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Journal articles on the topic 'Amplitude equation'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Rauf, S., and J. A. Tataronis. "Resonant four-wave mixing of finite-amplitude Alfvén waves." Journal of Plasma Physics 55, no. 2 (1996): 173–80. http://dx.doi.org/10.1017/s0022377800018766.

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Using the derivative nonlinear SchrÖdinger equation, resonant four-wave mixing of finite-amplitude Alfvén waves is explored in this paper. The evolution equations governing the amplitudes of the interacting waves and the conservation relations ale derived from the basic equation. These evolution equations are used to study parametric amplification and oscillation of two small-amplitude Alfvén waves due to two large-amplitude pump (Alfvén) waves. It is also shown that three pump waves can mix together to generate a low-frequency Alfven wave in a dissipative plasma.

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2

Zhang, Yu, Guanquan Zhang, and Norman Bleistein. "Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration." GEOPHYSICS 70, no. 4 (2005): E1—E10. http://dx.doi.org/10.1190/1.1988182.

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One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By “true-amplitude” one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in

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3

Vivas, Flor A., and Reynam C. Pestana. "True-amplitude one-way wave equation migration in the mixed domain." GEOPHYSICS 75, no. 5 (2010): S199—S209. http://dx.doi.org/10.1190/1.3478574.

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One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that

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4

Feng, Zongcai, and Gerard Schuster. "True-amplitude linearized waveform inversion with the quasi-elastic wave equation." GEOPHYSICS 84, no. 6 (2019): R827—R844. http://dx.doi.org/10.1190/geo2019-0116.1.

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We present a quasi-elastic wave equation as a function of the pressure variable, which can accurately model PP reflections with elastic amplitude variation with offset effects under the first-order Born approximation. The kinematic part of the quasi-elastic wave equation accurately models the propagation of P waves, whereas the virtual-source part, which models the amplitudes of reflections, is a function of the perturbations of density and Lamé parameters [Formula: see text] and [Formula: see text]. The quasi-elastic wave equation generates a scattering radiation pattern that is exactly the s

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5

Kovachev, Lubomir M. "Optical vortices in dispersive nonlinear Kerr-type media." International Journal of Mathematics and Mathematical Sciences 2004, no. 18 (2004): 949–67. http://dx.doi.org/10.1155/s0161171204301018.

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The applied method of slowly varying amplitudes gives us the possibility to reduce the nonlinear vector integrodifferential wave equation of the electrical and magnetic vector fields to the amplitude vector nonlinear differential equations. Using this approximation, different orders of dispersion of the linear and nonlinear susceptibility can be estimated. Critical values of parameters to observe different linear and nonlinear effects are determined. The obtained amplitude equations are a vector version of3D+1nonlinear Schrödinger equation (VNSE) describing the evolution of slowly varying ampl

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MOSLEM, W. M. "Propagation of ion acoustic waves in a warm multicomponent plasma with an electron beam." Journal of Plasma Physics 61, no. 2 (1999): 177–89. http://dx.doi.org/10.1017/s0022377898007429.

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The nonlinear wave structures of small-amplitude ion acoustic waves in a warm plasma with adiabatic negative-ion, positron and electron constituents traversed by a warm electron beam (with different temperatures) in the vicinity of the critical negative-ion density are investigated using reductive perturbation method. The basic set of equations is reduced to an evolution equation that includes quadratic and cubic nonlinearities. The effective potential of this equation agrees exactly, for small wave amplitudes, with the Sagdeev potential obtained from the original fluid equations using a pseud

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7

Hutahaean, Syawaluddin. "Wavelength and Wave Period Relationship with Wave Amplitude: A Velocity Potential Formulation." International Journal of Advanced Engineering Research and Science 9, no. 8 (2022): 387–93. http://dx.doi.org/10.22161/ijaers.98.44.

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In this study, the equation that expresses the explicit relationship between the wave number and wave amplitude, as well as wave period and wave amplitude are established. The wave number and the wave period are calculated solely using the input wave amplitude. The equation is formulated with the velocity potential of the solution to Laplace’s equation to the hydrodynamic conservation equations, such as the momentum equilibrium equation, Euler Equation for conservation of momentum, and by working on the kinematic bottom and free surface boundary condition.In this study, the equation that expre

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8

Meis, Constantin, and Pierre Richard Dahoo. "Vector potential quantization and the photon intrinsic electromagnetic properties: Towards nondestructive photon detection." International Journal of Quantum Information 15, no. 08 (2017): 1740003. http://dx.doi.org/10.1142/s0219749917400032.

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We employ here the enhancement of the vector potential amplitude quantization at a single photon state. The analysis of the general solution of the vector potential, obtained by resolving Maxwell’s equations, implies that the amplitude is proportional to the angular frequency. The photon vector potential function αkλ(r,t) can be written in the plane wave representation satisfying the classical wave propagation equation, Schrödinger’s equation for the energy with the relativistic massless field Hamiltonian and a linear time-dependent equation for the vector potential amplitude operator. Thus, t

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9

Angus, D. A. "True amplitude corrections for a narrow-angle one-way elastic wave equation." GEOPHYSICS 72, no. 2 (2007): T19—T26. http://dx.doi.org/10.1190/1.2430694.

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Wavefield extrapolators using one-way wave equations are computationally efficient methods for accurate traveltime modeling in laterally heterogeneous media, and have been used extensively in many seismic forward modeling and migration problems. However, most leading-order, one-way wave equations do not simulate waveform amplitudes accurately and this is primarily because energy flux is not accounted for correctly. I review the derivation of a leading-order, narrow-angle, one-way elastic wave equation for 3D media. I derive correction terms that enable energy-flux normalization and introduce a

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10

Zhang, Yu, Guanquan Zhang, and Norman Bleistein. "True amplitude wave equation migration arising from true amplitude one-way wave equations." Inverse Problems 19, no. 5 (2003): 1113–38. http://dx.doi.org/10.1088/0266-5611/19/5/307.

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11

Ipsen, M., F. Hynne, and P. G. Sørensen. "Amplitude Equations and Chemical Reaction–Diffusion Systems." International Journal of Bifurcation and Chaos 07, no. 07 (1997): 1539–54. http://dx.doi.org/10.1142/s0218127497001217.

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The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutio

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12

Zenyuk, Dmitry Alexeevich, and Georgii Gennadyevich Malinetskii. "Amplitude equation formalism for reaction—subdiffusion systems." Keldysh Institute Preprints, no. 93 (2021): 1–15. http://dx.doi.org/10.20948/prepr-2021-93.

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The paper presents derivation of the amplitude equation for the Hopf bifurcation in the two-component system with nonlinear chemical kinetics and subdiffusion. Anomalous diffusion transport is described via Caputo fractional derivatives. The obtained amplitude equation is much more complex compared to the case of normal diffusion because solutions of fractional order linear differential equations have inconvenient behavior.

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13

Yulida, Yuni, Haidir Ahsana, Muhammad Mahfuzh Shiddiq, and Muhammad Ahsar Karim. "APLIKASI PERSAMAAN GELOMBANG UNTUK MENENTUKAN KARAKTERISTIK GELOMBANG SENAR GITAR YANG DIPETIK." EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN 15, no. 2 (2022): 133. http://dx.doi.org/10.20527/epsilon.v15i2.4735.

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Partial differential equations are often used to explain physical phenomena, one of which is the wave equation. One application of the wave equation is in plucked strings. This study describes the formation of a wave equation from guitar strings, determines the solution to the wave equation by using the variable separation method and certain boundary conditions and initial conditions, determines the amplitude of the wave, and simulates the movement of the wave based on the initial position of the plucked string. The result obtained is the wave equation of the guitar strings. When the string is

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14

Piron, Robin, and Mikael Tacu. "An Explicit Numerical Scheme for Milne’s Phase–Amplitude Equations." Atoms 13, no. 6 (2025): 57. https://doi.org/10.3390/atoms13060057.

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We present an explicit numerical method to solve Milne’s phase–amplitude equations. Existing methods directly solve Milne’s nonlinear equation for amplitude. For that reason, they exhibit high sensitivity to errors and are prone to instability through the growth of a spurious, rapidly varying component of the amplitude. This makes the systematic use of these methods difficult. On the contrary, the present method is based on solving a linear third-order equation which is equivalent to the nonlinear amplitude equation. This linear equation was derived by Kiyokawa, who used it to obtain analytica

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15

Maiden, M. D., and M. A. Hoefer. "Modulations of viscous fluid conduit periodic waves." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2196 (2016): 20160533. http://dx.doi.org/10.1098/rspa.2016.0533.

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Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlin

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16

Didenkulova, Ekaterina, Efim Pelinovsky, and Marcelo V. Flamarion. "Bipolar Solitary Wave Interactions within the Schamel Equation." Mathematics 11, no. 22 (2023): 4649. http://dx.doi.org/10.3390/math11224649.

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Pair soliton interactions play a significant role in the dynamics of soliton turbulence. The interaction of solitons with different polarities is particularly crucial in the context of abnormally large wave formation, often referred to as freak or rogue waves, as these interactions result in an increase in the maximum wave field. In this article, we investigate the features and properties of bipolar solitary wave interactions within the framework of the non-integrable Schamel equation, contrasting them with the integrable modified Korteweg-de Vries (mKdV) equation. We show that in bipolar soli

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17

Blömker, D., S. Maier-Paape, and G. Schneider. "The stochastic Landau equation as an amplitude equation." Discrete & Continuous Dynamical Systems - B 1, no. 4 (2001): 527–41. http://dx.doi.org/10.3934/dcdsb.2001.1.527.

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18

Kim, Minjeong, Daseon Hong, and Sungsu Park. "A Study on the Amplitude Comparison Monopulse Algorithm." Applied Sciences 10, no. 11 (2020): 3966. http://dx.doi.org/10.3390/app10113966.

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This paper presents two amplitude comparison monopulse algorithms and their covariance prediction equation. The proposed algorithms are based on the iterated least-squares estimation method and include the conventional monopulse algorithm as a special case. The proposed covariance equation is simple but predicts RMS errors very accurately. This equation quantitatively states estimation accuracy in terms of major parameters of amplitude comparison monopulse radar, and is also applicable to the conventional monopulse algorithm. The proposed algorithms and covariance prediction equations are vali

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19

Sabeti, Ramtin, and Mohammad Heidarzadeh. "A new empirical equation for predicting the maximum initial amplitude of submarine landslide-generated waves." Landslides 19, no. 2 (2021): 491–503. http://dx.doi.org/10.1007/s10346-021-01747-w.

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AbstractThe accurate prediction of landslide tsunami amplitudes has been a challenging task given large uncertainties associated with landslide parameters and often the lack of enough information of geological and rheological characteristics. In this context, physical modelling and empirical equations have been instrumental in developing landslide tsunami science and engineering. This study is focused on developing a new empirical equation for estimating the maximum initial landslide tsunami amplitude for solid-block submarine mass movements. We are motivated by the fact that the predictions m

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20

JAFAROV, R. G. "THE MODEL BETHE-SALPETER EQUATIONS FOR THE HIGGS SCALARS SCATTERING AMPLITUDE WITH SOLUTIONS." Izvestiya vysshikh uchebnykh zavedenii. Fizika 67, no. 5 (2024): 20–26. https://doi.org/10.17223/00213411/67/5/3.

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The integral Bethe-Salpeter equation with minimally perturbative kernels for scattering amplitudes of Higgs scalar particles is studied using ladder (single-particle exchange) and two-particle irreducible diagram (bubble exchange) approximations. Asymptotic solutions of the corresponding equations for the scattering amplitudes of Higgs scalar particles are obtained. In case of approximation of the summation of ladder diagrams, a solution is found in the form of a power function s α in the Regge region of energy change s →∞. In the approximation of summation of two-particle irreducible diagrams

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21

Gray, Samuel H., and Norman Bleistein. "True-amplitude Gaussian-beam migration." GEOPHYSICS 74, no. 2 (2009): S11—S23. http://dx.doi.org/10.1190/1.3052116.

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Gaussian-beam depth migration and related beam migration methods can image multiple arrivals, so they provide an accurate, flexible alternative to conventional single-arrival Kirchhoff migration. Also, they are not subject to the steep-dip limitations of many (so-called wave-equation) methods that use a one-way wave equation in depth to downward-continue wavefields. Previous presentations of Gaussian-beam migration have emphasized its kinematic imaging capabilities without addressing its amplitude fidelity. We offer two true-amplitude versions of Gaussian-beam migration. The first version comb

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22

Arora, Rajan. "ASYMPTOTICAL SOLUTIONS FOR A VIBRATIONALLY RELAXING GAS." Mathematical Modelling and Analysis 14, no. 4 (2009): 423–34. http://dx.doi.org/10.3846/1392-6292.2009.14.423-434.

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Using the weakly non‐linear geometrical acoustics theory, we obtain the small amplitude high frequency asymptotic solution to the basic equations governing one dimensional unsteady planar, spherically and cylindrically symmetric flow in a vibrationally relaxing gas with Van der Waals equation of state. The transport equations for the amplitudes of resonantly interacting waves are derived. The evolutionary behavior of non‐resonant wave modes culminating into shock waves is also studied.

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23

Leibovich, S., and A. Kribus. "Large-amplitude wavetrains and solitary waves in vortices." Journal of Fluid Mechanics 216 (July 1990): 459–504. http://dx.doi.org/10.1017/s0022112090000507.

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Large-amplitude axisymmetric waves on columnar vortices, thought to be related to flow structures observed in vortex breakdown, are found as static bifurcations of the Bragg–Hawthorne equation. Solutions of this equation satisfy the steady, axisymmetric, Euler equations. Non-trivial solution branches bifurcate as the swirl ratio (the ratio of azimuthal to axial velocity) changes, and are followed into strongly nonlinear regimes using a numerical continuation method. Four types of solutions are found: multiple columnar solutions, corresponding to Benjamin's ‘conjugate flows’, with subcritical–s

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24

Grimshaw, R. H. J., and D. I. Pullin. "Stability of finite-amplitude interfacial waves. Part 1. Modulational instability for small-amplitude waves." Journal of Fluid Mechanics 160 (November 1985): 297–315. http://dx.doi.org/10.1017/s0022112085003494.

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In two previous papers (Pullin & Grimshaw 1983a, b) we studied the wave profile and other properties of finite-amplitude interfacial progressive waves in a two-layer fluid. In this and the following paper (Pullin & Grimshaw 1985) we discuss the stability of these waves to small perturbations. In this paper we obtain anatytical results for the long-wavelength modulational instability of small-amplitude waves. Using a multiscale expansion, we obtain a nonlinear Schrödinger equation coupled to a wave-induced mean-flow equation to describe slowly modulated waves. From these coupled equatio

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Liu, Houye, and Weiming Wang. "A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems." International Journal of Computational Models and Algorithms in Medicine 3, no. 4 (2012): 21–28. http://dx.doi.org/10.4018/ijcmam.2012100103.

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Amplitude equation may be used to study pattern formatio. In this article, the authors establish a new mechanical algorithm AE_Hopf for calculating the amplitude equation near Hopf bifurcation based on the method of normal form approach in Maple. The normal form approach needs a large number of variables and intricate calculations. As a result, deriving the amplitude equation from diffusion-reaction is a difficult task. Making use of our mechanical algorithm, we derived the amplitude equations from several biology and physics models. The results indicate that the algorithm is easy to apply and

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Huang, Jiandong, Dinghui Yang, and Xijun He. "Discontinuous Galerkin method for solving viscoacoustic wave equations with amplitude dissipation and phase dispersion separation in isotropic and anisotropic media." Geophysical Journal International 235, no. 3 (2023): 2339–60. http://dx.doi.org/10.1093/gji/ggad369.

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SUMMARY The standard-linear-solid (SLS) theory works well for viscoelasticity. However, the coupling of amplitude dissipation and phase dispersion makes it impossible to investigate their effects separately by the discontinuous Galerkin method (DGM). In this paper, we have derived newly viscoacoustic wave equations with amplitude dissipation and phase dispersion separation in isotropic and anisotropic media, based on a Fourier method, which is suitable for using a time–space-domain DGM on unstructured meshes. The basic framework of DGM is constructed and the amplitude-dissipation effect and th

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Park, Jungho, and Philip Strzelecki. "Bifurcation to traveling waves in the cubic–quintic complex Ginzburg–Landau equation." Analysis and Applications 13, no. 04 (2015): 395–411. http://dx.doi.org/10.1142/s0219530514500419.

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We consider the one-dimensional complex Ginzburg–Landau equation which is a generic modulation equation describing the nonlinear evolution of patterns in fluid dynamics. The existence of a Hopf bifurcation from the basic solution was proved by Park [Bifurcation and stability of the generalized complex Ginzburg–Landau equation, Pure Appl. Anal. 7(5) (2008) 1237–1253]. We prove in this paper that the solution bifurcates to traveling waves which have constant amplitudes. We also prove that there exist kink-profile traveling waves which have variable amplitudes. The structure of the traveling wave

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Hemlata, A. K. Upadhyay, and P. Jha. "Evolution of a circularly polarized laser beam in an obliquely magnetized plasma channel." Laser and Particle Beams 35, no. 4 (2017): 631–40. http://dx.doi.org/10.1017/s0263034617000581.

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AbstractThe evolution of the spot size and amplitude of a circularly polarized laser beam propagating in a plasma channel embedded in an obliquely applied magnetic field has been investigated. The wave equation describing the evolution of the radiation field is set up and a variational technique is used to obtain the equations governing the evolution of the spot size and amplitude. Numerical methods are used to analyze the evolution of the laser beam spot size and amplitude. It is seen that the amplitudes of the two transverse components of the electric field of the laser beam evolve different

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Guo, Peng, George A. McMechan, and Huimin Guan. "Comparison of two viscoacoustic propagators for Q-compensated reverse time migration." GEOPHYSICS 81, no. 5 (2016): S281—S297. http://dx.doi.org/10.1190/geo2015-0557.1.

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Without considering intrinsic attenuation, reverse time migration (RTM) of data from lossy media produces smeared migration images because of the [Formula: see text] effects (amplitude loss and velocity dispersion). To mitigate the [Formula: see text] effects during RTM, amplitudes need to be compensated and the propagation velocity of the compensated wavefield needs to be the same as in the attenuating wavefield. We have compared the decoupled constant [Formula: see text] (DCQ) viscoacoustic equation with the viscoacoustic equation based on the generalized standard linear solids (GSLS), for m

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Amazonas, Daniela, Rafael Aleixo, Gabriela Melo, Jörg Schleicher, Amélia Novais, and Jessé C. Costa. "Including lateral velocity variations into true-amplitude common-shot wave-equation migration." GEOPHYSICS 75, no. 5 (2010): S175—S186. http://dx.doi.org/10.1190/1.3481469.

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In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analyt

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Zhang, Yu, Sheng Xu, Norman Bleistein, and Guanquan Zhang. "True-amplitude, angle-domain, common-image gathers from one-way wave-equation migrations." GEOPHYSICS 72, no. 1 (2007): S49—S58. http://dx.doi.org/10.1190/1.2399371.

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True-amplitude wave-equation migration provides a quality migrated image of the earth’s interior. In addition, the amplitude of the output provides an estimate of the angular-dependent reflection coefficient, similar to the output of Kirchhoff inversion. Recently, true-amplitude wave-equation migration for common-shot data has been proposed to generate amplitude-reliable, shot-domain, common-image gathers in heterogeneous media. We present a method to directly produce angle-domain common-image gathers from both common-shot and shot-receiver wave-equation migration. Generating true-amplitude, s

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Wang, Yuanbin, Hu Ding, and Li-Qun Chen. "Kinematic Aspects in Modeling Large-Amplitude Vibration of Axially Moving Beams." International Journal of Applied Mechanics 11, no. 02 (2019): 1950021. http://dx.doi.org/10.1142/s1758825119500212.

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This paper clarified kinematic aspects of motion of axially moving beams undergoing large-amplitude vibration. The kinematics was formulated in the mixed Eulerian–Lagrangian framework. Based on the kinematic analysis, the governing equations of nonlinear vibration were derived from the extended Hamilton principle and the higher-order shear beam theory. The derivation considered the effects of material parameters on the beam deformation. The proposed governing equations were compared with a few previous governing equations. The comparisons show that proposed equations are with higher precision.

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Demiray, Hilmi. "Modulation of Electron-Acoustic Waves in a Plasma with Vortex Electron Distribution." International Journal of Nonlinear Sciences and Numerical Simulation 16, no. 2 (2015): 61–66. http://dx.doi.org/10.1515/ijnsns-2014-0017.

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AbstractIn the present work, employing a one-dimensional model of a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution and stationary ions, we study the amplitude modulation of electron-acoustic waves by use of the conventional reductive perturbation method. Employing the field equations with fractional power type of nonlinearity, we obtained the nonlinear Schrödinger equation as the evolution equation of the same order of nonlinearity. Seeking a harmonic wave solution with progressive wave amplitude to the evolution equation it is found that the

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Cheng, Xuansheng, De Li, Peijiang Li, Xiaoyan Zhang, and Guoliang Li. "Dynamic Response of Base-Isolated Concrete Rectangular Liquid-Storage Structure Under Large Amplitude Sloshing." Archives of Civil Engineering 63, no. 1 (2017): 33–45. http://dx.doi.org/10.1515/ace-2017-0003.

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AbstractConsidering concrete nonlinearity, the wave height limit between small and large amplitude sloshing is defined based on the Bernoulli equation. Based on Navier-Stokes equations, the mathematical model of large amplitude sloshing is established for a Concrete Rectangle Liquid-Storage Structure (CRLSS). The results show that the seismic response of a CRLSS increases with the increase of seismic intensity. Under different seismic fortification intensities, the change in trend of wave height, wallboard displacement, and stress are the same, but the amplitudes arc not. The areas of stress c

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Kiran, Palle, B. S. Bhadauria, and R. Roslan. "The Effect of Throughflow on Weakly Nonlinear Convection in a Viscoelastic Saturated Porous Medium." Journal of Nanofluids 9, no. 1 (2020): 36–46. http://dx.doi.org/10.1166/jon.2020.1724.

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In this paper we have investigated the effect of throughflow on thermal convection in a viscoelastic fluid saturated porous media. The governing equations are modelled in the presence of throughflow. These equations are made dimensionless and the obtained nonlinear problem solved numerically. There are two types of throughflow effects on thermal instability inflow and outflow investigated by finite amplitude analysis. This finite amplitude equation is obtained using the complex Ginzburg-Landau amplitude equation (CGLE) for a weak nonlinear oscillatory convection. The heat transport analysis is

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Noack, Bernd R., and Robert K. Niven. "Maximum-entropy closure for a Galerkin model of an incompressible periodic wake." Journal of Fluid Mechanics 700 (April 24, 2012): 187–213. http://dx.doi.org/10.1017/jfm.2012.125.

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AbstractA statistical closure is proposed for a Galerkin model of an incompressible periodic cylinder wake. This closure employs Jaynes’ maximum entropy principle to infer the probability distribution for mode amplitudes using exact statistical balance equations as side constraints. The analysis predicts mean amplitude values and modal energy levels in good agreement with direct Navier–Stokes simulation. In addition, it provides an analytical equation for the modal energy distribution.

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SHI, LEI, and HONGJUN GAO. "BIFURCATION ANALYSIS OF AN AMPLITUDE EQUATION." International Journal of Bifurcation and Chaos 23, no. 05 (2013): 1350081. http://dx.doi.org/10.1142/s0218127413500818.

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We study the bifurcation and stability of constant stationary solutions (u0, v0) of a particular system of parabolic partial differential equations as amplitude equations on a bounded domain (0, L) with Neumann boundary conditions. In this paper, the asymptotic behavior of the solutions (u0, v0) of the amplitude equations are considered. With the length L of the domain regarded as bifurcation parameter, branches of nontrivial solutions are shown by using the perturbation method. Moreover, local behavior of these branches are studied. We also analyze the stability of the bifurcated solutions.

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MARTÍNEZ-MARDONES, J., R. TIEMANN, W. ZELLER, and C. PÉREZ-GARCÍA. "AMPLITUDE EQUATION IN POLYMERIC FLUID CONVECTION." International Journal of Bifurcation and Chaos 04, no. 05 (1994): 1347–51. http://dx.doi.org/10.1142/s0218127494001052.

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The convective instabilities in viscoelastic polymeric Oldroyd-B models are studied. First, the nonlinear analysis of the stationary and oscillatory convection is carried out. Then, in the scope of weak nonlinear analysis, the coefficients of the amplitude equations are evaluated, in order to be in condition to estimate the possible behavior of stationary patterns and also travelling and standing waves. The values of these coefficients are determined by means of analytical and numerical techniques for convection in polymeric fluids.

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Bohlius, Stefan, Helmut R. Brand, and Harald Pleiner. "Amplitude Equation for the Rosensweig Instability." Progress of Theoretical Physics Supplement 175 (2008): 27–36. http://dx.doi.org/10.1143/ptps.175.27.

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Chow, C. C., S. J. Fromm, and H. Segur. "Analysis of a Hamiltonian Amplitude Equation." Journal of the Physical Society of Japan 62, no. 6 (1993): 1927–31. http://dx.doi.org/10.1143/jpsj.62.1927.

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Esposito, Giampiero, Emmanuele Battista, and Elisabetta Di Grezia. "Bicharacteristics and Fourier integral operators in Kasner spacetime." International Journal of Geometric Methods in Modern Physics 12, no. 05 (2015): 1550060. http://dx.doi.org/10.1142/s0219887815500607.

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The scalar wave equation in Kasner spacetime is solved, first for a particular choice of Kasner parameters, by relating the integrand in the wave packet to the Bessel functions. An alternative integral representation is also displayed, which relies upon the method of integration in the complex domain for the solution of hyperbolic equations with variable coefficients. In order to study the propagation of wave fronts, we integrate the equations of bicharacteristics which are null geodesics, and we are able to express them, for the first time in the literature, with the help of elliptic integral

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42

Wang, Kaier, Moira L. Steyn-Ross, D. Alistair Steyn-Ross, and Marcus T. Wilson. "Derivation of the Amplitude Equation for Reaction–Diffusion Systems via Computer-Aided Multiple-Scale Expansion." International Journal of Bifurcation and Chaos 24, no. 07 (2014): 1450101. http://dx.doi.org/10.1142/s0218127414501016.

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The amplitude equation describes a reduced form of a reaction–diffusion system, yet still retains its essential dynamical features. By approximating the analytic solution, the amplitude equation allows the examination of mode instability when the system is near a bifurcation point. Multiple-scale expansion (MSE) offers a straightforward way to systematically derive the amplitude equations. The method expresses the single independent variable as an asymptotic power series consisting of newly introduced independent variables with differing time and space scales. The amplitude equations are then

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Kolenda, Janusz. "On the fatigue-critical amplitude of random-amplitude stress." Polish Maritime Research 14, no. 2 (2007): 9–11. http://dx.doi.org/10.2478/v10012-007-0007-z.

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On the fatigue-critical amplitude of random-amplitude stress Uniaxial non-zero mean stress of constant circular frequency in the high-cycle fatigue regime is considered. It is assumed that equation of the S-N curve and modified Soderberg equation are applicable. For constant-amplitude stress, the fatigue-critical stress amplitude is defined as that which leads to failure during the required design life. For random-amplitude stress, expected values of the fatigue-critical stress amplitude and total fatigue damage accumulated during the required design life are estimated. It is found that the pr

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Ilgamov, M. A., and E. Sh Nasibullaeva. "Nonlinear analog of the Plesset equation for nonspherical motion of a gas bubble." Proceedings of the Mavlyutov Institute of Mechanics 3 (2003): 164–77. http://dx.doi.org/10.21662/uim2003.1.012.

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An equation analogous to Plesset's equation that describes the non-spherical motion of a gas bubble in an ideal incompressible fluid is derived. The terms containing the squares of the amplitude of the non-free perturbation were taken into account, whereas in the derivation of the Plesset equation only the linear terms are taken into account. On the basis of the obtained equations numerical calculations of the nonspherical oscillations of the bubble were carried out. A comparison of the obtained results and the Plesset equation showed that with a sufficient increase in the perturbation amplitu

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Reinfelds, Andrejs, Olgerts Dumbrajs, Harijs Kalis, Janis Cepitis, and Dana Constantinescu. "NUMERICAL EXPERIMENTS WITH SINGLE MODE GYROTRON EQUATIONS." Mathematical Modelling and Analysis 17, no. 2 (2012): 251–70. http://dx.doi.org/10.3846/13926292.2012.662659.

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Gyrotrons are microwave sources whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. This process is described by the system of two complex differential equations: nonlinear first order ordinary differential equation with parameter (averaged equation of electron motion) and second order partial differential equation for high frequency field (RF field) in resonator (Schrödinger type equation for the wave amplitude). The stationary problem of the single mode gyrotron equation in short time interval with real initial conditions was nu

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Qu, Yingming, Jianping Huang, Zhenchun Li, Zhe Guan, and Jinli Li. "Attenuation compensation in anisotropic least-squares reverse time migration." GEOPHYSICS 82, no. 6 (2017): S411—S423. http://dx.doi.org/10.1190/geo2016-0677.1.

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Anisotropic and attenuating properties of subsurface media cause amplitude loss and waveform distortion in seismic wave propagation, resulting in negative influence on seismic imaging. To correct the anisotropy effect and compensate amplitude attenuation, a compensated-amplitude vertical transverse isotropic (VTI) least-squares reverse time migration (LSRTM) method is adopted. In this method, the attenuation term of an attenuated acoustic wave equation is extended to a VTI quasi-differential wave equation, which takes care of effects from anisotropy and attenuation. The finite-difference metho

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Mace, R. L., M. A. Hellberg, R. Bharuthram, and S. Baboolal. "Electron-acoustic solitons in a weakly relativistic plasma." Journal of Plasma Physics 47, no. 1 (1992): 61–74. http://dx.doi.org/10.1017/s0022377800024089.

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Weakly relativistic electron-acoustic solitons are investigated in a two-electron-component plasma whose cool electrons form a relativistic beam. A general Korteweg-de Vries (KdV) equation is derived, in the small-|ø| domain, for a plasma consisting of an arbitrary number of relativistically streaming fluid components and a hot Boltzmann component. This equation is then applied to the specific case of electron-acoustic waves. In addition, the fully nonlinear system of fluid and Poisson equations is integrated to yield electron-acoustic solitons of arbitrary amplitude. It is shown that relativi

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Destrade, Michel, Alain Goriely, and Giuseppe Saccomandi. "Scalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2131 (2010): 1823–34. http://dx.doi.org/10.1098/rspa.2010.0508.

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We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy–Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materia

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Mohammed, Wael W. "Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation." Journal of the Egyptian Mathematical Society 23, no. 3 (2015): 482–89. http://dx.doi.org/10.1016/j.joems.2014.10.005.

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Klepel, Konrad, Dirk Blömker, and Wael W. Mohammed. "Amplitude equation for the generalized Swift–Hohenberg equation with noise." Zeitschrift für angewandte Mathematik und Physik 65, no. 6 (2013): 1107–26. http://dx.doi.org/10.1007/s00033-013-0371-8.

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