Dynamics of counterpropagating waves in parametrically driven systems: dispersion vs. advection

Martel, Carlos; Vega, José M.; Knobloch, Edgar

doi:10.1016/s0167-2789(02)00691-7

Cited by 17 publications

(15 citation statements)

References 20 publications

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“…The boundary condition (4.13) must be imposed because since we are including dispersión, (4.5) is second order in £. As explained by Martel et al (2003) (see also Martel & Vega 1996), the appropriate additional boundary condition is the compatibility condition for the hyperbolic equations obtained when dispersión is ignored, which in our case is precisely (4.13). Equation (4.14a) at £ = +L imposes no net mass flux across the lateral sidewalls.…”

Section: Harmonic Surface Waves At Large Aspect Ratio: L > Dmentioning

confidence: 97%

“…If dispersión is neglected, then the equations become hyperbolic, as in related dissipative systems (first considered by Daniels 1978, see also Martel & Vega 1998), where small diffusive terms lead to subtle effects (Martel & Vega 1996). Similarly, dispersión cannot be ignored a priori in counterpropagating surface waves (Lapuerta, Martel & Vega 2002;Martel, Vega & Knobloch 2003) because dispersive scales can be destabilized. There are two kinds of surface waves, namely harmonic and subharmonic, whose frequency is the forcing frequency and half of the forcing frequency, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Modulated surface waves in large-aspect-ratio horizontally vibrated containers

Varas

Vega

2007

J. Fluid Mech.

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We consider the harmonio and subharmonic modulated surface waves that appear upon horizontal vibration along the surface of the liquid in a two-dimensional large-aspect-ratio (length large compared to depth) container, whose depth is large compared to the wavelength of the surface waves. The analysis requires us also to consider an oscillatory bulk flow and a viscous mean flow. A weakly nonlinear description of the harmonic waves is made which provides the threshold forcing amplitude to trigger harmonic instabilities, which are of various qualitatively different kinds. A linear analysis provides the threshold amplitude for the appearance of subharmonic waves through a subharmonic instability. The results obtained are used to make several specific qualitative and quantitative predictions.

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Section: Harmonic Surface Waves At Large Aspect Ratio: L > Dmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Modulated surface waves in large-aspect-ratio horizontally vibrated containers

Varas

Vega

2007

J. Fluid Mech.

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“…In this section we will complete these results and give a detailed description of the CW instability regions given by (19).…”

Section: Cw Stability: Transport Scale Perturbationsmentioning

confidence: 99%

Dispersive destabilization of nonlinear light propagation in fiber Bragg gratings

Martel

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The effect of retaining the material dispersion terms in the nonlinear coupled mode equations (NLCME) that describe light propagation in fiber Bragg gratings is analyzed. It is found that dispersion is responsible for new instabilities of the uniform states and gives rise to new complex spatio temporal dynamics that is not captured by the standard NLCME formulation. A detailed analysis of the effect of dispersion on the linear stability characteristics of the uniform solutions is presented and some numerical integrations of the NLCME with dispersion are also performed in order to corroborate the theoretical results. * Electronic address: martel@fmetsia.upm.es 1

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“…It is also important to emphasize that this type of dynamics is not contained in the standard dispersion-less NLCME formulation for light propagation in FBG. Moreover, this behavior is just the result of the competition of two effects with different asymptotic order: transport and disper- sion, and this is a generic situation that applies to any propagative extended system unless some special care is taken to reduce the group velocity (similar effects have been previously described in the context of Hopf bifurcation in dissipative systems [13] and in parametrically forced surface waves [14]). …”

Section: )mentioning

confidence: 70%

Localized Dispersive States in Nonlinear Coupled Mode Equations for Light Propagation in Fiber Bragg Gratings

Martel¹,

Higuera²,

Carrasco³

2009

SIAM J. Appl. Dyn. Syst.

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Abstract. Dispersion effects induce new instabilities and dynamics in the weakly nonlinear description of light propagation in fiber Bragg gratings. A new family of dispersive localized pulses that propagate with the group velocity is numerically found and its stability is also analyzed. The unavoidable different asymptotic order of transport and dispersion effects plays a crucial role in the determination of these localized states. These results are also interesting from the point of view of general pattern formation since this asymptotic imbalance is a generic situation in any transport dominated (i.e., nonzero group velocity) spatially extended system.1. Introduction. Fiber Bragg gratings (FBG) are microstructured optical fibers that present a spatially periodic variation of the refractive index. The combination of the guiding properties of the periodic media with the Kerr nonlinearity of the fiber results in the very particular light propagation characteristic of these elements, which make them very promising for many technological applications that range from optical communications (wavelength division, dispersion management, optical buffers and storing devices, etc.) to fiber sensing (structural stress measure in aircraft components and buildings, temperature change detection, etc.), see, e.g., the recent review [8].The amplitude equations that are commonly used in the literature to model one dimensional light propagation in a FBG are the so-called nonlinear coupled mode equations (NLCME) [17,6,5,1,7], which, conveniently scaled,can be written as

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Dynamics of counterpropagating waves in parametrically driven systems: dispersion vs. advection

Cited by 17 publications

References 20 publications

Modulated surface waves in large-aspect-ratio horizontally vibrated containers

Modulated surface waves in large-aspect-ratio horizontally vibrated containers

Dispersive destabilization of nonlinear light propagation in fiber Bragg gratings

Localized Dispersive States in Nonlinear Coupled Mode Equations for Light Propagation in Fiber Bragg Gratings

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