][9][10][11] and the bibliography therein); recall that the influence of lower-order terms in the parabolic case can be of fundamental importance [12]. We analyze the behavior of the solution of the Cauchy problem for this equation as t → ∞ and show that, unlike the classical case of differential equations (e.g., see [13,14]), for differentialdifference equations the stabilization of solutions is replaced by a more general phenomenon described by theorems on closeness of solutions.…”

Section: Introductionmentioning

confidence: 99%

On the Asymptotics of the Solution of the Cauchy Problem for Some Differential-Difference Parabolic Equations

Muravnik

2005

Diff Equat

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“…Equations of the form (1) arise, for example, in problems of nonlinear optics (see [8][9][10]). The classical solvability of problem (1), (2) was proved and an integral representation of the solution was obtained in [11].…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of the solution of the cauchy problem for some differential-difference parabolic equations

Muravnik

2004

Diff Equat

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<title>Rotational multipetal waves in optical system with 2D feedback</title>

Cited by 31 publications

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Optical buffering in stationary structures

Optical buffering in stationary structures

On the Asymptotics of the Solution of the Cauchy Problem for Some Differential-Difference Parabolic Equations

Uniqueness of the solution of the cauchy problem for some differential-difference parabolic equations

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