THE MAPPING RELATION AMONG SOLUTION OF SOME NONLINEAR EQUATIONS

Among the solutions of three kinds of nonlinear equations in one dimensional systems, cubic nonlinear Klein-Gordon (including Φ~4), Sine-Gordon and double Sine-Gordon, some mapping relations exist. When a solution of any one equation is known, so are the other two.

Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to Nonlinear SchrSdinger Equations with Variable Coefficient

In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we obtain explicit solutions of nonlinear Schrodinger equation with variable-coefficient. These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some are found for the first time. Six figures are given to illustrate some features of these solutions. The method can be applied to other nonlinear evolution equations in mathematical physics.

SOLUTION OF DIFFERENT HOLES SHAPE BORDERS OF FIBRE REINFORCED COMPOSITE PLATES BY INTEGRAL EQUATIONS

Accurate boundary conditions of composite material plates with different holes are founded to settle boundary condition problems of complex holes by conformal mapping method upon the nonhomogeneous anisotropic elastic and complex function theory. And then the two stress functions required were founded on Cauchy integral by boundary conditions. The final stress distributions of opening structure and the analytical solution on composite material plate with rectangle hole and wing manholes were achieved. The influences on hole-edge stress concentration factors are discussed under different loads and fiber direction cases, and then contrast calculates are carried through FEM.

ON THE VERTEX PARTITION EQUATION OF ROOTED LOOPLESS PLANAR MAPS

This paper provides a functional equation astisfied by the generating function for enumerating rooted loopless planar maps with vertex partition. A kind of applications in enumerating, by providing explicit formulae, a type of rooted loopless planar maps with the maximum valency of vertices given are described. Meanwhile, the functional equation for enumerating rooted loopless planar maps (connected) with the edge number and the valency of root-vertex as the parameters is also derived directly.

Existence and algorithm of solutions for a system of generalized mixed implicit equilibrium problems in Banach spaces

A new system of generalized mixed implicit equilibrium problems is introduced and studied in Banach spaces. First, the notion of the Yosida proximal mapping for generalized mixed implicit equilibrium problems is introduced. By using the notion, a system of generalized equation problems is considered, and its equivalence with the system of generalized mixed implicit equilibrium problems is also proved. Next, by applying the system of generalized equation problems, we suggest and analyze an iterative algorithm to compute the approximate solutions of the system of generalized mixed implicit equilibrium problems. The strong convergence of the iterative sequences generated by the algorithm is proved under quite mild conditions. The results are new and unify and generalize some recent results in this field.

COHERENT STATES, INVERSE MAPPING FORMULA FOR WEYL TRANSFORMATIONS AND APPLICATION TO EQUATIONS OF KdV TYPE

In this paper, we consider the trace of generalized operators and inverse Weyl transformation.First of all we repeat the definition of test operators and generalized operators given in [18],denoting L~2(R) by H.

ON THE VERTEX PARTITION EQUATION OF LOOPLESS EULERIAN PLANAR MAPS

The functional equation satisfied by the vertex partition function of rooted loopless Eulerianplanar maps is provided. As applications, the enumerating equations for general and regular casesof this kind of maps are also discussed.

Analytical and approximate solutions of nonlinear Schrödinger equation with higher dimension in the anomalous dispersion regime

The generalized Riccati equation mapping method(GREMM)is used in this paper to obtain different types of soliton solutions for nonlinear Schrödinger equation with higher dimension that existed in the regimes of anomalous dispersion.Later,we use the q-homotopy analysis method combined with the Laplace transform(q-HATM)to obtain approximate solutions of the bright and dark optical solitons.The q-HATM illustrates the solutions as a rapid convergent series.In addition,to show the physical behavior of the solutions obtained by the proposed techniques,the graphical representation has been provided with some parameter values.The findings demonstrate that the proposed techniques are useful,efficient and reliable mathematical method for the extraction of soliton solutions.

New sub-equation method to construct solitons and other solutions for perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials

In a previous work,Zayed and Al-Nowehy have applied the Riccati equation mapping method combined with the generalized extended(G/G)-expansion method and found new exact solutions of the nonlinear KPP equation.In the present article,we propose a different method,namely,a new sub-equation method consists of the Riccati equation mapping method and the(G/G,1/G)-expansion method to find new exact solutions of the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials.This proposed method is not found elsewhere.Hyperbolic,trigonometric and rational function solutions are given.New solutions of the generalized Riccati equation are presented for the first time which are not reported previously.The solutions of the given nonlinear equation can be applied in ocean engineering for calculating the height of tides in the ocean.

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods:the Lie symmetry analysis,the generalized Riccati equation mapping approach,and the modified Kudryashov approach,we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation(NCDR)with power-law nonlinearity and density-dependent diffusion.Lie symmetry analysis is one of the pow-erful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables.By the Lie group technique,we obtain one-parameter in-variant transformations,determining equations and corresponding vectors for the considered convection-diffusion-reaction equation.By treating the parameters of the governing equation as constants,the ap-plied methods yield a variety of new closed-form solutions,including inverse function solutions,periodic solutions,exponential function solutions,dark solitons,singular solitons,combo bright-singular solitons,and the combine of bright-dark solitons and dark-bright solitons.Moreover,using the Bäcklund transfor-mation of the generalized Riccati equation and modified Kudryashov method,we can construct multiple solitons and other solutions of the considered equation.The obtained new solutions of this work demon-strate that the used approaches are powerful and effective in dealing with nonlinear equations,and that these solutions are required to explain many biological and physical phenomena.Comparing our obtained solutions of this paper with the ones obtained in the literature,we see that our solutions are new and not reported elsewhere.These newly formed soliton solutions will be more beneficial in the various dis-ciplines of ocean engineering,plasma physics,and nonlinear sciences.

Electron-positron pair production in the low-density approximation

Electron-positron pair creation is studied in the low-density approximation by solving the quantum Vlasov equation exactly and the mapping equation approximately. The simpler mapping equation is an approximate treatment of the quantum Vlasov equation in which the continuous external field is regarded as a series of delta kicks. Our study indicates that this new treatment is appropriate because the results of the two methods are in good agreement with each other. However, as the period number increases, interference and a complicated structure in the momentum distribution are observed. Furthermore, we also obtain the square power law relation of the number density to the applied electric field strength.

High-temperature deformation behavior of a beta Ti-3.0Al-3.5Cr-2.0Fe-0.1B alloy

The high-temperature deformation behavior of a beta Ti-3.0 Al-3.5 Cr-2.0 Fe-0.1 B alloy was investigated by a Gleeble-1500 D thermal simulator. The height reduction was 50%, corresponding to a true strain of 0.693. The strain rate ranging from 0.01 to 10.00 s^-1 and the deformation temperature ranging from 800 to 950 ℃ were considered.The flow stress and the apparent activation energy for deformation, along with the constitutive equation, were used to analyze the behavior of the Ti-3.0 Al-3.5 Cr-2.0 Fe-0.1 B alloy. The processing map was established. The effect of strain rate on the microstructure at 850 ℃ was evaluated.The flow stress-strain curves indicated that the peak flow stresses increased along with an increase in the strain rate and decreased as the deformation temperature increased.Based on the true stress-true strain curves, the constitutive equation was established and followed as the ε= 6.58×10-（10）[sinh（0.0113σ）]-（3.44）exp（-245481.3/RT）. The processing map exhibited the ＂unsafe＂ region at the strain rate of10 s^-1 and the temperature of 850 ℃,and the rest region was ＂safe＂. The deformation microstructure demonstrated that both dynamic recovery（DRV） and dynamic recrystallization（DRX） existed during deformation. At the lower strain rate of 0.01 s^-1, the main deformation mechanism was the DRV, and the DRX was the dominant deformation mechanism at the higher strain rate of 1.00 s^-1.

Constitutive Modeling and Hot Deformation Behavior of Duplex Structured Mg–Li–Al–Sr Alloy

Hot deformation behavior of an as-extruded duplex structured Mg-9Li-3Al-2.5Sr alloy is investigated via hot compression tests conducted at 200-350℃ with strain rate of 0.001-1 s^-1.The flow behavior of Mg-9Li-3Al-2.5Sr alloy can be described accurately by hyperbolic sine constitutive equation and the average activation energy for deformation is calculated as 143.5 k J/mol.Based on a dynamic materials model,the processing maps of Mg-9Li-3Al-2.5Sr alloy which describe the variation of power dissipation efficiency are constructed as a function of temperature and strain rate.The processing maps exhibit an area of discontinuous dynamic recrystallization occurring at 280-300℃ with strain rate of 0.001-0.01 s^-1,which corresponds to the optimum hot working conditions.

Hot deformation behavior of an antibacterial Co–29Cr–6Mo–1.8Cu alloy and its effect on mechanical property and corrosion resistance

In order to optimize the deformation processing, the hot deformation behavior of Co-Cr-Mo-Cu （here- after named as Co-Cu） alloy was studied in this paper at a deformation temperature range of 950-1150 ℃ and a strain rate range of 0.008-5 s^-1. Based on the true stress-true strain curves, a constitutive equation in hyperbolic sin function was established and a hot processing map was drawn. It was found that the flow stress of the Co-Cu alloy increased with the increase of the strain rate and decreased with the increase of the deforming temperature. The hot processing map indicated that there were two unstable regions and one well-processing region. The microstructure, the hardness distribution and the electro- chemical properties of the hot deformed sample were investigated in order to reveal the influence of the hot deformation. Microstructure observation indicated that the grain size increased with the increase of the deformation temperature but decreased with the increase of the strain rate. High temperature and low strain rate promoted the crystallization process but increased the grain size, which results in a reduction in the hardness. The hot deformation at high temperature （1100-1150 ℃） would reduce the corrosion resistance slightly. The final optimized deformation process was： a deformation temperature from 1050to 1100 ℃, and a strain rate from 0.008 to 0.2 s^-1, where a completely recrystallized and homogeneously distributed microstructure would be obtained.

High temperature deformation behavior and processing map of hot isostatically pressed Ti-47.5Al-2Cr-2Nb-0.2W-0.2B alloy using gas atomization powders

The hot compressive deformation behavior of hot isostatically pressed Ti-47.5Al-2Cr-2Nb-0.2W-0.2B alloy using gas atomization powders was systematically investigated and the processing map was obtained in the temperature range of 1323-1473 Kand strain rate range of 0.001-0.5s^（-1）.The calculated activation energy in the above variational ranges of temperature and strain rate possesses a low activation energy value of approximately 365.6kJ/mol based on the constitutive relationship models developed with the Arrhenius-type constitutive model respectively considering the strain rate and deformation temperature.The hot working flow behavior during the deformation process was analyzed combined with the microstructural evolution.Meanwhile,the processing maps during the deformation process were established based on the dynamic material model and Prasad instability criterion under different deformation conditions.Finally,the optimal hot processing window of this alloy corresponding to the wide temperature range of 1353-1453 Kand the low strain rate of 0.001-0.1s^（-1） was obtained.

Constitutive analysis and optimization on hot working parameters of as-cast high Cr ultra-super-critical rotor steel with columnar grains

Isothermal hot compression tests on the as-cast high-Cr ultra-super-critical rotor steel with columnar grains were carried out in the temperature range from 1223 to 1523Kand at strain rates from 0.001 to 1s^（-1）.The compression direction was parallel to the longitudinal direction of columnar grains.The constitutive equation based on Arrhenius model was presented,and the processing maps based on the dynamic material model were developed,correlating with microstructure observation.The main softening mechanism was dynamic recovery at 1223 Kunder strain rates from 0.1to 1s^（-1）,whereas it was dynamic recrystallization under other deformation conditions.The constitutive equation modified by strain compensation reasonably predicted the flow stresses.The processing maps and microstructure evolution mechanism schematic indicated that the optimum hot working parameters lay in the zone defined by the temperature range from 1423 to 1473Kand the strain rate range from 0.001 to 1s^（-1）.

Aerodynamic design optimization of nacelle/pylon position on an aircraft

The arbitrary space-shape free form deformation （FFD） method developed in this paper is based on non-uniform rational B-splines （NURBS） basis function and used for the integral parameterization of nacelle-pylon geometry. The multi-block structured grid deformation technique is established by Delaunay graph mapping method. The optimization objects of aerodynamic characteristics are evaluated by solving NavierStokes equations on the basis of multi-block structured grid. The advanced particle swarm optimization （PSO） is utilized as search algorithm, which com-bines the Kriging model as surrogate model during optimization. The optimization system is used for optimizing the nacelle location of DLR-F6 wing-body-pylon-nacelle. The results indicate that the aerodynamic interference between the parts is significantly reduced. The optimization design system established in this paper has extensive applications and engineering value.