Featured Researches

Pattern Formation And Solitons

An analytical stability theory for Faraday waves and the observation of the harmonic surface response

We present an analytical stability theory for the onset of the Faraday instability, applying over a wide frequency range between shallow water gravity and deep water capillary waves. For sufficiently thin fluid layers the surface is predicted to occur in harmonic rather than subharmonic resonance with the forcing. An experimental confirmation of this result is given. PACS: this http URL, 47.20.Gv, 47.15.Cb

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Pattern Formation And Solitons

An integrable shallow water equation with peaked solitons

We derive a new completely integrable dispersive shallow water equation that is biHamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.

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Pattern Formation And Solitons

Analysis of Optimal Velocity Model with Explicit Delay

We analyze Optimal Velocity Model (OVM) with explicit delay. The properties of congestion and the delay time of car motion are investigated by analytical and numerical methods. It is shown that the small explicit delay time has almost no effects. In the case of the large explicit delay time, a new phase of congestion pattern of OVM seems to appear.

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Pattern Formation And Solitons

Annular electroconvection with shear

We report experiments on convection driven by a radial electrical force in suspended annular smectic A liquid crystal films. In the absence of an externally imposed azimuthal shear, a stationary one-dimensional (1D) pattern consisting of symmetric vortex pairs is formed via a supercritical transition at the onset of convection. Shearing reduces the symmetries of the base state and produces a traveling 1D pattern whose basic periodic unit is a pair of asymmetric vortices. For a sufficiently large shear, the primary bifurcation changes from supercritical to subcritical. We describe measurements of the resulting hysteresis as a function of the shear at radius ratio η∼0.8 . This simple pattern forming system has an unusual combination of symmetries and control parameters and should be amenable to quantitative theoretical analysis.

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Pattern Formation And Solitons

Approximate solutions and scaling transformations for quadratic solitons

We study quadratic solitons supported by two- and three-wave parametric interactions in chi-2 nonlinear media. Both planar and two-dimensional cases are considered. We obtain very accurate, 'almost exact', explicit analytical solutions, matching the actual bright soliton profiles, with the help of a specially-developed approach, based on analysis of the scaling properties. Additionally, we use these approximations to describe the linear tails of solitary waves which are related to the properties of the soliton bound states.

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Pattern Formation And Solitons

Asymptotic description of transients and synchronized states of globally coupled oscillators

A two-time scale asymptotic method has been introduced to analyze the multimodal mean-field Kuramoto-Sakaguchi model of oscillator synchronization in the high-frequency limit. The method allows to uncouple the probability density in different components corresponding to the different peaks of the oscillator frequency distribution. Each component evolves toward a stationary state in a comoving frame and the overall order parameter can be reconstructed by combining them. Synchronized phases are a combination of traveling waves and incoherent solutions depending on parameter values. Our results agree very well with direct numerical simulations of the nonlinear Fokker-Planck equation for the probability density. Numerical results have been obtained by finite differences and a spectral method in the particular case of bimodal (symmetric and asymmetric) frequency distribution with or without external field. We also recover in a very easy and intuitive way the only other known analytical results: those corresponding to reflection-symmetric bimodal frequency distributions near bifurcation points.

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Pattern Formation And Solitons

Asymptotic expansion for layer solutions of a singularly perturbed reaction-diffusion system

For a singularly perturbed system of reaction--diffusion equations, assuming that the 0th order solutions in regular and singular regions are all stable, we construct matched asymptotic expansions for formal solutions to any desired order in ϵ . The formal solution shows that there is an invariant manifold of wave-front-like solutions that attracts other nearby solutions. With an additional assumption on the sign of the wave speed, the wave-front-like solutions converge slowly to stable stationary solutions on that manifold.

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Pattern Formation And Solitons

Attracting Manifold for a Viscous Topology Transition

An analytical method is developed describing the approach to a finite-time singularity associated with collapse of a narrow fluid layer in an unstable Hele-Shaw flow. Under the separation of time scales near a bifurcation point, a long-wavelength mode entrains higher-frequency modes, as described by a version of Hill's equation. In the slaved dynamics, the initial-value problem is solved explicitly, yielding the time and analytical structure of a singularity which is associated with the motion of zeroes in the complex plane. This suggests a general mechanism of singularity formation in this system.

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Pattern Formation And Solitons

Attractive Interaction Between Pulses in a Model for Binary-Mixture Convection

Recent experiments on convection in binary mixtures have shown that the interaction between localized waves (pulses) can be repulsive as well as {\it attractive} and depends strongly on the relative {\it orientation} of the pulses. It is demonstrated that the concentration mode, which is characteristic of the extended Ginzburg-Landau equations introduced recently, allows a natural understanding of that result. Within the standard complex Ginzburg-Landau equation this would not be possible.

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Pattern Formation And Solitons

Bekki-Nozaki Amplitude Holes in Hydrothermal Nonlinear Waves

We present and analyze experimental results on the dynamics of hydrothermal waves occuring in a laterally-heated fluid layer. We argue that the large-scale modulations of the waves are governed by a one-dimensional complex Ginzburg-Landau equation (CGLE). We determine quantitatively all the coefficients of this amplitude equation using the localized amplitude holes observed in the experiment, which we show to be well described as Bekki-Nozaki hole solutions of the CGLE.

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