Computing spectra of linear operators using the Floquet–Fourier–Hill method

Deconinck, Bernard; Kutz, J. Nathan

doi:10.1016/j.jcp.2006.03.020

Cited by 168 publications

(239 citation statements)

References 30 publications

(66 reference statements)

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“…To continue the simulations accurately, a prohibitively large number of Fourier modes would be required (we recall that the wave profiles in figure 10 required N = 1400 modes), and consequently, an impractically small time step would be needed; moreover the inversion of the Jacobian in the Newton iterations at each time step becomes computationally very expensive. Simulations carried out with the surfactant diffusion present, taking the Péclet numbers to be finite in (44), produce qualitatively similar results, but suffer from similar difficulties when attempting to integrate to large times. Further evidence that the k = 0.7 travelling-wave is unstable is provided in the next section.…”

Section: Time-dependent Simulationsmentioning

confidence: 79%

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Thompson

Blyth

2016

Phys. Rev. Fluids

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Multilayer film flow down an inclined plane in the presence of an insoluble surfactant is investigated with particular emphasis on determining flow stability and investigating the possibility of travelling wave solutions. The investigation is conducted for two or three layers under conditions of Stokes flow and, separately, on the basis of a long-wave assumption. A normal mode linear stability analysis for Stokes flow shows that adding surfactant to one of the film surfaces can destabilise an otherwise stable flow configuration. For the long-wave system, periodic travelling-wave branches are detected and traced, revealing solutions with pulse-like solitary waves on each film surface travelling in phase with each other, travelling waves with capillary ridge structures, and solutions with two of the film surfaces almost in contact. Time-periodic travelling wave solutions are also found. The stability of the travelling waves is determined by solving initial value problems and by computing eigenvalue spectra. Boundary element simulations for Stokes flow confirm the existence of travelling waves outside of the longwave regime.

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Section: Time-dependent Simulationsmentioning

confidence: 79%

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Thompson

Blyth

2016

Phys. Rev. Fluids

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“…[76] for e carry out thi can investiga ing the system ing over one matrix. Holzl [78] have disc The eigenmo periodically-s modes), and Without are likely to gorithms, the only to prov …”

Section: Discussionmentioning

confidence: 99%

“…When an equilibrium is found, we can investigate the stability of the system by perturbing the system using a complete set of modes, integrating over one round trip, and creating a transformation matrix. Holzlöhner et al [77] and Deconninck and Kutz [78] have discussed algorithms for carrying out this task. The eigenmodes of this transformation matrix are the periodically-stationary eigenmodes (BlochFloquet-Hill modes), and their eigenvalues determine the stability.…”

Section: Discussionmentioning

confidence: 99%

Spectral Methods for Determining the Stability and Noise Performance of Passively Modelocked Lasers

Menyuk

Wang

2016

Nanophotonics

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Abstract:We describe spectral or dynamical methods that can be used to determine the stability and noise performance of modelocked lasers. We first review methods that have been used to date to theoretically and computationally study passively modelocked lasers, contrasting evolutionary and dynamical approaches and their application to full, averaged, and reduced models. We then develop the spectral methods and show how they can be used to determine the stability and to calculate the timing jitter and power spectral density for any averaged model with any equilibrium pulse shape. We review work that has been done on soliton lasers using soliton perturbation theory from this dynamical perspective, and we contrast the simplicity and generality of our methods to prior work. We close with a discussion of how to extend our approach from averaged models to full models.

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“…The Numerical Computation of the Spectrum. We use Hill's Method as presented in [15]. Since the coefficients of L can be approximated by functions with period 2L we represent them as Fourier series.…”

Section: Nontrivial-phase Solutionsmentioning

confidence: 99%

On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation

Deconinck

Sheils

Nguyen

et al. 2013

J. Phys. A: Math. Theor.

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Abstract. We consider a system of coupled cubic nonlinear Schrödinger (NLS) equationswhere the interaction coefficients α jk are real. The spectral stability of solitary wave solutions (both bright and dark) is examined both analytically and numerically. Our results build on preceding work by Nguyen et al. and others. Specifically, we present closed-form solitary wave solutions with trivial and non-trivial phase profiles. Their spectral stability is examined analytically by determining the locus of their essential spectrum. Their full stability spectrum is computed numerically using a large-period limit of Hill's method. We find that all nontrivial-phase solutions are unstable while some trivial-phase solutions are spectrally stable. To our knowledge, this paper presents the first investigation of the stability of the solitary waves of the coupled cubic NLS equation without the restriction that all components ψ j are proportional to sech.

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Computing spectra of linear operators using the Floquet–Fourier–Hill method

Cited by 168 publications

References 30 publications

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Spectral Methods for Determining the Stability and Noise Performance of Passively Modelocked Lasers

On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation

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