Direct Characterization of Spectral Stability of Small-Amplitude Periodic Waves in Scalar Hamiltonian Problems via Dispersion Relation

This paper gives an extensive characterization of the range of validity of the Compton and Raman approximations to the exact free electron laser dispersion relation for a cold, relativistic electron beam propagating through a constantamplitude helical wiggler magnetic field. The electron beam is treated as infinite in transverse extent. Specific properties of the exact and approximate dispersion relations are investigated analytically and numerically. In particular, a detailed numerical analysis is carried out to determine the range of validity of the Compton approximation.

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ON THE INSTABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS—REVISITED

Nagoya Mathematical Journal ◽

10.1017/nmj.2021.9 ◽

2021 ◽

pp. 1-23

Author(s):

FÁBIO NATALI ◽

SABRINA AMARAL

Keyword(s):

Traveling Wave ◽

Orbital Stability ◽

Traveling Wave Solutions ◽

Spectral Stability ◽

Dispersive Equations ◽

Periodic Waves ◽

General Proof ◽

Wave Solutions ◽

Periodic Traveling Wave

Abstract The purpose of this paper is to present an extension of the results in [8]. We establish a more general proof for the moving kernel formula to prove the spectral stability of periodic traveling wave solutions for the regularized Benjamin–Bona–Mahony type equations. As applications of our analysis, we show the spectral instability for the quintic Benjamin–Bona–Mahony equation and the spectral (orbital) stability for the regularized Benjamin–Ono equation.

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Long-term and Short-term Spectral Stability Characterization of Supercontinuum Laser Sources

2006 IEEE LEOS Annual Meeting Conference Proceedings ◽

10.1109/leos.2006.278898 ◽

2006 ◽

Author(s):

Roberto Llorente ◽

Marta Beltran ◽

Joaquin Perez ◽

Noboru Uehara ◽

Md. Khan ◽

...

Keyword(s):

Spectral Stability ◽

Short Term ◽

Laser Sources ◽

Supercontinuum Laser

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SOLITON EXCITATIONS AND PERIODIC WAVES WITHOUT DISPERSION RELATION IN (2+1)-DIMENSIONAL DISPERSIVE LONG WAVE EQUATION

Nonlinear Evolution Equations and Dynamical Systems ◽

10.1142/9789812795366_0001 ◽

2003 ◽

Author(s):

CHUN-LI CHEN ◽

SEN-YUE LOU

Keyword(s):

Wave Equation ◽

Dispersion Relation ◽

Periodic Waves ◽

Long Wave

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Dispersion of discontinuous periodic waves

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2012.0407 ◽

2013 ◽

Vol 469 (2149) ◽

pp. 20120407 ◽

Cited By ~ 9

Author(s):

Gong Chen ◽

Peter J. Olver

Keyword(s):

Dispersion Relation ◽

Numerical Experiments ◽

Polynomial Growth ◽

Dynamic Evolution ◽

Bounded Domains ◽

Periodic Waves ◽

Talbot Effect ◽

Asymptotically Linear ◽

Dispersive Waves ◽

Wave Phenomena

The dynamic evolution of linearly dispersive waves on periodic domains with discontinuous initial profiles is shown to depend remarkedly upon the asymptotics of the dispersion relation at large wavenumbers. Asymptotically linear or sublinear dispersion relations produce slowly changing waves, while those with polynomial growth exhibit dispersive quantization, a.k.a. the Talbot effect, being (approximately) quantized at rational times, but a non-differentiable fractal at irrational times. Numerical experiments suggest that such effects persist into the nonlinear regime, for both integrable and non-integrable systems. Implications for the successful modelling of wave phenomena on bounded domains and numerical challenges are discussed.

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Structural Characterization of Composition-Modulated ZnSel-xTex Epitaxial Films

MRS Proceedings ◽

10.1557/proc-417-295 ◽

1995 ◽

Vol 417 ◽

Author(s):

S. P. Ahrenkiel ◽

M. H. Bode ◽

M. M. Al-Jassim ◽

H. Luo ◽

S. H. Xin ◽

...

Keyword(s):

Cross Sections ◽

Small Amplitude ◽

Molecular Beam ◽

Growth Direction ◽

Epitaxial Films ◽

Modulation Amplitude ◽

Transmission Electron ◽

Short Period ◽

Strain Modulation

AbstractWe examine the microstructure of short-period (14–31 Å) composition modulations in epitaxial ZnSel-xTex ([email protected]) films grown by molecular-beam epitaxy (MBE) on vicinal (001) GaAs. Transmission electron microscope (TEM) images of cross-sections reveal highly-periodic contrast along the growth direction throughout the full thicknesses of the films (over 2 μm) that corresponds to a nearly sinusoidal variation between Se- and Te-rich compositions. Growth of ZnSe1-xTex at 285°C on substrates tilted 4° toward [1111 maximizes the strength and regularity of the modulation. Using dynamical electron-diffraction simulations, we estimate a modulation amplitude of [email protected](7) in a sample showing strong modulation. We assume a small amplitude of strain modulation to fit the experimental data.

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Capillary–gravity waves of solitary type and envelope solitons on deep water

Journal of Fluid Mechanics ◽

10.1017/s0022112093003945 ◽

1993 ◽

Vol 252 ◽

pp. 703-711 ◽

Cited By ~ 42

Author(s):

Michael S. Longuet-Higgins

Keyword(s):

Surface Tension ◽

Dispersion Relation ◽

Gravity Waves ◽

Deep Water ◽

Small Amplitude ◽

Finite Amplitude ◽

Surface Slope ◽

Carrier Wave ◽

Envelope Solitons ◽

Special Case

The existence of steady solitary waves on deep water was suggested on physical grounds by Longuet-Higgins (1988) and later confirmed by numerical computation (Longuet-Higgins 1989; Vanden-Broeck & Dias 1992). Their numerical methods are accurate only for waves of finite amplitude. In this paper we show that solitary capillary-gravity waves of small amplitude are in fact a special case of envelope solitons, namely those having a carrier wave of length 2π(T/ρg)1½2 (g = gravity, T = surface tension, ρ = density). The dispersion relation $c^2 = 2(1-\frac{11}{32}\alpha^2_{\max)$ between the speed c and the maximum surface slope αmax is derived from the nonlinear Schrödinger equation for deep-water solitons (Djordjevik & Redekopp 1977) and is found to provide a good asymptote for the numerical calculations.

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RUNUP RECIPE FOR PERIODIC WAVES ON UNIFORMLY SLOPING BEACHES

Coastal Engineering Proceedings ◽

10.9753/icce.v10.21 ◽

1966 ◽

Vol 1 (10) ◽

pp. 21 ◽

Cited By ~ 2

Author(s):

Wm. G. Van Dorn

Keyword(s):

Small Amplitude ◽

Breaking Waves ◽

Wave Crest ◽

Periodic Waves ◽

Conservation Of Energy ◽

Wave Channel ◽

Wave Trains ◽

Elevation Measurement ◽

Steep Slopes ◽

Modified Theory

The shoaling enhancement of small-amplitude, dispersive wave trains traveling over uniform, impermeable slopes was observed in a specially-constructed wave channel, where the reproducible wave elevation measurement accuracy was about .0005-in. These observations substantially support the enhancement predicted from linear theory (conservation of energy flux) except in very shallow water and on very steep slopes, where accelerative effects become important. On the hypothesis that small-amplitude runup theory might be similarly valid for periodic waves of finite height, provided that the positive incident wave amplitude Is replaced by the local crest height above still water, this theory was modified to include the effect of the superelevation under a wave crest due to profile asymmetry. The modified theory is shown to agree acceptably with runup observations of larger waves previously reported - both for breaking and non-breaking waves. Because solutions to the modified theory cannot conveniently be obtained by manual calculation, a nomograph chart is included, from which runup predictions can be easily made, given only the wave height, period, and water depth a wavelength or so from shore, and the beach slope.

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