Branching random motions, nonlinear hyperbolic systems and traveling waves

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independient of random motion, and intensities of reverses are defined by a particle's current direction. A soluton of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) have a so-called McKean representation via such processes. Commonly this system possesses traveling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed.This Paper realizes the McKean programme for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.

Generation of inertia-gravity waves in the rotating thermal annulus by a localised boundary layer instability

Waves with periods shorter than the inertial period exist in the atmosphere (as inertia-gravity waves) and in the oceans (as Poincaré and internal gravity waves). Such waves owe their origin to various mechanisms, but of particular interest are those arising either from local secondary instabilities or spontaneous emission due to loss of balance. These phenomena have been studied in the laboratory, both in the mechanically-forced and the thermally-forced rotating annulus. Their generation mechanisms, especially in the latter system, have not yet been fully understood, however. Here we examine short period waves in a numerical model of the rotating thermal annulus, and show how the results are consistent with those from earlier laboratory experiments. We then show how these waves are consistent with being inertia-gravity waves generated by a localised instability within the thermal boundary layer, the location of which is determined by regions of strong shear and downwelling at certain points within a large-scale baroclinic wave flow. The resulting instability launches small-scale inertia-gravity waves into the geostrophic interior of the flow. Their behaviour is captured in fully nonlinear numerical simulations in a finite-difference, 3D Boussinesq Navier-Stokes model. Such a mechanism has many similarities with those responsible for launching small- and meso-scale inertia-gravity waves in the atmosphere from fronts and local convection.

Inverting recurrent neural networks for internal model control of nonlinear systems

In this paper, we show how a set of recently derived theoretical results for recurrent neural networks can be applied to the production of an internal model control system for a nonlinear plant. The results include determination of the relative order of a recurrent neural network and invertibility of such a network. A closed loop controller is produced without the need to retrain the neural network plant model. Stability of the closed-loop controller is also demonstrated.

Internal model control of nonlinear systems through the inversion of recurrent neural networks

Recurrent neural networks can be used for both the identification and control of nonlinear systems. This paper takes a previously derived set of theoretical results about recurrent neural networks and applies them to the task of providing internal model control for a nonlinear plant. Using the theoretical results, we show how an inverse controller can be produced from a neural network model of the plant, without the need to train an additional network to perform the inverse control.

Mountain waves in two-layer sheared flows: Critical-level effects, wave reflection, and drag enhancement

Internal gravity waves generated in two-layer stratified shear flows over mountains are investigated here using linear theory and numerical simulations. The impact on the gravity wave drag of wind profiles with constant unidirectional or directional shear up to a certain height and zero shear above, with and without critical levels, is evaluated. This kind of wind profile, which is more realistic than the constant shear extending indefinitely assumed in many analytical studies, leads to important modifications in the drag behavior due to wave reflection at the shear discontinuity and wave filtering by critical levels. In inviscid, nonrotating, and hydrostatic conditions, linear theory predicts that the drag behaves asymmetrically for backward and forward shear flows. These differences primarily depend on the fraction of wavenumbers that pass through their critical level before they are reflected by the shear discontinuity. If this fraction is large, the drag variation is not too different from that predicted for an unbounded shear layer, while if it is small the differences are marked, with the drag being enhanced by a considerable factor at low Richardson numbers (Ri). The drag may be further enhanced by nonlinear processes, but its qualitative variation for relatively low Ri is essentially unchanged. However, nonlinear processes seem to interact constructively with shear, so that the drag for a noninfinite but relatively high Ri is considerably larger than the drag without any shear at all.

Periodic waves and solitons in a nonlinear fibre with resonant impurities

We shall consider a coupled nonlinear Schrodinger equation- Bloch system of equations describing the propagation of a single pulse through a nonlinear dispersive waveguide in the presence of resonances; this could be, for example, a doped optical fibre. By making use of the integrability of the dynamic equations, we shall apply the finite-gap integration method to obtain periodic solutions for this system. Next, we consider the problem of the formation of solitons at a sharp front pulse and, by means of the Whitham modulational theory, we derive the amplitude and velocity of the largest soliton.

Formation of soliton trains in Bose-Einstein condensates as a nonlinear Fresnel diffraction of matter waves

The problem of generation of atomic soliton trains in elongated Bose-Einstein condensates is considered in framework of Whitham theory of modulations of nonlinear waves. Complete analytical solution is presented for the case when the initial density distribution has sharp enough boundaries. In this case the process of soliton train formation can be viewed as a nonlinear Fresnel diffraction of matter waves. Theoretical predictions are compared with results of numerical simulations of one- and three-dimensional Gross-Pitaevskii equation and with experimental data on formation of Bose-Einstein bright solitons in cigar-shaped traps. (C) 2003 Elsevier B.V. All rights reserved.

The nonlinear essence of gravitational waves

A critical review of gravitational wave theory is made. It is pointed out that the usual linear approach to the gravitational wave theory is neither conceptually consistent nor mathematically justified. Relying upon that analysis it is argued that-analogously to a Yang-Mills propagating field, which must be nonlinear to carry its gauge charge-a gravitational wave must necessarily be nonlinear to transport its own charge-that is, energy-momentum.

Remarks on Parametric Surface Waves in a Nonlinear and Non-Ideally Excited Tank

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Effects of a temperature dependent viscosity in surface nonlinear waves propagating in a shallow fluid heated from below

The effects of a temperature dependent viscosity in surface nonlinear waves propagating in a shallow fluid heated from below are investigated. It is shown that the (2+1)-dimensional Burgers equation may appear as the equation governing the upper free surface perturbations of a Bénard system, even when the viscosity is assumed to depend on temperature. The critical Rayleigh number for the appearance of waves governed by the Kadomtsev-Petviashvili equation, however, will be smaller than R=30, which is the critical number obtained for a constant viscosity. © 1992.

Nonlinear dynamics of short traveling capillary-gravity waves

We establish a Green-Nagdhi model equation for capillary-gravity waves in (2+1) dimensions. Through the derivation of an asymptotic equation governing short-wave dynamics, we show that this system possesses (1 + 1) traveling-wave solutions for almost all the values of the Bond number θ (the special case θ=1/3 is not studied). These waves become singular when their amplitude is larger than a threshold value, related to the velocity of the wave. The limit angle at the crest is then calculated. The stability of a wave train is also studied via a Benjamin-Feir modulational analysis. ©2005 The American Physical Society.

Dynamical analysis of turbulence in fusion plasmas and nonlinear waves

Turbulence is one of the key problems of classical physics, and it has been the object of intense research in the last decades in a large spectrum of problems involving fluids, plasmas, and waves. In order to review some advances in theoretical and experimental investigations on turbulence a mini-symposium on this subject was organized in the Dynamics Days South America 2010 Conference. The main goal of this mini-symposium was to present recent developments in both fundamental aspects and dynamical analysis of turbulence in nonlinear waves and fusion plasmas. In this paper we present a summary of the works presented at this mini-symposium. Among the questions to be addressed were the onset and control of turbulence and spatio-temporal chaos. (C) 2011 Elsevier B. V. All rights reserved.

Remarks on Parametric Surface Waves in a Nonlinear and Non-Ideally Excited Tank

We studied free surface oscillations of a fluid in a cylinder tank excited by an electric motor with limited power supply. We investigated the possibility of parametric resonance in this system, showing that the excitation mechanism can generate chaotic response. Numerical experiments are carried out to present the existence of several types of regular and chaotic attractors. For the first time powers (power of the motor, power consumed by the damping force under fluid free surface oscillations, and a total power) are calculated, investigated, and shown for different regimes, regular and chaotic ones for parametric resonance interactions. [DOI: 10.1115/1.4005844]