A Schamel equation for ion acoustic waves in superthermal plasmas

An investigation of the propagation of ion acoustic waves in nonthermal plasmas in the presence of trapped electrons has been undertaken. This has been motivated by space and laboratory plasma observations of plasmas containing energetic particles, resulting in long-tailed distributions, in combination with trapped particles, whereby some of the plasma particles are confined to a finite region of phase space. An unmagnetized collisionless electron-ion plasma is considered, featuring a non-Maxwellian-trapped electron distribution, which is modelled by a kappa distribution function combined with a Schamel distribution. The effect of particle trapping has been considered, resulting in an expression for the electron density. Reductive perturbation theory has been used to construct a KdV-like Schamel equation, and examine its behaviour. The relevant configurational parameters in our study include the superthermality index κ and the characteristic trapping parameter β. A pulse-shaped family of solutions is proposed, also depending on the weak soliton speed increment u₀. The main modification due to an increase in particle trapping is an increase in the amplitude of solitary waves, yet leaving their spatial width practically unaffected. With enhanced superthermality, there is a decrease in both amplitude and width of solitary waves, for any given values of the trapping parameter and of the incremental soliton speed. Only positive polarity excitations were observed in our parametric investigation. 

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

Effect of mean stress on multiaxial ratcheting life: a simplified life prediction model based on average equivalent stress amplitude

This article proposes a model to predict uniaxial and multiaxial ratcheting life by addressing the three primary parameters that influence failure life: fatigue damage, ratcheting damage and the multiaxial loading path. These three factors are addressed in the present model by (a) the stress amplitude for fatigue damage, (b) mean stress-dependent Goodman equation for ratcheting damage and (c) an inherent weight factor based on average equivalent stress to account for the multiaxial loading. The proposed model requires only two material constants which can be easily determined from uniaxial symmetric stress-controlled fatigue tests. Experimental ratcheting life data collected from the literature for 1025 and 42CrMo steel under multiaxial proportional and nonproportional constant amplitude loading ratcheting with triangular sinusoidal and trapezoidal waveform (i.e. linear, rhombic, circular, elliptical and square stress paths) have shown good agreement with the proposed model.

Short-wave instabilities in the Benjamin-Bona-Mahoney-Peregrine equation: theory and numerics

In this paper we discuss the nonlinear propagation of waves of short wavelength in dispersive systems. We propose a family of equations that is likely to describe the asymptotic behaviour of a large class of systems. We then restrict our attention to the analysis of the simplest nonlinear short-wave dynamics given by U-0 xi tau, = U-0 - 3(U-0)(2). We integrate numerically this equation for periodic and non-periodic boundary conditions, and we find that short waves may exist only if the amplitude of the initial profile is not too large.

Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Fatigue crack growth of Al 7475 T761 under constant and variable amplitude loading

Experimental programs in constant and variable amplitude loading were performed to obtain a x N curves and to study retardation in fatigue crack growth due to overloads. The main aim of this research program was to analyse the effect of overload ratio and number of overload peaks. The effect of underloads, before and after the overload blocks was also studied. The generalised equation of Paris-Erdogan type was used for modelling of obtained data on crack propagation under constant amplitude load.

Migração por equação de onda em meios anisotrópicos com correção de amplitude

Este trabalho discute dois aspectos da migração em profundidade através da continuação para baixo dos campos de onda: o tratamento de modos evanescentes e a correção da amplitude dos eventos migrados. Estes dois aspectos são discutidos em meios isotrópicos e para uma classe de meios anisotrópicos. Migrações por diferenças finitas (FD) e por diferenças finitas e Fourier (FFD) podem ser instáveis em meios com forte variação lateral de velocidade. Estes métodos utilizam aproximações de Padé reais para representar o operador que descreve a propagação de ondas descendentes. Estas abordagens não são capazes de tratar corretamente os modos evanescentes, o que pode levar à instabilidades numéricas em meios com forte variação lateral de velocidade. Uma solução possível para esse problema é utilizar aproximação de Padé complexa, que consegue melhor representar os modos evanescentes associados às reflexões pós-críticas, e neste trabalho esta aproximação é utilizada para obter algoritmos FD e híbrido FD/FFD estáveis para migração em meios transversalmente isotrópicos com eixo de simetria vertical (VTI), mesmo na presença de forte variação nas propriedades elásticas do meio. A estabilidade dos algoritmos propostos para meios VTI foi validada através da resposta ao impulso do operador de migração e pela sua aplicação na migração de dados sintéticos, em meios fortemente heterogêneos. Métodos de migração por equação de onda em meios heterogêneos não tratam corretamente a amplitude dos eventos durante a propagação. As equações de onda unidirecionais tradicionais descrevem corretamente apenas a parte cinemática da propagação do campo de onda. Assim, para uma descrição correta das amplitudes deve-se usar as equações de onda unidirecionais de amplitude verdadeira. Em meios verticalmente heterogêneos, as equações de onda unidirecionais de amplitude verdadeira podem ser resolvidas analiticamente. Em meios lateralmente heterogêneos, essas equações não possuem uma solução analítica. Mesmo soluções numéricas tendem a ser instáveis. Para melhorar a compensação de amplitude na migração, em meios com variação lateral de velocidade, é proposto uma aproximação estável para solução da equação de onda unidirecional de amplitude verdadeira. Esta nova aproximação é implementada nas migrações split-step e diferenças finitas e Fourier (FFD). O algoritmo split-step com correção de amplitude foi estendido para meios VTI. A migração pré e pós-empilhamento de dados sintéticos, em meios isotrópicos e anisotrópicos, confirmam o melhor tratamento das amplitudes e estabilidade dos algoritmos propostos.

Limit on the pion distribution amplitude

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Modelling relativistic solitary wave interactions in over-dense plasmas: a perturbed nonlinear Schröndinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrödinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrödinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude

Electron acoustic soliton energy of the Kadomtsev-Petviashvili equation in the Earth’s magnetotail region at critical ion density

The nonlinear properties of small amplitude electron-acoustic solitary waves (EAWs) in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili (KP) equation. At the critical ion density, the KP equation is not appropriate for describing the system. Hence, a new set of stretched coordinates
is considered to derive the modified KP equation. Moreover, the solitary solution, soliton energy and the associated electric field at the critical ion density were computed. The present investigation can be of relevance to the electrostatic solitary structures observed in various space plasma environments, such as Earth’s magnetotail region.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.