Formation of Saturated Solitons in a Nonlinear Dispersive System with Instability and Dissipation

Kawahara, Takuji

doi:10.1103/physrevlett.51.381

Cited by 205 publications

(115 citation statements)

References 8 publications

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“…(2) will change L s on a slow time scale, until they balance each other (this essential role of the small dissipative terms was revealed in Ref. [9] for the 1D case). This selects the soliton of L s ∼ 1, which results in c ∼ 1 and η ∼ 1 as well, independent of ǫ.…”

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confidence: 99%

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Mutually penetrating motion of self-organized two-dimensional patterns of solitonlike structures

Indireshkumar

Frenkel

1997

Phys. Rev. E

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Results of numerical simulations of a recently derived most general dissipative-dispersive PDE describing evolution of a film flowing down an inclined plane are presented. They indicate that a novel complex type of spatiotemporal patterns can exist for strange attractors of nonequilibrium systems. It is suggested that real-life experiments satisfying the validity conditions of the theory are possible: the required sufficiently viscous liquids are readily available.PACS numbers: 05.45.+b, 47.20.Ky, 03.40Gc The phenomenon of pattern formation in nonequilibrium dissipative systems is currently a topic of active experimental and theoretical research (see e.g.[1] for a recent progress review). Here we report our theoretical studies and numerical simulations of a two-dimensional (2D) evolution PDE approximating a flow down an inclined plane; it exhibits self-organization of a remarkably complex spatiotemporal pattern which then persists indefinetly in this dissipative-dispersive system. In certain cases discussed below, such a pattern consists of two subpatterns of two-dimensionally (2D) localized surface structures. One of these subpatterns is an essentially 1D arrangement of larger-amplitude bulges on the film surface which are nearly equidistantly aligned on (a number of) straight-line segments; those are surrounded by smaller-amplitude bumps, which constitute the second, lattice-like subpattern filling up essentially the entire flow domain. Each of the two subpatterns moves as a whole; their velocities are different, and every elementary structure (a bulge as well as a bump) periodically collides with those of the other kind. In the collision of a bump with a bulge (or with a pair of neighboring bulges), the two structures pass through each other similar to the wellknown 1D Korteweg-deVries solitons, returning to their pre-collisional shapes and speeds after the interaction.Studies of wavy film flows on solid surfaces (the "Kapitza problem") have a considerable history. However, the nonlinear dynamics of wavy films is far from being fully understood (see e.g. [2]; see [3] and [4] for recent progress reviews). Fortunately, the nonlinear coupled-PDE Navier-Stokes (NS) problem, additionally complicated with a free boundary, can be reduced to simpler approximate descriptions of the wave dynamics for certain domains of the parameter space. In the most favorable cases, such a description reduces to a single partial differential equation (PDE) governing the evolution of film thickness. Recently, we (see [3]) have derived the most general evolution equation (EE) capable of all-time-valid description of a wavy liquid film (of a constant density ρ, kinematic viscosity ν, surface tension σ, and average thickness h 0 ) flowing down an inclined plane. Its dimensionless form isHere η is the deviation of film thickness from its average value of 1, z and y are the streamwise and spanwise coordinates, and ∇ 2 = ∂ 2 /∂z 2 + ∂ 2 /∂y 2 . We have defined δ ≡ (4R/5 − cot θ), where θ is the angle of inclination of the plane and R = ...

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“…(2) is essentially the well-studied (see e.g. references in [8]) equation first introduced by Kawahara [9].…”

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confidence: 99%

Mutually penetrating motion of self-organized two-dimensional patterns of solitonlike structures

Indireshkumar

Frenkel

1997

Phys. Rev. E

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“…For sufficiently small dispersion, it exhibits spatio-temporal chaos while sufficiently large dispersion "regularizes" the solution in favor of a train of solitary pulses which continuously interact with each other [4]. Such "dissipative solitons", as they are referred to by Christov and Velarde [4], are due to a precise balance between nonlinearity, energy supply at long wavelengths, and energy dissipation at short ones and they appear in many different contexts [5]. Coherent-structures theories have been formulated based on the idea of weak interaction between neighboring structures, e.g., Ref.…”

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confidence: 99%

“…This equation has been postulated in the literature as a prototype for the study of pattern-formation dynamics and spatio-temporal complexity in active dispersivedissipative media. For sufficiently small dispersion, it exhibits spatio-temporal chaos while sufficiently large dispersion "regularizes" the solution in favor of a train of solitary pulses which continuously interact with each other [4]. Such "dissipative solitons", as they are referred to by Christov and Velarde [4], are due to a precise balance between nonlinearity, energy supply at long wavelengths, and energy dissipation at short ones and they appear in many different contexts [5].…”

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confidence: 99%

Liquid Film Coating a Fiber as a Model System for the Formation of Bound States in Active Dispersive-Dissipative Nonlinear Media

et al. 2009

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We analyze coherent-structures interaction and formation of bound states in active dispersivedissipative nonlinear media using a viscous film coating a vertical fiber as a prototype. The coherent structures in this case are drop-like pulses that dominate the evolution of the film. We study experimentally the interaction dynamics and show evidence for formation of bound states. A theoretical explanation is provided through a coherent structures theory of a simple model for the flow.

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“…The Benney equation thus interpolates between chaotic and regular patterns. It was studied numerically under a periodic boundary condition by Kawahara and coworkers [27]. When d is sufficiently large (jdj .…”

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Control of Chaotic Wandering of an Isolated Step by the Drift of Adatoms

1998

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The drift of adatoms strongly influences the wandering pattern of an isolated step moving in a surface diffusion field. When the drift velocity has a component against the step motion and exceeds a critical value, the straight step becomes unstable with long wavelength fluctuations, and wanders. This wandering pattern can be controlled by changing the direction of the drift. When the drift has no component parallel to the step edge, the unstable step obeys the Kuramoto-Sivashinsky equation and shows a chaotic pattern. When the drift has a component parallel to the step edge, the step obeys the Benney equation. If the parallel component is sufficiently large, the step shows a regular pattern.[ S0031-9007(98)

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Formation of Saturated Solitons in a Nonlinear Dispersive System with Instability and Dissipation

Cited by 205 publications

References 8 publications

Mutually penetrating motion of self-organized two-dimensional patterns of solitonlike structures

Mutually penetrating motion of self-organized two-dimensional patterns of solitonlike structures

Liquid Film Coating a Fiber as a Model System for the Formation of Bound States in Active Dispersive-Dissipative Nonlinear Media

Control of Chaotic Wandering of an Isolated Step by the Drift of Adatoms

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