Absolute instabilities of standing pulses

Sandstede, Björn; Scheel, Arnd

doi:10.1088/0951-7715/18/1/017

Cited by 18 publications

(15 citation statements)

References 30 publications

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“…Analogous bifurcations can also occur at fronts when the rest state ahead of the front destabilizes [48]. Last, we proved in [51] that both flip-flops and one-dimensional target patterns (see Figure 1.2(i) and (ii)) will bifurcate from standing pulses whose background states undergo a Hopf instability. This appears to be the mechanism that creates the chemical flip-flops observed in [38].…”

Section: Examplesmentioning

confidence: 62%

Defects in Oscillatory Media: Toward a Classification

Sandstede

Scheel

2004

SIAM J. Appl. Dyn. Syst.

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115

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Abstract. We investigate, in a systematic fashion, coherent structures, or defects, which serve as interfaces between wave trains with possibly different wavenumbers in reaction-diffusion systems. We propose a classification of defects into four different defect classes which have all been observed experimentally. The characteristic distinguishing these classes is the sign of the group velocities of the wave trains to either side of the defect, measured relative to the speed of the defect. Using a spatial-dynamics description in which defects correspond to homoclinic and heteroclinic connections of an ill-posed pseudoelliptic equation, we then relate robustness properties of defects to their spectral stability properties. Last, we illustrate that all four types of defects occur in the one-dimensional cubicquintic Ginzburg-Landau equation as a perturbation of the phase-slip vortex.

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Section: Examplesmentioning

confidence: 62%

Defects in Oscillatory Media: Toward a Classification

Sandstede

Scheel

2004

SIAM J. Appl. Dyn. Syst.

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115

180

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“…Information about small eigenvalues is drawn from the derivatives of the Evans function with respect to λ and ǫ near λ = 0 and ǫ = 0. However, computational formulas become more and more involved when higher-order derivatives of the Evans function are needed and this has caused some miscalculations in the past (see Appendix A in [44]). …”

Section: Eigenfunctions and Eigenvalues Of Kinksmentioning

confidence: 99%

Dark solitons in external potentials

Pelinovsky

Kevrekidis

2007

Z. angew. Math. Phys.

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We consider the persistence and stability of dark solitons in the Gross-Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation with cubic nonlinearity. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton's particle law for its position.

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“…This has been investigated in a variety of previous studies, including [Leg01,PSAK95,CM92,KR00,SS05,LF97]. Partial analytical results can be found in [KR00,SS05]. Numerical and asymptotic evidence in [CM92,PSAK95] suggests that the sources are stable in an open region of parameter space near the NLS limit of (1.1), which corresponds to the limit |α|, |β| → ∞ and γ 1 , γ 2 → 0.…”

Section: Introductionmentioning

confidence: 96%

“…To find a spectrally stable source, one needs to find parameter values for which both the unstable eigenvalue (from the rGL limit) and the perturbed zero eigenvalue (from the cCGL limit) become stable. This has been investigated in a variety of previous studies, including [Leg01,PSAK95,CM92,KR00,SS05,LF97]. Partial analytical results can be found in [KR00,SS05].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

et al. 2014

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In an appropriate moving coordinate frame, source defects are time-periodic solutions to reactiondiffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg-Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for the Green's function, which allow one to close a nonlinear iteration scheme.

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Absolute instabilities of standing pulses

Cited by 18 publications

References 30 publications

Defects in Oscillatory Media: Toward a Classification

Defects in Oscillatory Media: Toward a Classification

Dark solitons in external potentials

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

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