Self-Diffraction of Bessel Light Beams in a Nonlinear Medium

The phenomenon of self-diffraction of Bessel light beams (BLB) in a nonlinear liquid medium has been studied experimentally and theoretically for the first time. Diffraction maxima which do not correspond to integer orders for an induced periodic structure have been registered. It has been shown that the appearance of these maxima is due to the initial BLB modulation, which can be caused by the departure of the axicon refracting surface from the ideal conical surface, as well as by the imperfection of the form of the Gaussian beam incident on the axicon.     

Nonlinear Wave Propagation in a Disordered Medium

In this paper we consider the problem of solitary wave propagation in a weakly disordered potential. Through a series of careful numerical experiments we have observed behavior which is in agreement with the theoretical predictions of Kivshar et al., Bronski, and Gamier. In particular we observe numerically the existence of two regimes of propagation. In the first regime the mass of the solitary wave decays exponentially, while the velocity of the solitary wave approaches a constant. This exponential decay is what one would expect from known results in the theory of localization for the linear Schrödinger equation. In the second regime, where nonlinear effects dominate, we observe the anomalous behavior which was originally predicted by Kivshar et al. In this regime the mass of the solitary wave approaches a constant, while the velocity of the solitary wave displays an anomalously slow decay. For sufficiently small velocities (when the theory is no longer valid) we observe phenomena of total reflection and trapping.     

Dynamics of Electronic Rydberg Wave Packets in the Presence of Laser-induced Core Transitions

Recent theoretical work on the dynamics of electronic Rydberg wave packets under the influence of laser-induced core transitions is reviewed. The discussion focuses on the intricate interplay between laser-modified electron correlation effects, radiative damping by the ionic core and the time evolution of electronic Rydberg wave packets. Via the stimulated light force this interplay manifests itself also in the atomic center of mass motion. A unified theoretical framework is provided by combining methods of quantum defect theory, stochastic techniques and semiclassical path expansions.     

Wave Packets in a Magnetic Field

A Gaussian wave packet confined to move on a plane perpendicular to a magnetic field remains a Gaussian wave packet in its time evolution. The average position and momentum follow the Ehrenfest equations which are identical to the classical Hamilton equations. A set of nonlinear equations decoupled from the Ehrenfest equation is derived for the parameters describing the time evolution of the density distribution and phases of a wave packet. Explicit solutions are then obtained when the "internal" angular momentum of the wave packet vanishes. In this case it is shown that the motion of the wave packet is a superposition of a translational motion, a rotation and a vibration.     

Bessel–Gaussian Beams in an Inhomogeneous Medium

Solutions of the parabolic equation which describes propagation of Bessel–Gaussian beams along the axis of an axially-symmetric medium having quadratic inhomogeneity have been obtained and investigated. It is shown that in the process of propagation the parameters of the beams change nonmonotonically. In the case of a transparent medium the beams periodically recover their initial structure.     

Modulated Dust-Acoustic Wave Packets in an Opposite Polarity Dusty Plasma System

The nonlinear propagation of the dust-acoustic bright and dark envelope solitons in an opposite polarity dusty plasma(OPDP) system(composed of non-extensive q-distributed electrons, iso-thermal ions, and positively as well as negatively charged warm dust) has been theoretically investigated. The reductive perturbation method(which is valid for a small, but finite amplitude limit) is employed to derive the nonlinear Schr¨odinger equation. Two types of modes, namely, fast and slow dust-acoustic(DA) modes, have been observed. The conditions for the modulational instability(MI) and its growth rate in the unstable regime of the DA waves are significantly modified by the effects of non-extensive electrons, dust mass, and temperatures of different plasma species, etc. The implications of the obtained results from our current investigation in space and laboratory OPDP medium are briefly discussed.     

Michelangelo Wave Packets and Interference Forces

The spreading of a quantum mechanical particle in the absence of a classical force is a well-known effect.However, there exist situations when this phenomenon is suppressed or even completely stimulated. In the present talk we first briefly summarize various non-spreading wave packets emphasizing in particular the Michelangelo wave packets which have recently been verified experimentally. We then turn to the example of contracting wave packets in D = 2 dimensions. Here the shrinking effect results from quantum interference which is very peculiar in D = 2. In particular, we show that this interference force can be understood in terms of correlations of position and momentum which do not exist in classical physics.     

Modulated Wave Packets in DNA and Impact of Viscosity

We study the nonlinear dynamics of a DNA molecular system at physiological temperature in a viscous media by using the Peyrard-Bishop model. The nonlinear dynamics of the above system is shown to be governed by the discrete complex Cinzburg-Landau equation. In the non-viscous limit, the equation reduces to the nonlinear Schroedinger equation. Modulational instability criteria are derived for both the cases. On the basis of these criteria, numerical simulations are made, which confirm the analytical predictions. The planar wave solution used as the initial condition makes localized oscillations of base pairs and causes energy localization. The results also show that the viscosity of the solvent in the surrounding damps out the amplitude of wave patterns.     

克尔型介质平板波导中的弱非线性横向电场波

对芯子为克尔型非线性介质、覆盖层和衬底为线性介质的平板波导，在弱非线性近似下，导出了这类波导横向电场波满足的色散方程和场分布，其物理意义清晰，且大大降低了数值计算的强度。     

Wave Packets in Honeycomb Structures and Two-Dimensional Dirac Equations

In a recent article (Fefferman and Weinstein, in J Am Math Soc 25:1169–1220, 2012), the authors proved that the non-relativistic Schrödinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located at the vertices of the Brillouin zone, a regular hexagon. In this paper, we study the time-evolution of wave-packets, which are spectrally concentrated near such conical points. We prove that the large, but finite, time dynamics is governed by the two-dimensional Dirac equations.     

激发介质相对不应态对螺旋波动力学行为的影响

采用元胞自动机模型研究激发介质相对不应态对螺旋波动力学行为的影响。数值模拟表明：元胞激发阈值存在一临界区间,该区间的螺旋波周期会突然增加,并存在一最大周期,在合适的系统尺寸和状态数下,螺旋波周期不再受相对不应态的影响而只取决于系统的激发阈值;相对不应态导致螺旋波“Z”型漫游、小范围无规律漫游、花瓣状漫游、锯齿状漫游、风车状漫游等复杂的波头运动。观察到稳定螺旋波、漫游螺旋波和螺旋波消失,并对产生这些现象的机制作简要的解释。     

Solitary Wave Dynamics in an External Potential

We study the behavior of solitary-wave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We consider solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newtons equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.Supported by NSERC grant 22R80976The support of Wenner-Gren Foundation is gratefully acknowledgedSupported partially by NSERC under NA7601 and by NSF under DMS-0400526Acknowledgement. B.L.G.J. and I.M.S. are grateful to J. Colliander for useful discussions and remarks and to ETH-Zürich for hospitality during their work on this paper. J.F. thanks T.-P. Tsai and H.-T. Yau for very useful discussions and correspondence which led to the results in [16,17].     

Quantum Tomography, Wave Packets, and Solitons

The wave packets, both linear and nonlinear (like solitons) signals described by a complex time-dependent function, are mapped onto positive probability distributions (tomograms). The quasidistributions, wavelets, and tomograms are shown to have an intrinsic connection. The analysis is extended to signals obeying to the von Neumann-like equation. For solitons (nonlinear signals) obeying the nonlinear Schrödinger equation, the tomographic probability representation is introduced. It is shown that in the probability representation the soliton satisfies a nonlinear generalization of the Fokker–Planck equation. Solutions to the Gross–Pitaevskii equation corresponding to solitons in a Bose–Einstein condensate are considered.     

Diffusion of Wave Packets in a Markov Random Potential

We consider the evolution of a tight binding wave packet propagating in a time dependent potential. If the potential evolves according to a stationary Markov process, we show that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.     

Propagation of General Wave Packets in Some Classical and Quantum Systems

In quantum mechanics the center of a wave packet is precisely defined as the center of probability. The center-of-probability velocity describes the entire motion of the wave packet. In classical physics there is no precise counterpart to the center-of-probability velocity of quantum mechanics, in spite of the fact that there exist in the literature at least eight different velocities for the electromagnetic wave. We propose a center-of-energy velocity to describe the entire motion of general wave packets in classical physical systems. It is a measurable quantity, and is well defined for both continuous and discrete systems. For electromagnetic wave packets it is a generalization of the velocity of energy transport. General wave packets in several classical systems are studied and the center-of-energy velocity is calculated and expressed in terms of the dispersion relation and the Fourier coefficients. These systems include string subject to an external force, monatomic chain and diatomic chain in one dimension, and classical Heisenberg model in one dimension. In most cases the center-of-energy velocity reduces to the group velocity for quasi-monochromatic wave packets. Thus it also appears to be the generalization of the group velocity. Wave packets of the relativistic Dirac equation are discussed briefly.