Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

We give a detailed analysis of the interaction of two NLS-described wave packets with different carrier waves for a nonlinear wave equation. By separating the internal dynamics of each wave packet from the dynamics caused by the interaction we prove that there is almost no interaction of such wave packets. We also prove the validity of a formula for the envelope shift caused by the interaction of the wave packets.     

Energy and coherence loss rates in a one-dimensional vibrational system interacting with a bath

We consider the problem of the dynamics of a Gaussian wave packet in a one-dimensional harmonic ocsillator interacting with a bath. This problem arises in many chemical and biochemical applications related to the dynamics of chemical reactions. We take the bath-oscillator interaction into account in the framework of the Redfield theory. We obtain closed expressions for Redfield-tensor elements, which allows finding the explicit time dependence of the average vibrational energy. We show that the energy loss rate is temperature-independent, is the same for all wave packets, and depends only on the spectral function of the bath. We determine the degree of coherence of the vibrational motion as the trace of the density-matrix projection on a coherently moving wave packet. We find an explicit expression for the initial coherence loss rate, which depends on the wave packet width and is directly proportional to the intensity of the interaction with the bath. The minimum coherence loss rate is observed for a “coherent” Gaussian wave packet whose width corresponds to the oscillator frequency. We calculate the limiting value of the degree of coherence for large times and show that it is independent of the structural characteristics of the bath and depends only on the parameters of the wave packet and on the temperature. It is possible that residual coherence can be preserved at low temperatures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 130–144, October, 2007.     

Dynamical properties of a particle in a wave packet: Scaling invariance and boundary crisis

Some dynamical properties present in a problem concerning the acceleration of particles in a wave packet are studied. The dynamics of the model is described in terms of a two-dimensional area preserving map. We show that the phase space is mixed in the sense that there are regular and chaotic regions coexisting. We use a connection with the standard map in order to find the position of the first invariant spanning curve which borders the chaotic sea. We find that the position of the first invariant spanning curve increases as a power of the control parameter with the exponent 2/3. The standard deviation of the kinetic energy of an ensemble of initial conditions obeys a power law as a function of time, and saturates after some crossover. Scaling formalism is used in order to characterise the chaotic region close to the transition from integrability to nonintegrability and a relationship between the power law exponents is derived. The formalism can be applied in many different systems with mixed phase space. Then, dissipation is introduced into the model and therefore the property of area preservation is broken, and consequently attractors are observed. We show that after a small change of the dissipation, the chaotic attractor as well as its basin of attraction are destroyed, thus leading the system to experience a boundary crisis. The transient after the crisis follows a power law with exponent −2.     

Rogue waves in baroclinic flows

We investigate an AB system, which can be used to describe marginally unstable baroclinic wave packets in a geophysical fluid. Using the generalized Darboux transformation, we obtain higher-order rogue wave solutions and analyze rogue wave propagation and interaction. We obtain bright rogue waves with one and two peaks. For the wave packet amplitude and the mean-flow correction resulting from the self-rectification of the nonlinear wave, the positions and values of the wave crests and troughs are expressed in terms of a parameter describing the state of the basic flow, in terms of a parameter responsible for the interaction of the wave packet and the mean flow, and in terms of the group velocity. We show that the interaction of the wave packet and mean flow and also the group velocity affect the propagation and interaction of the amplitude of the wave packet and the self-rectification of the nonlinear wave.     

Controlling unpredictability in the randomly driven Hénon–Heiles system

Noisy scattering dynamics in the randomly driven Hénon–Heiles system is investigated in the range of initial energies where the motion is unbounded. In this paper we study, with the help of the exit basins and the escape time distributions, how an external perturbation, be it dissipation or periodic forcing with a random phase, can enhance or mitigate the unpredictability of a system that exhibit chaotic scattering. In fact, if basin boundaries have the Wada property, predictability becomes very complicated, since the basin boundaries start to intermingle, what means that there are points of different basins close to each other. The main responsible of this unpredictability is the external forcing with random phase, while the dissipation can recompose the basin boundaries and turn the system more predictable. Therefore, we do the necessary simulations to find out the values of dissipation and external forcing for which the exit basins present the Wada property. Through these numerical simulations, we show that the presence of the Wada basins have a specific relation with the damping, the forcing amplitude and the energy value. Our approach consists on investigating the dynamics of the system in order to gain knowledge able to control the unpredictability due to the Wada basins.     

Computation of dynamical phase transitions in solids

This paper deals with the dynamics of phase boundaries in a nonlinear elastic two-phase material. We consider the elasticity system in 1D and the equations of anti-plane shear motion in 2D, where effects of viscosity and capillarity are neglected. These first-order conservation laws allow to represent phase boundaries as shock-like sharp interfaces. However, in contrast to what is known for homogeneous materials, the entropy inequality does not select a unique solution, and an additional criterion, the so-called kinetic relation, is required.Based on a scheme introduced by Hou, Rosakis and LeFloch [T. Hou, Ph. Rosakis, P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics, J. Comput. Phys. 150 (1999) 302–331] we focus on the numerical solution of a specific model system. Using a level-set technique to enforce the kinetic relation on the discrete level leads to a reformulation of the original system in the form of a system of conservation laws coupled to a Hamilton–Jacobi equation for each phase boundary. The numerical method for the reformulated system is constructed for unstructured meshes (in 2D), and a self-adaptive algorithm is introduced.In the 1D-case we show that the reformulated system has a solution that corresponds to exact dynamical phase boundaries of the elasticity system which obey the kinetic relation. To validate the method in 2D, we present computations on the interaction of a plane wave with a phase boundary. The efficiency of the adaptation mechanism is demonstrated by an example showing the development of microstructures by twinning.     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

New Approximations and Tests of Linear Fluctuation-Response for Chaotic Nonlinear Forced-Dissipative Dynamical Systems

We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation–dissipation theorem. Unlike the earlier work in developing fluctuation–dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation–dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation–dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.     

耗散对广义的WBK型耗散方程行波解的影响

运用平面动力系统理论对广义的WBK型耗散方程所对应的动力系统作了定性分析,给出了其在不同参数条件下的全局相图.研究了该方程行波解的性态与耗散系数r之间的关系,得到当耗散作用较大时行波表现为扭状孤波,当耗散作用较小时行波表现为衰减振荡解的结论.     

Monitoring the packet gap of real-time packet traffic

Real-time packet traffic is characterized by a strict deadline on the end-to-end time delay and an upper bound on the information loss. Due to the high correlation among consecutive packets, the individual packet loss does not well characterize the performance of real-time packet sessions. An additional measure of packet loss is necessary to adequately assess the quality of each real-time connection. The additional measure considered here is the average number of consecutively lost packets, also called the average packet gap. We derive a closed form for the average packet gap for the multiclassG/G/m/B queueing system in equilibrium and show that it only depends on the loss behavior of two consecutive packets. This result considerably simplifies the monitoring process of real-time packet traffic sessions. If the packet loss process is markovian, the consecutive packet loss has a geometric distribution.     

On the transition mode characterizing the triggering of a vibrator in the subsonic boundary layer on a plate

The problem of the development of two-dimensional linear perturbations in a boundary layer, generated by the triggering of a vibrator, is considered. Fourier transformations in the longitudinal coordinate and a Laplace transform in time are used to construct the solution. The inverse transforms are evaluated for large values of the characteristic time t and all values of the longitudinal coordinate x. Domains located downstream of the vibrator are studied in the first of which the perturbations will have the form of Tollmien-Schlichting waves that go over into a wave packet in the second domain. The identity in the structure of the wave packets, which are orthonormalized to the maximum amplitude for this packet for different frequencies of vibrator oscillation is noted.     

Propagation of wave packets through resonant quantum systems

We use a previously proposed modified saddle point method to describe the tunneling of a rectangular wave packet through a resonant quantum system. We calculate the shape of the wave packet obtained at the quantum system output analytically for different values of the level width. The result of propagation is a wave packet that is the superposition of complementary error functions. Comparing the result with the exact numerical solution obtained without using any asymptotic methods shows a rather good coincidence. We study the propagation of Gaussian and rectangular wave packets in detail for large values of the resonance level width.     

Uniform stabilization to equilibrium of a nonlinear fluid–structure interaction model

We consider uniform stability to a nontrivial equilibrium of a nonlinear fluid–structure interaction (FSI) defined on a two or three dimensional bounded domain. Stabilization is achieved via boundary and/or interior feedback controls implemented on both the fluid and the structure. The interior damping on the fluid combining with the viscosity effect stabilizes the dynamics of fluid. However, this dissipation propagated from the fluid alone is not sufficient to drive uniformly to equilibrium the entire coupled system. Therefore, additional interior damping on the wave component or boundary porous like damping on the interface is considered. A geometric condition on the interface is needed if only boundary damping on the wave is active. The main technical difficulty is the mismatch of regularity of hyperbolic and parabolic component of the coupled system. This is overcome by considering special multipliers constructed from Stokes solvers. The uniform stabilization result obtained in this article is global for the fully coupled FSI model.     

地形对正压大气Rossby波非线性相互作用的影响

本文采用弱非线性近似，推导出地形和Ｅｋｍａｎ摩擦共同作用下连续谱正压Ｒｏｓｓｂｙ波的非线性时空演化方程．根据这组方程，我们研究了窄角谱Ｒｏｓｓｂｙ波包的波波相互作用问题，当一个大振幅Ｒｏｓｓｂｙ波包通过大气传播时，如果它的振幅超过某个阈值，非线性相互作用会使一个尺度比它大的Ｒｏｓｓｂｙ波包和一个尺度比它小的Ｒｏｓｓｂｙ波包的振幅随时间指数增长，这两个次级波的本征频率会发生改变，Ｅｋｍａｎ摩擦、频率不匹配、地形坡度以及波包的空间演变共同决定了主波振幅的阈值及次级波本征频率的改变量．     

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

We consider the nonlinear Schrödinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrödinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.     

Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.     

Stability analysis of coupled structural acoustics PDE models under thermal effects and with no additional dissipation

In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber Ω. This acoustic wave equation is coupled on a boundary interface Γ₀ to a two dimensional system of thermoelasticity: this thermoelastic PDE is composed in part of a structural beam or plate equation, which governs the vibrations of flexible wall portion Γ₀ of the chamber Ω. Moreover, this elastic dynamics is coupled to a heat equation which also evolves on Γ₀, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface Γ₀, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of any additive feedback dissipation on the hard walls Γ₁ of the boundary . Under a certain geometric assumption on Γ₁, an assumption which has appeared in the literature in connection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.     

Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium

The phase transformation of the first kind in a non-linearly elastic heat conducting medium is simulated by the relationships on a strong discontinuity. A generalization of the Stefan formulation is given. An existence condition for stationary flow, analogous to the Gibbs phase equilibrium condition, is obtained for non-equilibrium phase boundaries. A pure dilatational phase transition in a compressible fluid and pure shear transformation of the twinning type in non-linearly elastic crystals are considered as model examples. The problem of the structure is solved for closure of the system of relationships on the shock.

A phase transformation ordinarily turns out to be localized in a narrow domain of space and it can be simulated in terms of the conditions on a strong discontinuity /1/. Formulation of the problem of the static equilibrium of liquid phases as well as of liquid and (non-linearly elastic) solid phases was given by Gibbs, who proposed a phase equilibrium criterion and formulated appropriate conditions on the shock; the extension of the Gibbs conditions to the case of the equilibrium of two solid phases is known in both the linear /2/ and non-linear /3/ theories of elasticity. The dynamic problem of the propagation of the equilibrium phase boundary is considered in the Stefan formulation as a rule, including the assumption about the continuity of the density (the strain tensor component) on the shock; the thermal problem is here separated from the mechanical one. Simulating the interphasal surface on the shock the temperature fields are merged by using the well-known Stefan conditions as well as the phase equilibrium condition that reduces to giving the temperature on the front.

The purpose of this paper is to extend the Stefan-Gibbs formulation to the case of the motion of a coherent isothermal phase boundary in a non-linearly elastic heat conducting medium and to derive the dynamic analogue of the phase equilibrium condition (and the Stefan conditions) with possible dissipation at the transformation front. Two dissipative mechanisms are examined, viscous and kinetic. The case of equilibrium phase boundaries was investigated in /4–6/.     

On a class of packet-switching networks on an infinite graph

A class of open queueing networks with packet switching is discussed. The configuration graph of the network may be finite or infinite. The external messages are divided into standard pieces (packets) each of which is transmitted as a single message. The messages are addressed, as a rule, to nearest neighbours and thereby the network may be treated as a small perturbation of the collection of isolated servers. The switching rule adopted admits overtaking: packets which appeared later may reach the delivery node earlier. The transmission of a message is completed when its last packet reaches the destination node. The main result of this paper is that the network possesses a unique stationary working regime. Weak dependence properties of this regime are established as well as the continuity with respect to the parameters of the external message flows.     

Markov-modulated queueing systems

Markov-modulated queueing systems are those in which the primary arrival and service mechanisms are influenced by changes of phase in a secondary Markov process. This influence may be external or internal, and may represent factors such as changes in environment or service interruptions. An important example of such a model arises in packet switching, where the calls generating packets are identified as customers being served at an infinite server system. In this paper we first survey a number of different models for Markov-modulated queueing systems. We then analyze a model in which the workload process and the secondary process together constitute a Markov compound Poisson process. We derive the properties of the waiting time, idle time and busy period, using techniques based on infinitesimal generators. This model was first investigated by G.J.K. Regterschot and J.H.A. de Smit using Wiener-Hopf techniques, their primary interest being the queue-length and waiting time.