Exact Solutions of Forced Schrödinger Equation and How to Choose the External Forces

Schrödinger equations are very common equations in physics and mathematics for nonlinear physics to model the dynamics of wave propagation in waveguides such as power lines, atomic chains, optical fibers, and even in quantum mechanics. But all these equations are most often studied without worrying about what would happen if this equation were maintained, that is to say, had a second member synonymous with an external force. It is true that on a physical level, such equations can be considered as describing the generation of waves on a waveguide using an external force. However, the in-depth analysis of this aspect is not at the center of our reflection in this article, but for us, it is a question of proposing exact solutions to this type of equation and above all proposing the general form of the external force so that the obtaining exact solutions is possible.

UNIFORM ATTRACTOR FOR NONAUTONOMOUS INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH A NEW CLASS OF EXTERNAL FORCES

This paper is joint with [27]. The authors prove in this article the existence and reveal its structure of uniform attractor for a two-dimensional nonautonomous incompressible non-Newtonian fluid with a new class of external forces.

Nonstationary Time Series Prediction by Incorporating External Forces

Almost all climate time series have some degree of nonstationarity due to external forces of the observed system. Therefore, these external forces should be taken into account when reconstructing the climate dy- namics. This paper presents a novel technique in predicting nonstationary time series. The main difference of this new technique from some previous methods is that it incorporates the driving forces in the pre- diction model. To appraise its effectiveness, three prediction experiments were carried out using the data generated from some known classical dynamical models and a climate model with multiple external forces. Experimental results indicate that this technique is able to improve the prediction skill effectively.

Adaptive Nonlinear PD Controller of Two-Wheeled Self-Balancing Robot with External Force

This paper proposes an adaptive nonlinear proportional-derivative(ANPD)controller for a two-wheeled self-balancing robot(TWSB)modeled by the Lagrange equation with external forces.The proposed control scheme is designed based on the combination of a nonlinear proportional-derivative(NPD)controller and a genetic algorithm,in which the proportional-derivative(PD)parameters are updated online based on the tracking error and the preset error threshold.In addition,the genetic algorithm is employed to adaptively select initial controller parameters,contributing to system stability and improved control accuracy.The proposed controller is basic in design yet simple to implement.The ANPD controller has the advantage of being computationally lightweight and providing high robustness against external forces.The stability of the closed-loop system is rigorously analyzed and verified using Lyapunov theory,providing theoretical assurance of its robustness.Simulations and experimental results show that the TWSB robot with the proposed ANPD controller achieves quick balance and tracks target values with very small errors,demonstrating the effectiveness and performance of the proposed controller.The proposed ANPD controller demonstrates significant improvements in balancing and tracking performance for two-wheeled self-balancing robots,which has great applicability in the field of robot control systems.This represents a promising solution for applications requiring precise and stable motion control under varying external conditions.

Nonresonance and Global Existence of Nonlinear Elastic Wave Equations with External Forces

This paper establishes the global existence of classical solution to the system of homogeneous,isotropic hyperelasticity with time-independent external force,provided that the nonlinear term obeys a type of null condition.The authors first prove the existence and uniqueness of the stationary solution.Then they show that the solution to the dynamical system converges to the stationary solution as time goes to infinity.

BIFURCATION IN A NONLINEAR DUFFING SYSTEM WITHMULTI-FREQUENCY EXTERNAL PERIODIC FORCES

By introducing nonlinear frequency, using Floquet theory and referring to the characteristics of the solution when it passes through the transition boundaries, all kinds of bifurcation modes and their transition boundaries of Duffing equation with two periodic excitations as well as the possible ways to chaos are studied in this paper.

Establishment and validation of standardized animal models of spinal cord injury by normal external force-caused fracture dislocation

The duplication of animal models plays a key role in spinal cord injury research; however, there has been limited study into normal, external force-derived fracture dislocation. This study adopted experimental devices, designed in-house, to construct standardized ventral and dorsal spinal cord injury animal models of 6 g and 17 g falling from a height of 2, 4, and 10 cm, and 15, 30 or 50 g transversal compression on the spinal cord. The results showed that gradual increases in the degree of histopathological injury led to decreased Tarlov and Basso, Beattie and Bresnahan scores for the behavioral test, and increased Ashworth scores for the hind limb. Furthermore, there was a gradual decline in the slope test in the rats with dorsal spinal cord injury that correlated to increases in the falling substance weight or falling height. Similar alterations were observed in the ventral spinal cord injured rats, proportional to the increase in compression weight. Our experimental findings indicate that the standardized experimental rat models of dorsal and ventral spinal cord injury are stable, reliable and reproducible.

CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H^3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H^3 and bounded in L6/5.

NUMERICAL SIMULATION ANALYSIS OF EXTERNAL FLOW FIELD OF WAGON-SHAPED CAR AT THE MOMENT OF PASSING

In the course of studying on aerodynamic change and its effect on steering stability and controllability of an automobile in passing, because of multi interaction streams, it is difficult to use traditional methods, such as wind tunnel test and road test. If the passing process of an automobile is divided into many time segments, so as to avoid the use of moving mesh which takes large calculation resource and CPU processing time in calculating, the segments are simulated with computational fluid dynamics （CFD） method, then the approximate computational results about external flow field will be obtained. On the basis of the idea, the change of external flow field of wagon-shaped car at the moment of passing is simulated through solving three-dimensional, steady and uncompressible N-S equations with finite volume method. Numerical simulation analysis of side force coefficient, stream lines, body surface pressure distribution of wagon-shaped car are presented and a preliminary discussion of aerodynamic characteristics of correlative situations is obtained. Finally, the C3 -x/l curve of side force coefficient（C3） of car following relative distance （x/l） between cars is obtained. By comparison, the curve is coincident well with the experimental data, which shows creditability of numerical simulation methods presented.

Self-Organization of Charged Particulates in the Presence of External Force

Evolution of spatial distribution of charged particulates under the action of an external force is investigated. It is found that starting from a homogeneous Maxwellian distribution of particulates, clusters can form and aggregate. The evolution process, as well as the asymptotic number and configuration of the clusters formed, depends strongly on the strength of the external force. The particulates in most of the final clusters are in the crystal state, as can also be deduced from the corresponding velocity and auto-correlation functions.

Numerical Quantification Model and Experiment of External Force on Roller Hemming of Curved Edge Aluminium Alloy with Adhesive

Accurate quantification of external force is the key to improve the high-precision hemming of autobody closure panels.However,the mechanism of external force on forming quality of complex contour sheet metal with adhesive is not clear subjected to geometric curvature and materials.In the present study,taking the curved edge aluminum sheet as the research object,SPH(smooth particle hydrodynamics)is introduced to simulate the viscous adhesive,and the SPH-FEM(Finite element method)coupling model of adhesive and panels considering the viscosity-pressure effect is established.The numerical simulation of the roller hemming process is carried out,then the validity and reliability of the proposed method are verified by measuring the external force in real time using triaxial force sensor.The multi-step forming process and the effect of external force on the roll in/out,surface wave and plastic strain of aluminum alloy sheet under the viscosity-pressure effect are studied,and the relationship between process parameters and external force is discussed.Results show that the coupling SPH-FEM model can well reflect the hemming process of curved edge structure.The normal force is about 2–3 times of the tangential force in the pre and final hemming process.Compared with the case without adhesive,the surface wave of flange part of the hemming with adhesive is slightly larger.The normal force and the tangential force increase about 90 N and 30 N respectively,when the height increases by 1 mm.It provides an important basis for the accurate control of hemming trajectory and the improvement of manufacturing quality of autobody closure panels.

DYNAMIC MODEL OF CROP GROWTH SYSTEM AND NUMERICAL SIMULATION OF CROP GROWTH PROCESS UNDER THE MULTI-ENVIRONMENT EXTERNAL FORCE ACTION

According to the biomechanic theory and method, the dynamic mechanism of crop growth under the external force action of multi_environment factors (light, temperature,soil and nutrients etc.) was comprehensively explored.Continuous_time Markov(CTM) approach was adopted to build the dynamic model of the crop growth system and the simulated numerical method. The growth rate responses to the variation of the external force and the change of biomass saturation value were studied. The crop grew in the semiarid area was taken as an example to carry out the numerical simulation analysis, therefore the results provide the quantity basis for the field management. Comparing the dynamic model with the other plant growth model, the superiority of the former is that it displays multi_dimension of resource utilization by means of combining macroscopic with microcosmic and reveals the process of resource transition. The simulation method of crop growth system is advanced and manipulated. A real simulation result is well identical with field observational results.

Long-time evolution of charged grains in plasma under harmonic external force and after being withdrawn

Evolution of the charged grains in a two-dimensional dusty plasma under a spatially harmonic external force,in particular,their long-time behaviors after the force has been withdrawn,is studied by using molecular dynamics simulation.Under an external force and a grain–grain interaction force,initially homogeneously distributed grains can reach a quasistationary state in the form of a disk crystal.After the external force is withdrawn,the disk moves initially with its size and shape nearly unchanged until it rapidly stops moving,and eventually the disk grain rotates like a vortex.The time needed to reach the final state increases with the strength of the initial external force increasing.

Oscillating Propagation of Kink in Nondissipative Frenkel-Kontorova Chain Due to External DC Force

We report the oscillating propagation of kink in a nondissipative Frenkel-Kontorova （FK） chain driven by external DC force, which is different from the usual propagation of localized modes with equal speed. When the kink moves in the opposite direction of the external DC force, the kink will be accelerated and the potential of the FK chain in the external force field is transformed to be the kinetic energy of the kink. If the kink reaches the boundary of the FK chain, the kink will be bounced back and moves in the opposite direction, then the kink will be decelerated gradually and the kinetic energy of the kink ＆ transformed to be the potential of the FK chain in the external force field. If the speed of the kink reaches zero, the kink will move in the opposite direction again driven by the external DC force, and a new oscillating cycle begins. Simulation result demonstrates exactly the transformation between the kinetic energy of the kink and the potential of the FK chain in the external force field. The interesting energy exchange is induced by the special topology of kinks, and other localized modes, such as breathers and envelope solitons, have no the interesting phenomenon.

PARTITION RATIO AT SOLID-LIQUID INTERFACE UNDER INFLUENCE OF TEMPERATURE GRADIENT AND EXTERNAL FORCE FIELD

The relationship between the partition ratio at a solid-liquid interface and the temperature gradient or the external force field has been theoretically analysed.It is shown that under the influence of a temperature gradient or an external force field,the partition ratio at a solid-liquid interface will deviate from the equilibrium value.

EXISTENCE OF INFINITE ENERGY SOLUTION TO THE INELASTIC BOLTZMANN EQUATION WITH EXTERNAL FORCE

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy. More precisely, under the assumptions on the bicharacteristic generated by external force, we prove the global existence of solution for small initial data compared to the local Maxwellian exp（-p｜x - v｜^2）, which has infinite mass and energy.

Factors of Pumping Capacity in Some Different Types of Channel Brownian Pumps

The motion of particles in different channel Brownian pumps can be described by Langevin equations,and the pumping capacity is a useful indicator to demonstrate the strength of a pump’s transportation ability.Via the simulation,there is always an optimal value of temperature and unbiased external force for different pumps which make the concentration ratio between the right tube and left tube derive its maximum and minimum in two asymmetric tubes respectively.Besides,the concentration ratio will keep 1 regardless of radius,temperature or magnitude of force in the tube in a symmetric tube.To obtain more information about pumping capacity,exploring the average probability current(APC) of tubes in different conditions is necessary.Results indicate that as the concentration ratio is 1,the change of the APC when x_(0)=0 is similar to that when x_(0)=π.Also,when the concentration ratio is more than 1,there are optimal values of temperature,radius and magnitude of force where the APC gains a maximum,and the maximum decreases as the concentration in the right tube increases when x_(0)=0.

ROBUSTNESS OF MULTIPLE-LIMB MANIPULATION SYSTEMS

The characteristic that multiple limb manipulation systems can resist disturbance external forces without relying on the feedback control of joint torques, called the robustness, has been addressed by several researchers. Based on their results, a further study of robustness is presented in this paper. By decomposing the space of external forces into two subspaces, the necessary and sufficient condition for a system to have robustness is given and a definition of robustness which is applicable to any multiple limb systems is also proposed. For the purpose of the evaluation of robustness, a new quality measure — critical disturbance external force and its algorithm are put forward. Finally, two examples of robustness analysis are presented.

Variation of Surface Temperature during the Last Millennium in a Simulation with the FGOALS-gl Climate System Model

A reasonable past millennial climate simulation relies heavily on the specified external forcings, including both natural and anthropogenic forcing agents. In this paper, we examine the surface temperature responses to specified external forcing agents in a millennium-scale transient climate simulation with the fast version of LASG IAP Flexible Global Ocean-Atmosphere-Land System model （FGOALS-gl） developed in the State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics （LASG/IAP）. The model presents a reasonable performance in comparison with reconstructions of surface temperature. Differentiated from significant changes in the 20th century at the global scale, changes during the natural-forcing-dominant period are mainly manifested in the Northern Hemisphere. Seasonally, modeled significant changes are more pronounced during the wintertime at higher latitudes. This may be a manifestation of polar amplification associated with sea-ice-temperature positive feedback. The climate responses to total external forcings can explain about half of the climate variance during the whole millennium period, especially at decadal timescales. Surface temperature in the Antarctic shows heterogeneous and insignificant changes during the preindustrial period and the climate response to external forcings is undetectable due to the strong internal variability. The model response to specified external forcings is modulated by cloud radiative forcing （CRF）. The CRF acts against the fluctuations of external forcings. Effects of clouds are manifested in shortwave radiation by changes in cloud water during the natural-forcing-dominant period, but mainly in longwave radiation by a decrease in cloud amount in the ant hropogenic- forcing-dominant period.

Decreasing Trend in Global Land Monsoon Precipitation over the Past 50 Years Simulated by a Coupled Climate Model

The authors examine the effects of external forcing agents such as greenhouse gases （GHGs） and aerosols, as well as solar variability and ozone, on global land monsoon precipitation by using a coupled climate model HadGEM1, which was developed by the Met Office Hadley Centre for Climate Research. The results indicate that HadGEM1 performs well in simulating the observed decreasing trend of global land monsoon precipitation over the past 50 years. This trend mainly occurred in the Northern Hemisphere and is significantly different from the trend of natural variability due to ocean-atmosphere-land interactions. The coherence between the simulation and the observations indicates that the specified external forcing agents, including GHGs and aerosols as well as solar variability and ozone, are important factors that have affected the decreasing trend of global land monsoon precipitation in the past 50 years.