The pth-order periodic solutions for a family of N-coupled nonlinear SchrSdinger equations

By using the solutions of an auxiliary Lame equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schrodinger equations. The abundant higher-order exact periodic solutions of a family of N-coupled nonlinear Schrodinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.

Explicit One-Step P-Stable Methods for Second Order Periodic Initial Value Problems

In this paper, we present an explicit one-step method for solving periodic initial value problems of second order ordinary differential equations. The method is P-stable, and of first algebraic order and high phase-lag order. To improve the algebraic order, we give a composition second order scheme with the proposed method and its adjoint. We report some numerical results to illustrate the efficiency of our methods.

Microstructure characterization and corrosion behavior of Mg-Y-Zn alloys with different long period stacking ordered structures

Mg-Y-Zn alloys with long period stacking ordered(LPSO)structure have received much attention recently and exhibit great potential in applications such as automotive,aerospace and in bio-medical fields.This paper aimed to investigate the effect of different phase constitution of LPSO structures on corrosion rate of bio-medical Mg-Y-Zn alloys.The results showed that as-cast Mg98.5Y1Zn0.5 alloys containing only 18R structure exhibited the highest corrosion resistance with the corrosion rate of 2.78 mm/year.The precipitation of 14H lamellas within a-Mg grains during solid solution treatment introduced the crystallographic orientation corrosion by accelerating micro-galvanic corrosion.The increase of 18R/14H interfaces deteriorated the corrosion resistance,and the grain boundaries also suffered from severe electrochemical dissolution.This work suggested that Mg-Y-Zn alloys with single LPSO structure(either 18R or 14H)exhibited better corrosion resistance than alloys with co-existence 18R and I4H LPSO structures.

Recent developments on corrosion behaviors of Mg alloys with stacking fault or long period stacking ordered structures

Corrosion is one of the most drawbacks which restricts the wide applications of Mg alloys.In the last decade,the corrosion behaviors of Mg alloys with stacking fault(SF)and/or long period stacking ordered(LPSO)structures have obtained increasing attention.However,the corrosion mechanism of the SF–or LPSO–containing Mg alloys has not been well illustrated and even reverse results have been reported.In this paper,we have reviewed recent reports on corrosion behaviors of SF–or LPSO–containing Mg alloys to better clarify and understand the significance and mechanism.Moreover,some deficiencies are presented and advises are proposed for the development of corrosion resistant Mg alloys with SF or LPSO structures.

Effect of Li on formation of long period stacking ordered phases and mechanical properties of Mg-Gd-Zn alloy

Alloys with composition of Mg_(96-x)Gd_3Zn_1Li_x(at.%)(x=0, 2, 4, and 6) were prepared by conventional casting. The microstructures of these alloys under as-cast and solid-solution conditions have been observed, and the mechanical properties were investigated. The results showed that Li is an effective element to refine the grains and break the eutectic networks in as-cast MgGd_3Zn_1 alloy. During solid solution treatment, these broken eutectic networks are spheroidized and highly dispersed. In addition, plentiful lamellar long period stacking ordered(LPSO) phases are precipitated in an α-Mg matrix when the Li addition is not more than 4%. Solid-solution treated Mg_(92)Gd_3Zn_1Li_4 alloy exhibits an optimal ultimate tensile strength(UTS) of 226 MPa and elongation of 5.8%. The strength of MgGd_3Zn_1 alloy is improved significantly, meanwhile, the toughness is apparently increased.

Effect of long period stacking ordered phase on damping capacities of as-cast Mg-Zn-Y alloys

The microstructure and damping capacities of MgZnxYi.33x(x=l-4at.%)alloys were discussed and researched.The main phase composition of the alloys consists of a_Mg and long-period stacking ordered(LPSO)phase.Due to increasedLPSO phase,grain size was refined.LPSO phase was advantageous to the damping properties of the Mg-Zn-Y alloys.Mg-7%Zn-12.8%Y has the highest damping capacity up to0.04.Due to stacking fault probability,the LPSO phase in the Mg-Zn-Yalloys could be new damping source to dissipate energy so as to contribute to the improvement of damping capacities.

Review on long-period stacking-ordered structures in Mg-Zn-RE alloys

The recent development of high-strength magnesium alloys is focused on the role of the strengthening phases with a novel long-period stacking-ordered (LPSO) structure. This review detailed the main factors influencing the formation of LPSO phases, including alloying ele-ments, preparation methods, and heat treatments. Furthermore, process control in structure types, formation and transformation behavior, strengthening and toughening mechanisms of the LPSO phase were discussed. Finally, the current problems and development trends of high-strength Mg-Zn-RE alloys were also put forward.

Microstructure and mechanical properties of a compound reinforced Mg95Y2.5Zn2.5 alloy with long period stacking ordered phase and W phase

The microstructure evolution of Mg100-2xYxZnx （x=2, 2.5, 3, 3.5） alloys was investigated. Results show that the Mg100-2xYxZnx alloys are composed of a-Mg, long period stacking ordered （LPSO） phase and eutectic structure phase （W phase）, and the Mg95Y2.5Zn2.5 alloy has the best comprehensive mechanical properties. Subsequently, the microstructure evolution of the optimized alloy Mg95Y2.5Zn2.5 during solidification and heat treatment processes was analyzed and discussed by means of OM, SEM, TEM, XRD and DTA. After heat treatment, the lamellar phase 14H-LPSO precipitated in a-Mg and W phase transforms into particle phase （MgyZn2）. Due to the compound reinforcement effect of the particle phase and LPSO phase （18R＋14H）, the mechanical properties of the alloy are enhanced. The tensile strength and elongation of the Mg95Y2.5Zn2.5 alloy is improved by 9.1% and 31.3% to 215 MPa and 10.5%, respectively, after solid-solution treatment.

A PRIORI BOUNDS FOR PERIODIC SOLUTIONS OF A KIND OF THIRD-ORDER DELAY DIFFERENTIAL EQUATION

By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.

EXISTENCE OF SOLUTIONS FOR PERIODIC BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

By establishing a comparison result and using monotone iterative methods, the theorem of existence for minimal and maximal solutions of periodic boundary value problems for second-order nonlinear integro-differential equations in Banach spaces is proved.

Periodic solutions of non－autonomous second order systems with subconvex potential

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Dilute long period stacking/order(LPSO)-variant phases along the composition gradient in a Mg-Ho-Cu alloy

We have systematically investigated the microstructures of as-cast Mg_(97.49)Ho_(1.99)Cu_(0.43)Zr_(0.09)alloy by atomic resolution high-angle annular dark field scanning transmission electron microscopy(HAADF-STEM), revealing the coexistence of 18R, 14H and 24R long period stacking/order(LPSO) phases with fully coherent interfaces along step-like composition gradient in a blocky intermetallic compound distributed at grain boundary. The short-range order(SRO) L1_(2)-type Cu_(6)Ho_(8)clusters embedded across AB’C’A-stacking fault layers are directly revealed at atomic scale. Importantly, the order degree of SRO clusters in the present dilute alloy is significant lower than previous 6M and 7M in-plane order reported in ternary Mg-TM(transition metal)-RE(rare earth) alloys, which can be well matched by 9M in-plane order. This directly demonstrates that SRO in-plane L1_(2)-type clusters can be expanded into more dilute composition regions bounded along the definite TM/RE ratio of 3/4. In addition, the estimated chemical compositions of solute enriched stacking fault(SESF) in all LPSO variants are almost identical with the ideal SESF composition of 9M in-plane order, regardless of the type of LPSO phases. The results further support the viewpoint that robust L1_(2)-type TM_(6)RE_(8)clusters play an important role in governing LPSO phase formation.

EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS TO THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM

The existence and multiplicity of positive solutions to a periodic boundary value problem for nonlinear third-order ordinary differential equation are established, based on the zero point theorem concerning cone expansion and compression of order type. Our main approach is different from the previous papers on the existence of multiple positive solutions to the similar problem.

Re-study on Recurrence Period of Stokes Wave Train with High Order Spectral Method

Owing to the Benjamin-Feir instability, the Stokes wave train experiences a modulation-demodulation process, and presents a recurrence characteristics. Stiassnie and Shemer researched the unstable evolution process and provided a theoretical formulation for the recurrence period in 1985 on the basis of the nonlinear cubic Schrodinger equation （NLS）. However, NLS has limitations on the narrow band and the weak nonlinearity. The recurrence period is re-investigated in this paper by using a highly efficient High Order Spectral （HOS） method, which can be applied for the direct phase- resolved simulation of the nonlinear wave train evolution. It is found that the Stiassnie and Shemer＇s formula should be modified in the cases with most unstable initial conditions, which is important for such topics as the generation mechanisms of freak waves. A new recurrence period formula is presented and some new evolution characteristics of the Stokes wave train are also discussed in details.

Finding Periodic Solutions of High Order Duffing Equations via Homotopy Method

This paper presents a detailed analysis of finding the periodic solutions for the high order Duffing equation x^（2n） ＋ g（x） = e（t） （n ≥ 1）. Firstly, we give a constructive proof for the existence of periodic solutions via the homotopy method. Then we establish an efficient and global convergence method to find periodic solutions numerically.

ON THE BOUNDEDNESS AND PERIODICITY OF THESOLUTIONS OF A CERTAIN VECTOR DIFFERENTIALEQUATION OF THIRD-ORDER

There are given sufficient conditions for the ultimate boundedness of solutions and for the existence of periodic solutions of a certain vector differential equation of third-order.

CRYSTAL STRUCTURE OF A LONG－PERIOD ORDERED PHASE IN Fe－C MARTENSITE AND COMPUTER SIMULATION OF ITS ELECTRON DIFFRACTION PATTERNS

Different structure models of a long-period ordered phase in Fe-C martenstie formed during aging have been checked by computer simulation of electron diffraction(ED) patterns based on these models.The results showed that the simulated ED pattern of γ'-FexC(Ⅱ) model proposed by the present authors is in good agreement with experimentally observed ED pattern.It was also confirmed that the incommensurate superperiod stems from the coexistence of several γ'-Fe_xC(H) phases with different superperiods.The Fe(144)C(24)(Fe6C) model proposed by Uwakweh et al.generated ED patterns remarkably different from the experimental ones.

ON THE ORDER OF APPROXIMATION BY PERIODIC NEURAL NETWORKS BASED ON SCATTERED NODES

The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degree of approximation by the periodic neural network with one hidden layer and scattered threshold value nodes is increased with the increase of the number of neurons hid in hidden layer and the smoothness of excitation function.

Periodic Solutions of Even Order Nonlinear Differential Equations

This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodic solutions. This criterion is a complement of the well known Lazer type results.

The Periodic Solitary Wave Solutions for the (2 + 1)-Dimensional Fifth-Order KdV Equation

The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.