Semiclassical Limit of the Gross-Pitaevskii Equation in an Exterior Domain

In this paper, we study the semiclassical limit of the Gross-Pitaevskii equation (a cubic nonlinear Schrödinger equation) with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches 0.     

Singular behaviour and asymptotic stress field of a crack terminating at a bimaterial interface

A crack terminating at an interface of two dissimilar elastic materials is investigated. It is found that the asymptotic stress field near the crack tip is in general composed of two parts with each part being characterized by one singularity. The detailed relation of the two singularities with the bimaterial properties is given for some special cases of the crack.     

Molecular dynamics simulation of crack-tip processes in copper

The crack tip processes in copper under mode II loading have been simulated by a molecular dynamics method. The nucleation, emission, dislocation free zone (DFZ) and pile-up of the dislocations are analyzed by using a suitable atom lattice configuration and Finnis & Sinclair potential. The simulated results show that the dislocation emitted always exhibits a dissociated fashion. The stress intensity factor for dislocation nucleation, DFZ and dissociated width of partial dislocations are strongly dependent on the loading rate. The stress distributions are in agreement with the elasticity solution before the dislocation emission, but are not in agreement after the emission. The dislocation can move at subsonic wave speed (less than the shear wave speed) or at transonic speed (greater than the shear wave speed but less than the longitudinal wave speed), but at the longitudinal wave speed the atom lattice breaks down. The project supported by the National Natural Science Foundation of China     

Convective coupled map for simulating spatiotemporal chaos in flows

A coupled map lattices with convective nonlinearity or, for short, Convective Coupled Map (CCM) is proposed in this paper to simulate spatiotemporal chaos in fluid flows. It is found that the parameter region of spatiotemporal chaos can be determined by the maximal Liapunov exponent of its complexity time series. This simple model implies a similar physical mechanism for turbulence such that the route to spatiotemporal chaos in fluid flows can be envisaged. The study is supported by “Nonlinear Sciences Project” from the State Science and Technology Commission of China.     

The motion of a small bubble in waves

The motion of a single spherical small bubble due to buoyancy in the ideal fluid with waves is investigated theoretically and experimentally in this article. Assuming that the bubble has no effect on the wave field, equations of a bubble motion are obtained and solved. It is found that the nonlinear effect increases with the increase of the bubble radius and the rising time. The rising time and the motion orbit are given by calculations and experiments. When the radius of a bubble is smaller than 0.5mm and the distance from the free surface is greater than the wave height, the results of the present theory are in close agreement with measurements. The project supported by the National Natural Science Foundation of China     

Periodic solution of the system with impulsive state feedback control

The order-1 periodic solution of the system with impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semi-continuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincarè criterion. Finally, we analyze numerically the theoretical results obtained.     

Rh-catalyzed intramolecular aromatic C–H insertion of α-diazo β-ketoesters: synthesis of 4-carbonyl chroman derivatives

A Rh-catalyzed intramolecular aromatic C–H insertion of α-diazo β-ketoesters was developed. This protocol offers a practical strategy for the synthesis of 4-carbonyl chroman derivatives with high yield and is compatible with a wide variety of substituents. Synthetic applications of the 4-carbonyl chroman were also demonstrated.