A Stability Criterion for Envelope Equations

Lange, Charles G.; Newell, Alan C.

doi:10.1137/0127034

Cited by 130 publications

(53 citation statements)

References 10 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 .…”

supporting

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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“…Under proper balance conditions a CW can be formed as an equilibrium that can be either modulationally stable or unstable. Such non-conservative systems have been studied in terms of the Ginzburg-Landau (GL) equation [7][8][9] for the case of homogeneous gain and loss mechanisms.…”

Section: Introductionmentioning

confidence: 99%

Stability through asymmetry: Modulationally stable nonlinear supermodes of asymmetric non-Hermitian optical couplers

2017

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We analyze the stability of a non-Hermitian coupler with respect to modulational inhomogeneous perturbations in the presence of unbalanced gain and loss. At the parity-time (PT) symmetry point the coupler is unstable. Suitable symmetry breakings lead to an asymmetric coupler, which hosts nonlinear supermodes. A subset of these broken symmetry cases finally yields nonlinear supermodes which are stable against modulational perturbations. The lack of symmetry requirements is expected to facilitate experimental implementations and relevant photonics applications.

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“…The nonlinear evolution of A under this equation has been comprehensively discussed by Lange and Newell (1974) and Newell (1974). By contrast, the second category of instabilities is the "dispersive" type and occurs when no dissipation is present.…”

Section: Introductionmentioning

confidence: 99%

Quasiperiodicity and chaos in the nonlinear evolution of the Kelvin-Helmholtz instability of supersonic anisotropic tangential velocity discontinuities

Choudhury¹,

Brown²

2003

Quart. Appl. Math.

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Abstract.A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, strongly anisotropic, magnetized, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations.For modes near the critical point of the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein-Gordon equation.The nonlinear coefficient turns out to be complex, which is, to the best of our knowledge, unlike previously considered cases and leads to completely different dynamics from that reported earlier. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability. In particular, the complex nonlinearity allows the existence of quasiperiodic and chaotic wave envelopes, unlike in earlier physical models governed by nonlinear Klein-Gordon equations. In addition, numerical diagnostics reveal the onset of chaos as a consequence of modulation of the external magnetic field.

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A Stability Criterion for Envelope Equations

Cited by 130 publications

References 10 publications

Stability of plane wave solutions of complex Ginzburg-Landau equations

Stability of plane wave solutions of complex Ginzburg-Landau equations

Stability through asymmetry: Modulationally stable nonlinear supermodes of asymmetric non-Hermitian optical couplers

Quasiperiodicity and chaos in the nonlinear evolution of the Kelvin-Helmholtz instability of supersonic anisotropic tangential velocity discontinuities

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