The dynamics of modulated wave trains

Doelman, Arjen; Sandstede, Björn; Scheel, Arnd; Schneider, Guido

doi:10.1090/memo/0934

Cited by 109 publications

(189 citation statements)

References 55 publications

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“…The group velocity is important as it is the speed with which small localized perturbations of the wave train propagate as functions of time, and we refer to [DSSS09] for a rigorous justification of this.…”

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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Section: Introductionmentioning

confidence: 99%

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

et al. 2014

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“…However, a KdV equation can appear in the RD setting when the wavetrain undergoes modulation instability, and then the modulation equation changes to a KdV equation (van Harten 1995;Doelman et al 2009). The appearance of the KdV equation in the setting of RD equations is more remarkable because the KdV equation is Hamiltonian and RD equations are, in general, dissipative.…”

Section: ) For the Complex-valued Function A(x T) An Example Of A mentioning

confidence: 99%

Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns

Bridges

2012

Proc. R. Soc. A.

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A dark solitary wave, in one space dimension and time, is a wave that is bi-asymptotic to a periodic state, with a phase shift, and with localized modulation in between. The most well-known case of dark solitary waves is the exact solution of the defocusing nonlinear Schrödinger equation. In this paper, our interest is in developing a mechanism for the emergence of dark solitary waves in general, and not necessarily integrable, Hamiltonian PDEs. The focus is on the periodic state at infinity as the generator. It is shown that a natural mechanism for the emergence is a transition between one periodic state that is (spatially) elliptic and another one that is (spatially) hyperbolic. It is shown that the emergence is governed by a Korteweg-de Vries (KdV) equation for the perturbation wavenumber on a periodic background. A novelty in the result is that the three coefficients in the KdV equation are determined by the Krein signature of the elliptic periodic orbit, the curvature of the wave action flux and the slope of the wave action, with the last two evaluated at the critical periodic state.

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“…Oscillatory media support phase waves u(kx − ωt; k), u(ξ; k) = u(ξ + 2π; k), which may nucleate at inhomogeneities or boundaries; see for instance [19]. Wave numbers vary in an admissible band, where the frequency is a function of the wave number, called the dispersion relation ω = ω(k) [10,31]. The limit k = 0 corresponds to the spatially homogeneous oscillation.…”

Section: Excitable and Oscillatory Mediamentioning

confidence: 99%

Coherent structures near the boundary between excitable and oscillatory media

Bellay

Scheel

2009

Dynamical Systems

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We investigate reaction-diffusion systems near parameter values that mark the transition from an excitable to an oscillatory medium. We analyze existence and stability of traveling waves near a steep pulse that arises as the limit of excitation pulses when parameters cross into the oscillatory regime. Traveling waves near this limiting profile are obtained by analyzing a codimension-two homoclinic saddle-node/orbit-flip bifurcation. The main result shows that there are precisely two generic scenarios for such a transition, distinguished by the sign of an interaction coefficient between pulses. In both scenarios, we find stable fast fronts, unstable slow fronts, stable excitation pulses, and trigger and phase waves. Both trigger and phase waves are stable for repulsive interaction and both are unstable for attractive interaction.

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The dynamics of modulated wave trains

Cited by 109 publications

References 55 publications

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

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

Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns

Coherent structures near the boundary between excitable and oscillatory media

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