The Eckhaus and Benjamin-Feir resonance mechanisms

Stuart, J. T.; DiPrima, R. C.

doi:10.1098/rspa.1978.0118

Cited by 265 publications

(57 citation statements)

References 19 publications

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“…Our results for scalar equations extend those in [1,8,24], In [6] the stability of traveling wave solutions with respect to long-wavelength perturbations, that is small k , is discussed. In fact, however, caution must be exercised in extrapolating the result for stability to LW perturbations to larger values of k .…”

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confidence: 52%

Stability of plane wave solutions of complex Ginzburg-Landau equations

Matkowsky¹,

Volpert²

1993

Quart. Appl. Math.

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Abstract. We consider the stability of plane wave solutions of both single and coupled complex Ginzburg-Landau equations and determine stability domains in the space of coefficients of the equations. [6,11] and the references therein).Actually, CGL equations describe the evolution of the amplitudes of unstable modes for any process exhibiting Hopf bifurcation, for which the continuous band of unstable wave numbers is taken into account. Therefore, the equations have become a self-significant object of study.Coupled CGL equations govern the amplitudes, on slow spatial and temporal scales, of traveling waves propagating in opposite directions on the faster temporal and spatial scales associated with the original problem from which the GinzburgLandau equations were derived. The case when one of the amplitudes equals zero corresponds to a traveling wave solution of the original problem, while the case when the moduli of the two amplitudes are equal corresponds to a standing wave.The simplest solutions of CGL equations are plane wave solutions and the present paper is devoted to an analysis of their stability. Stability of traveling wave solutions of a single CGL equation, which arises when the most unstable wave number is

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confidence: 52%

Stability of plane wave solutions of complex Ginzburg-Landau equations

Matkowsky¹,

Volpert²

1993

Quart. Appl. Math.

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“…This can be seen from (9). In the limit of large amplitudes the diffusion coefficient becomes negative in the band center if c r d r + c i d i > 0 which is the condition for BenjaminFeir instability of the unforced waves [32]. This implies that either the parametrically forced waves become unstable at all wave numbers or the stable band has split into subbands.…”

Section: Discussionmentioning

confidence: 99%

Parametric forcing of waves with a nonmonotonic dispersion relation: Domain structures in ferrofluids

Raitt

Riecke

1997

Phys. Rev. E

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Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a nonmonotonic dispersion relation. The effect of a parametric driving on such waves is studied within suitable coupled Ginzburg-Landau equations. Due to the non-monotonicity the neutral curve for the excitation of standing waves can have up to three minima. The stability of the waves with respect to long-wave perturbations is determined via a phase-diffusion equation. It shows that the band of stable wave numbers can split up into two or three sub-bands. The resulting competition between the wave numbers corresponding to the respective sub-bands leads quite naturally to patterns consisting of multiple domains of standing waves which differ in their wave number. The coarsening dynamics of such domain structures is addressed.October 28, 2018 05.45.+b, 05.90.+m, 47.20. Tg

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“…The eventual instability is of the type discovered first by Benjamin and Feir in the frame of water waves [16]. It has been shown that this kind of instability involves the nonlinear interaction of two modulation terms with conjugated phases with the square A 2 of the constant solution [17]. Thus we write u as:…”

Section: Different Regimes For the Cgl Equation 321 The Stationary mentioning

confidence: 99%

Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser

et al. 2002

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We consider an Yb-doped double-clad fiber laser in a unidirectional ring cavity containing a polarizer placed between two half-wave plates. Depending on the orientation of the phase plates, the laser operates in continuous, Qswitch, mode-lock or unstable self-pulsing regime. An experimental study of the stability of the mode locking regime is realized versus the orientation of the half-wave plates. A model for the stability of self-mode-locking and cw operation is developed starting from two coupled nonlinear Schrödinger equations in a gain medium. The model is reduced to a master equation in which the coefficients are explicitly dependent on the orientation angles of the phase plates. Analytical solutions are given together with their stability versus the angles.

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The Eckhaus and Benjamin-Feir resonance mechanisms

Cited by 265 publications

References 19 publications

Stability of plane wave solutions of complex Ginzburg-Landau equations

Stability of plane wave solutions of complex Ginzburg-Landau equations

Parametric forcing of waves with a nonmonotonic dispersion relation: Domain structures in ferrofluids

Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser

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