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In this paper, we try to understand the geometry for a nonlocal nonlinear Schrödinger equation (nonlocal NLS) and its discrete version introduced by Ablowitz and Musslimani, Phys. Rev. Lett. 110, 064105 (2013); Phys. Rev. E 90, 042912 (2014). We show that, under the gauge transformations, the nonlocal focusing NLS and the nonlocal defocusing NLS are, respectively, gauge equivalent to a Heisenberg-like equation and a modified Heisenberg-like equation, and their discrete versions are, respectively, gauge equivalent to a discrete Heisenberg-like equation and a discrete modified Heisenberg-like equation. Although the geometry related to the nonlocal NLS and its discrete version is not very clear, from the gauge equivalence, we can see that the properties between the nonlocal NLS and its discrete version and NLS and discrete NLS have significant difference. By constructing the Darboux transformation for discrete nonlocal NLS equations including the cases of focusing and defocusing, we derive their discrete soliton solutions, which differ from the ones obtained by using the inverse scattering transformation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate nonintegrable semidiscrete Hirota equations, including the nonintegrable semidiscrete Hirota- equation and the nonintegrable semidiscrete Hirota+ equation. We focus on the topics on gauge-equivalent structures and dynamical behaviors for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that, under the discrete gauge transformations, the nonintegrable semidiscrete Hirota- equation and the nonintegrable semidiscrete Hirota+ equation are, respectively, gauge equivalent to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We prove that the two discrete gauge transformations are reversible. We study the dynamical properties for the two nonintegrable semidiscrete Hirota equations. The exact spatial period solutions of the two nonintegrable semidiscrete Hirota equations are obtained through the constructions of period orbits of the stationary discrete Hirota equations. We discuss the topic regarding whether the spatial period property of the solution to the nonintegrable semidiscrete Hirota equation is preserved to that of the corresponding gauge-equivalent nonintegrable semidiscrete equations under the action of discrete gauge transformation. By using the gauge equivalent, we obtain the exact solutions to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We also give the numerical simulations for the stationary discrete Hirota equations. We find that their dynamics are much richer than the ones of stationary discrete nonlinear Schrödinger equations. [ABSTRACT FROM AUTHOR]

This paper examines the evolution of the size distribution of Chinese cities. Since the relaxation of restrictions on rural-urban migration in the 1980s, China has experienced rapid urban growth. However, cities of different sizes have experienced varying patterns of growth. First, the evolution of city size distribution in China is described by documenting the growth in city size and in the number of existing cities. Then, focusing on the period from 1990 to 2000, the urban evolutionary trend is analysed by means of the Pareto law estimation and the mobility of cities between different size groups is examined with the Markov transition matrix. The convergence hypothesis in the city population growth process is also tested. The results suggest that, contrary to the expected dominance of large city growth, the Chinese city size distribution evened out during the 1990s, with small cities growing more rapidly than large cities. [ABSTRACT FROM AUTHOR]

Using the data from the longitudinal study "L'Établissement des nouveaux immigrants (ENI)," this paper analyses the residential trajectories of new immigrants in Montreal during the 1990s. Two methodologies were applied. First, survival models were used to identify the factors that determine the probabilities of moving and of access to homeownership. Second, the immigrants' mobility within the Montreal metropolitan area was analysed with conditional Logit models. The results show that highly-skilled immigrants have a better situation in the housing market, and that there is a strong relation between employment and housing. We also notice a residential dispersion from downtown to the suburbs, especially in spaces where the houses are older and the average income lower. Apart from that, our results illustrate the process of residential concentration, for immigrants from industrial countries, as well as for people from third-world countries. [ABSTRACT FROM AUTHOR]

In this paper, we consider the (2+1)-dimensional discrete fourth-order nonisospectral problem. By using the Lax technique, three new (2+1)-dimensional nonisospectral four-field integrable lattice hierarchies are constructed. Their reductions yield three (1+1)-dimensional isospectral four-field integrable lattice hierarchies due to Mlaszak-Marciniak. We make a comparison between the (1+1)-dimensional discrete fourth-order nonisospectral problem and the third-order nonisospectral problem. We found that the integrable lattice hierarchies related to the discrete fourth-order nonisospectral problem have new characteristics. [ABSTRACT FROM AUTHOR]

The higher-order rogue wave (RW) for a spatial discrete Hirota equation is investigated by the generalized (1,)-fold Darboux transformation. We obtain the higher-order discrete RW solution to the spatial discrete Hirota equation. The fundamental RWs exhibit different amplitudes and shapes associated with the spectral parameters. The higher-order RWs display triangular patterns and pentagons with different peaks. We show the differences between the RW of the spatially discrete Hirota equation and the discrete nonlinear Schrödinger equation. Using the contour line method, we study the localization characters including the length, width, and area of the first-order RWs of the spatially discrete Hirota equation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an integrable nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity 29, 915-946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one can see that there exists significant difference between the nonlocal complex mKdV equation and the classical complex mKdV equation. Through constructing the Darboux transformation for nonlocal complex mKdV equation, a variety of exact solutions including dark soliton,W-type soliton, M-type soliton, and periodic solutions are derived. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new semidiscrete Hirota equation which yields the Hirota equation in the continuum limit. We focus on the topic of how the discrete space step δ affects the simulation for the soliton solution to the Hirota equation. The Darboux transformation and explicit solution for the semidiscrete Hirota equation are constructed. We show that the continuum limit for the semidiscrete Hirota equation, including the Lax pair, the Darboux transformation and the explicit solution, yields the corresponding results for the Hirota equation as δ→0. [ABSTRACT FROM AUTHOR]

The flow characteristics of liquids in microtubes driven by a high pressure ranging from 1 MPa to 30 MPa are studied in this paper. The diameter of the microtube is from 3 μm to 10 μm and liquids composed of simple small molecules are chosen as the working fluids. The Reynolds number ranges from 0.1 to 24. The behavior of isopropanol and carbon tetrachloride under high pressure is found different from the prediction from conventional Hagen–Poiseuille (HP) equation. The normalized friction coefficient C* increases significantly with the pressure. From an analysis of the microtube deformation, liquid compressibility, viscous heating and wall slip, it may be seen that the viscosity at high pressure plays an important role here. An exponential function of viscosity vs pressure is introduced into the HP equation to counteract the difference between experimental and theoretical values. However, this difference is not so marked for di-water. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Examines the crystallization segregation of copolymers. Modification of the characteristics of polyolefins; Molecular packing of polyethylenes for meeting the different application requirements; Value of pure polypropylene.

The nonlinear Schrödinger (NLS) equation and Boussinesq equation are two very important integrable equations. They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright–bright, bright–dark, and dark–dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright–bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright–bright or bright–dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems. [ABSTRACT FROM AUTHOR]

Temperature dependence of the electron diffusion in metallic targets, where the electron–electron collision is the dominant process, is investigated with the help of an extended two-temperature model. In sharp contrast to the low electron temperature case, where only the electron–phonon collisions are commonly considered, the electron diffusion process underlying the high electron temperatures evolves dramatically different in both temporal and spatial domains. Calculated results of the ablation yield at different pulse durations are presented for a copper plate impinged by ultrashort laser pulses with energy fluences ranging from 0.1 J/cm2 to 10 J/cm2. The excellent agreement between the simulation results and the experimental data indicates the significant role of electron–electron collisions in material ablations using intense ultrashort laser pulses. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate highly efficient anti-Stokes signal generation inside an irregular multicore microstructure fiber, using unamplified femtosecond laser pulses. For this special type of fiber, there are randomly distributed cores with a diameter ranging from 0.6 to 1.5 μm. We have observed a large ratio of the amplitude of the anti-Stokes wave to that of the pump wave in the output spectrum. Our simulation results indicate that the features of the two zero-dispersion wavelengths of the fiber are responsible for meeting the phase-matching condition of four-wave mixing, which produces the observed large anti-Stokes wave. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 389–393, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21359 [ABSTRACT FROM AUTHOR]

We study a nonintegrable discrete nonlinear Schrödinger (dNLS) equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term. [ABSTRACT FROM AUTHOR]