Narrow Results

We study nonlinear dispersive wave systems described by hyperbolic PDE's in ℝ^d and difference equations on the lattice ℤ^d. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

We discuss the intermediate wave-packet formalism for analytically quantifying the energy dependence of the two-flavor conversion formula that is usually considered for analyzing neutrino oscillations and adjusting the focusing horn, target position and/or detector location of some flavor conversion experiments. Following a sequence of analytical approximations where we consider the second-order corrections in a power series expansion of the energy, we point out a residual time-dependent phase which, in addition to some well-known wave-packet effects, can subtly modify the oscillation parameters and limits. In the present precision era of neutrino oscillation experiments where higher precision measurements are required, we quantify some small corrections in neutrino flavor conversion formulas which lead to a modified energy-dependence for ν_μ ↔ ν_e oscillations.

We consider a class of models for the relativistic covariant wave packets (WPs) which can be used as asymptotically free in and out states in the quantum field theoretical formalisms for description of the neutrino flavor oscillation phenomenon. We demonstrate that the new “asymmetric” wave packet (AWP) is an appropriate alternative to the more conventional “symmetric” WPs, like the so-called relativistic Gaussian packet (RGP) widely used in the quantum field theory (QFT)-based approaches to neutrino oscillations. We show that RGP is not a particular case of AWP, although many properties of these models are almost identical in the quasistable regime. We discuss some features of AWP distinguishing it from RGP.

Some recent theoretical studies tended to employ analytically-continuous gaussian, or infinite-bandwidth step pulses to examine tunneling process. The stationary phase method is often employed to this aim. However, gaussian functions do not have a well-defined front, such that their speed of propagation becomes ambiguous. Also, infinite bandwidth signals cannot propagate through any real physical medium (whose transfer function is therefore finite) without pulse distortion, which also leads to ambiguities in determining propagation velocity during the tunneling process. In this manuscript, we appoint some incompatibilities with the application of the method of stationary phase in deriving tunneling times which, eventually, can ruin the superluminal interpretation of transition times.

An approach to carrying out accurate quantum dynamics simulations of chemical reactions, termed the real wave packet (RWP) method, is outlined. The method focuses on propagation of just the real part of a complex-valued wave packet, halving computational memory and effort requirements in comparison with comparable high accuracy quantum propagation methods. Applications to 3-atom and 4-atom chemical reactions are reviewed. Potential future directions are indicated.

In this work, the nonadiabatic effects in the femtosecond photoionization of NaI molecule driven by pump and probe pulses are investigated theoretically using the accurate quantum wave packet method. The calculation with the inclusion of nonadiabatic coupling remarkably improves the agreement with experimental results of Jouvet et al. [J Phys Chen A101: 2555, 1997], indicating the importance of the nonadiabatic effect. Moreover, the dynamical evolutions of wave packets and their corresponding time-resolved photoelectron spectroscopy are presented both on the adiabatic and diabatic potentials. These comparisons contribute to further understanding of the influences of nonadiabatic effects in the femtosecond photoionization of NaI molecule.

We show that for every nonzero vector $\varphi$ in ${ℂ}^N$, the discrete wave packet system ${\{D_a{M}_l{T}_k\varphi \}}_{a\in U(N),l,k\in {𝕀}_N}$ constitutes a frame for the unitary space ${ℂ}^N$. An application of this result is given, where frame conditions cannot be derived from discrete wavelet systems in ${ℂ}^{\mathbb{N}}$. The canonical dual of discrete wave packet frame is also discussed.

It is demonstrated that the Cartesian momenta and positions offer a proper description for the motion of particles constrained on the surface of a sphere. In classical limit, their mean values on the classical wave packet go over to classical quantities.

We study Gaussian beams for the wave equation on a Riemannian manifold. For the transport equation we geometrize the leading term at the center of the Gaussian beam. More precisely, if

In this study, a radical hypothesis concerning the wave-particle duality exhibited by extremely small objects in nature is explored by developing a Planck lattice model of space–time that grapples with the uncertain dynamic quantum structure of space–time at the Planck scale. Upon applying the Planck lattice model to two notable experiments that most clearly demonstrate the essence of wave-particle duality, one immediately finds that it successfully shows in a relatively straightforward and physically consistent manner how parcels of matter and energy, i.e., electrons and photons, respectively, can behave like waves. The heuristic concepts regarding the underlying structure of space–time contained herein are intended to show the classical particle description of matter and energy as fundamental, while at the same time doing away with the widely held notion of a continuous space–time.

That branch of semiotics called semantics deals with the relationships between meanings and representations. In my view meanings exist only in brains, which have no representations in them. A meaning is the focus of an activity pattern that may occupy the entire available brain. It is constructed by intentional action, that is followed by learning from the consequences of that action. Communication between brains requires that meanings be represented by construction of words, gestures, symbolse, etc., which elicit meanings in other brains. A representation, is a material object or process, that has no meaning in itself. EEG data indicate that meaning is carried by spatiotemporal patterns of neural activity in frames like a motion picture. The discrete steps occur by cortical phase transitions in the 2-D arrays of neurons interacting synaptically to form wave packets. The rapid exchanges of discrete wave packets between interactive cortical domains generate self-organized dynamics controlling behavior including making representations of meaning. The dynamics of neural arrays is described by sets of differential equations, leading to the possibility of constructing intelligent machines that have the capacity to generate and represent meanings that are comparable to those existing in small animals in machines currently under study in situated robotics.