Dispersion of Singularities of Solutions for Schrödinger Equations

We consider the singularities of solutions for the Schrödinger evolution equation associated with where Q is a d×d real symmetric matrix with the eigenvalues ₁,,_d, and WC(R^d,R) satisfies W(x)=o(|x|²) as |x|. Under additional conditions, we show the dispersion of microlocal singularities of solutions due to the principal symbol in all directions at time and in the nondegenerate directions at t. We also show the weaker dispersion of microlocal singularities of solutions due to the subprincipal symbol W in the degenerate directions at t if W satisfies W(x)=O(|x|¹⁺) as |x| for some 0<<1 and additional conditions. In particular, we prove the dispersion of microlocal singularities of solutions at resonant times when H is a perturbed harmonic oscillator.Partly supported by Grand-in-Aid for Young Scientists (B) 14740110, Japan Society of the Promotion of Science; and Mathematical Sciences Research Institute in BerkeleyDedicated to Professor Mitsuru Ikawa on his sixtieth birthday     

“Principle of Indistinguishability” and Equations of Motion for Particles with Spin

In this work we review the derivation of Dirac and Weinberg equations based on a principle of indistinguishability for the (j,0) and (0,j) irreducible representations (irreps) of the homogeneous Lorentz group (HLG). We generalize this principle and explore its consequences for other irreps containing j1. We rederive Ahluwalia–Kirchbach equation using this principle and conclude that it yields equations of motion for any representation containing spin j and lower spins. We also use the obtained generators of the HLG for a given representation to explore the possibility of the existence of first order equations for that representation. We show that, except for j= , there exists no Dirac-like equation for the (j,0)(0,j) representation nor for the ( , ) representation. We rederive Kemmer–Duffin–Petieau (KDP) equation for the (1,0)( , )(0,1) representation by this method and show that the (1, )( ,1) representation satisfies a Dirac-like equation which describes a multiplet of with masses m and m/2, respectively.     

Transformation Formula of the “Second” Order Mock Theta Function

We give a transformation formula for the “second order” mock theta function

Manifestation of Three-Body Forces in Three-Body Bethe–Salpeter and Light-Front Equations

Bethe–Salpeter and light-front bound state equations for three scalar particles interacting by scalar exchange-bosons are solved in ladder truncation. In contrast to two-body systems, the three-body binding energies obtained in these two approaches differ significantly from each other: the ladder kernel in light-front dynamics underbinds by approximately a factor of two compared to the ladder Bethe–Salpeter equation. By taking into account three-body forces in the light-front approach, generated by two exchange-bosons in flight, we find that most of this difference disappears; for small exchange masses, the obtained binding energies coincide with each other.     

Is It Possible to Derive Newtonian Equations of Motion with Memory?

In this paper for a given example we proved that the Riemann-Liouville fractional integral term appears naturally and relates the external force with acceleration within the fractional Newtonian equation. The consideration of some self-similar process that leads to the fractional integral as well as some possible generalizations of the proposed model was discussed.     

Symmetry Classification of the One–Dimensional Second Order Equation of a Hydrodynamic Type

Abstract

The paper contains a symmetry classification of the one–dimensional second order equation of a hydrodynamical type L(Lu)+λLu=F (u), where L ≡ ? _t+u? _x. Some classes of exact solutions of this equation are given.     

Investigations of Pulse‐Plasma‐Polymer Coating on TiO2 Nanoparticles and Their Dispersion

An amorphous acrylic acid (AA) polymer coating was generated on TiO₂ nanoparticles through pulse radio frequency (RF) plasma polymerization. The AA plasma synthesis mechanism was studied by its optical emission spectrum. The chemical structures of AA–plasma‐polymer were carefully investigated by Fourier transform infrared spectroscopy (FTIR). The dispersion behaviors of AA‐coated and uncoated TiO₂ nanoparticles in glycol solution were characterized by ultraviolet absorbency and particle size distribution measurements. The results showed that the aggregation of TiO₂ nanoparticles in glycol solution was effectively lowered and the dispersion was improved a lot after AA–plasma‐polymer coating. The pulse plasma coating parameters played an important role in the dispersion enhancement of TiO₂ nanoparticles. By properly regulating the pulse discharge parameters, the system could gain the highest radical–monomer reactions rate, the most compatible functional groups on the nanoparticles, and the best dispersion in the background media.     

Reduced System of Bethe Equations for Three‐Boson Interaction

The reduced system of Bethe equations is found for the threeboson interaction. The invariants and surfaces in the space of Bethe parameters connected with this system are constructed. It is shown that on these surfaces the eigenvectors can be presented in a power form. The connection between Bethe ansatz and polynomial extensions of Lie algebras is briefly discussed. The computational procedures corresponding to the original Bethe equations and the method of polynomial algebras are compared.     

Topological Solitons of the Nonlinear Schrödinger’s Equation with Fourth Order Dispersion

This paper obtains the topological 1-soliton solution of the nonlinear Schrödinger’s equation, in a non-Kerr law media, with fourth order dispersion. An exact 1-soliton solution is obtained. The types of nonlinearity that are studied in this paper are Kerr law and power law.     

Two -Body Correlation Transport Theory for Heavy Ion Collisions Ⅰ. Theory Model: Construction of Coherent State Representation and Basic Equations of Motion

A two-body correlation transport theory(TBCTT) for describing the dynamical process in heavy ion collisions(HIC)is established by means of time-dependent coherent single particle state representations and the two-body correlation dynamics.Containing time-dependent mean field effect, Pauli block, and two-body collisions,this model is capable of describing the time-evolution of nonuniform nuclear matter,fluctuation effects and dynamical formation of fragments in HIC.     

Constrained Correlation Dynamics of QED in Canonical Form(Ⅰ) Equations of Motion for Correlation Green's Functions

With the aid of generating-functional technique and in the framework of correlation dynamics,we establish the constrained correlation dynamics of QED in the temporal gauge and canonical quantization form,and obtain a closed set of equations of motion for correlation Green's functions.     

Thermal Cross‐Linking of Poly(Vinyl Methyl Ether). I. Effect of Cross‐Linking Process on the Viscoelastic Properties

Abstract

Thermal cross‐linking of poly(vinyl methyl ether) (PVME) in the absence of cross‐linking agent, was detected rheologically. The linear viscoelastic properties of PVME were found to be greatly changed by the onset of the cross‐linking process. The viscoelastic material functions, such as dynamic shear moduli, G′ and G″, complex shear viscosity, η*, and loss tangent, tan δ, were found to be sensitive to the structure changes during the cross‐linking process and the formation of a three‐dimensional polymer network. At the onset temperature of the cross‐linking process, an abrupt increase in G′, G″, and η* (several orders of magnitude) during dynamic temperature ramps (2°C/min heating rate) was observed with some frequency dependence. The temperature dependence of tan δ was found to be frequency independent at the gel‐point, T _gel, that is, the crossover in tan δ regardless of the value of frequency can be taken as an accurate method for determination of T _gel. The coincidence of G′ and G″ at the gel‐point cannot be considered a general method for evaluation of T _gel due to its high frequency dependence, that is, T _gel determined from the crossover of G′ and G″ in the dynamic temperature ramp at 1 rad/sec is about 20°C less than at 100 rad/sec. Furthermore, a dramatic increase in η₀ above the minimum (“v” shape) was observed at T = T _gel in agreement with the value obtained from tan δ vs. T (190°C). The time–temperature‐superposition principle was found to be valid only for temperatures lower than the T _gel (190°C); the principle failed at T ≥ 190°C. This was clearly seen in the low‐frequency region as a deviation from the terminal slope in the G′ curve. Similar behavior was observed in the modified Cole–Cole analyses (G″ vs. G′) that is, the curves start to deviate at 190°C.     

Conditional Lie-Bcklund Symmetry and New Variable Separation Solutions of the Third Order KdV-Type Equations

The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-B¨acklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.     

Rheology and Dispersion Behavior of Ethylene‐Vinyl Acetate Copolymers/TiO2 Masterbatches Prepared by Melt‐Compounding

The rheology and dispersion behavior of ethylene‐vinyl acetate (EVA) copolymer/TiO₂ masterbatches prepared by melt‐compounding were investigated. The pure EVA exhibits obviously pseudoplastic behavior and the apparent viscosity decreases remarkably at experimental temperatures, especially in the range of 100–500 s^?1. The EVA/TiO₂ masterbatches exhibit similar shear rheology behavior with pure EVA and the apparent viscosities are obviously higher than that of pure EVA when the TiO₂ content is above 10 wt.%. Field‐emission scanning electron microscopy (FE‐SEM) and energy dispersive x‐ray spectroscopy (EDX) show that relatively low TiO₂ loading and moderate shear rate are helpful for the improvement of dispersion behavior of TiO₂ nanoparticles; moreover, the dispersion behavior of TiO₂ greatly influences the melt viscosity. The extensional rheology of pure EVA decreases with increasing extension rate, especially at low melt temperatures. EVA/TiO₂ masterbatches have similar extensional rheology behavior as pure EVA and the TiO₂ loading has almost no influence on the extensional viscosity of materbatches.     

Enhanced Compression of Fundamental Solitons in Dispersion Decreasing Fibers with Negative Third-order Dispersion Ⅱ. Effects of Fiber Characteristics and Initial Soliton Width

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Enhanced Compression of Fundamental Solitons in Dispersion Decreasing Fibers with Negative Third-order Dispersion Ⅱ. Effects of Fiber Characteristics and Initial Soliton Width

Previous work have shown that the combined effects of negative third-order dispersion (TOD) and Raman self-scattering (RSS) can significantly enhance soliton compression in dispersion decreasing fibers (DDFs). In this paper, the effects of the negative TOD coefficient and the effective amplification of DDF′s on the performance of soliton compression are investigated. It is shown that for a given initial soliton width and a given effective amplification, there exists an optimum value of the negative TOD coefficient of the DDF at which the enhancement in soliton compression is maximum. It is also shown that the compression enhancement saturates when the effective amplification exceeds a certain value, which has been explained as a compromise between the higher-order effects induced increase of the ratio of input to output group-velocity dispersion coefficients of the DDF and the TOD induced non-adiabatic compression characteristics. The dependence of the compression enhancement on the initial soliton width have also be studied and the scheme is found works well for solitons with initial widths less than 3 ps.     

Decay Estimates of Solutions for the Equations of Motion of Compressible Viscous and Heat-Conductive Gases in an Exterior Domain in ℝ3

We consider the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ³. We give the L_^qmL_^p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem.     

Generation of Ultrashort Light Bullets in Dispersive Kerr Media Using the Fourth—Order Dispersion—Dependent Spatiotemporal Instability

A new method for generation of a train of ultrashort pulses or a sequence of ultrashort light bullets is proposed. This method is based on the fourth-order dispersion-dependent spatiotemporal instability in dispersive Kerr media. The repetition-rate of the generated bullets can be made quite large by increasing the corresponding spatial modulation frequency locating in the new instability region resulted from fourth-order dispersion.