Free vibration analysis of piezoelectric coupled circular plate with open circuit

A vibration analysis of a circular steel substrate surface bonded by a piezoelectric layer with open circuit is presented. A solution for the electrical potential along the thickness direction of the piezoelectric layer satisfying the open circuit electric boundary condition is developed for the first time. The mechanical model and solutions for the vibration analysis of the piezoelectric coupled circular plate are then established based on the developed electrical potential, the Kirchhoff plate model, and the Maxwell equation. The first four mode shapes and the corresponding resonant frequencies of the plate with two standard boundary conditions are presented in numerical simulations and compared with those of a piezoelectric coupled plate with the closed circuit condition. The simulations show that the resonant frequencies of the open circuit piezoelectric coupled plate are higher than those of the closed circuit piezoelectric coupled plate. Corresponding discussions are thus provided for the higher piezoelectric effect from the open circuit piezoelectric layer.     

Selection criteria for two-parameter solutions to scalar-tensor gravity

We investigate the generic properties of static, spherically symmetric, asymptotically flat solutions to the field equations describing gravity minimally coupled to a nonlinear self-gravitating real scalar field. Five corollaries and a theorem on selection criteria for two- and one-parametric solutions are proven and conditions for obtaining particle-like solutions, black holes or naked singularities are derived. A series of exact solutions in closed forms including different black holes, naked singularities and particle-like solutions, all of which obey the weak energy condition at spatial infinity, are provided. Two further corollaries elaborate on the behavior of solutions at spatial infinity, critical mass, mass density, pressure and energy conditions.     

Pfaffian Solutions and Resonant Interaction Properties of a Coupled BKP Lattice

In this paper, we give a coupled lattice equation with the help of Hirota operators, which comes from a special BKP lattice. Two-soliton and three-soliton solutions to the coupled system are constructed. Furthermore,resonant interaction of the two-soliton solution is analyzed in detail. Under some special resonant condition, it is shown that low soliton can propagate faster than high one. Finally, the N-soliton solution is presented in the Pfaffian form.     

Pfaffian Solutions and Resonant Interaction Properties of a Coupled BKP Lattice

In this paper, we give a coupled lattice equation with the help of Hirota operators, which comes from a special BKP lattice. Two-soliton and three-soliton solutions to the coupled system are constructed. Furthermore, resonant interaction of the two-soliton solution is analyzed in detail. Under some special resonant condition, it is shown that low soliton can propagate faster than high one. Finally, the N-soliton solution is presented in the Pfaffian form.     

飞秒脉冲的二次谐波理论和实验研究

本文分析了小信号情况下飞秒脉冲的倍频特性,在非耗尽近似下,得出了飞秒脉冲二次谐波波形及效率的解析解,在时间域内,对一般条件的耦合波方程进行了数值解,分析了相位失配对飞秒脉冲倍频波形及效率的影响。 用0.5mm厚的一类匹配LBO晶体对碰撞锁模激光器产生的80fs超短脉冲进行了腔外倍频实验,实验结果与理论计算符合得很好。     

Phase-locked solutions and their stability in the presence of propagation delays

We investigate phase-locked solutions of a continuum field of nonlocally coupled identical phase oscillators with distance-dependent propagation delays. Equilibrium relations for both synchronous and travelling wave solutions in the parameter space characterizing the nonlocality and time delay are delineated. For the synchronous states a comprehensive stability diagram is presented that provides a heuristic synchronization condition as well as an analytic relation for the marginal stability curve. The relation yields simple stability expressions in the limiting cases of local and global coupling of phase oscillators.     

非线性耦合微分方程组的精确解析解

提出了利用耦合的Riccati方程组的某些特解构造非线性微分方程组精确解析解的一种方法.应用这种方法研究了两个耦合的常微分方程组,系统地获得了它们的一些精确解.给出了非线性浅水波近似方程组和非线性Schr?dinger-KdV方程组若干新的孤波解. 关键词： 非线性耦合方程组 Riccati方程组 符号计算 孤波     

Exact periodic wave solutions to the coupled Schrödinger-KdV equation and DS equations

The exact solutions for the coupled non-linear partial differential equations are studied by means of the mapping method proposed recently by the author. Taking the coupled Schrödinger-KdV equation and DS equations as examples, abundant periodic wave solutions in terms of Jacobi elliptic functions are obtained. Under the limit conditions, soliton wave solutions are given.     

A Pair of Coupled Equations for High Frequency Langmuir Wave and Low Frequency Potential

A coupled pair of nonlinear equations for the amplitude modulated Langmuir wave and the associated low frequency electric potential wave have been derived by relaxing quasineutrality condition of the frequency motion. Solitary wave solutions exist. It is found that the small correction term due to ion pressure changes the amplitude of the solitary wave.     

Point-force excited vibrations of a thin,infinite panel separating a fluid layer from a fluid half-space

A coupled system consisting of a thin, infinite panel separating a fluid layer and a fluid half-space is considered. The response of the panel when driven by a point harmonic force is evaluated by using Hankel transform analysis in conjunction with complex variable integration techniques. The emphasis is placed on frequencies well below the panel critical frequency, on dense fluid loading, and on the condition of shallow layer depth. The analysis yields the free propagation wave numbers of the coupled system: below the critical frequency, all but one are complex. From the solutions of the response integrals, the panel flexural velocity, the acoustic pressures, the power transmitted into the structure and into the fluid layer, and the radiated acoustic power, are obtained.     

Euclidean Wormholes with Phantom Field and Phantom Field Accompanied by Perfect Fluid

We study the classical Euclidean wormhole solutions for the gravitational systems with minimally coupled pure Phantom field and minimally coupled Phantom field accompanied by perfect fluid. It is shown that such solutions do exist and then the general forms of the Phantom field potential are obtained for which there are classical Euclidean wormhole solutions.     

Grammian solutions and Pfaffianization of the non-isospectral modified Kadomtsev-Petviashvili equation

The Grammian determinant solutions of the non-isospectral modified Kadomtsev-Petviashvili (mKP) equation are presented. Moreover, a new non-isospectral coupled system is constructed by using the Pfaffianization procedure. Furthermore, Gramm-type Pfaffian solutions of the non-isospectral coupled system are obtained.     

Integrability of two coupled Kadomtsev–Petviashvili equations

The integrability of two coupled KP equations is studied. The simplified Hereman form of Hirota’s bilinear method is used to examine the integrability of each coupled equation. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation.     

New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.     

Exact Solutions of a Coupled Burgers System

The exact solutions of a new coupled Burgers system are studied in three different ways. The first type of solutions are found thanks to the coupled Burgers system possessing a simple single Burgers reduction. The second type of multiple soliton solutions are revealed via the decouple procedure. The third type of exact solutions are found by means of a prior ansatz and solutions of the heat conduction equation. Two different kinds of soliton fission phenomena of the model are discovered and a special type of completely elastic soliton collision without phase shift of the model is also displayed.     

An integrable coupled Alice–Bob modified Korteweg de-Vries system: Lax pairs,Bäcklund transformations,residual symmetries and exact solutions

ABSTRACT

A coupled Alice–Bob modified Korteweg de-Vries (mKdV) system is established from the mKdV equation in this paper, which is nonlocal and suitable to model two-place entangled events. The Lax integrability of the coupled Alice–Bob mKdV system is proved by demonstrating three types of Lax pairs. By means of the truncated Painlevé expansion, auto-Bäcklund transformation of the coupled Alice–Bob mKdV system and Bäcklund transformation between the coupled Alice–Bob mKdV system and the Schwarzian mKdV equation are demonstrated. Nonlocal residual symmetries of the coupled Alice–Bob mKdV system are researched. To obtain localized Lie point symmetries of residual symmetries, the coupled Alice–Bob mKdV system is extended to a system consisting six equations. Calculation on the prolonged system shows that it is invariant under the scaling transformations, space-time translations, phase translations and Galilean translations. One-parameter group transformation and one-parameter subgroup invariant solutions are obtained. The consistent Riccati expansion (CRE) solvability of the coupled Alice–Bob mKdV system is proved and some interaction structures between soliton–cnoidal waves are obtained by CRE. Moreover, Jacobi periodic wave solutions, solitary wave solutions and singular solutions are obtained by elliptic function expansion and exponential function expansion.     

Nonsingular positon and complexiton solutions for the coupled KdV system

Taking the coupled KdV system as a simple example, analytical and nonsingular complexiton solutions are firstly discovered in this Letter for integrable systems. Additionally, the analytical and nonsingular positon–negaton interaction solutions are also firstly found for S-integrable model. The new analytical positon, negaton and complexiton solutions of the coupled KdV system are given out both analytically and graphically by means of the iterative Darboux transformations.     

两自由度带电耦合振子系统的守恒量与近似解

由于两自由度带电耦合振子系统的Lagrange函数中存在耦合项,从而导致其运动微分方程是非线性耦合的.先通过坐标变换消去Lagrange函数中的耦合项,用直接积分法求得系统的守恒量,用Adomian分解法求得系统的近似解,再通过坐标反变换求得系统在原坐标下的守恒量与近似解,并对近似解作了讨论.     

On the possibility of realizing a periodic low-density spatial distribution of excitons

The solutions of a system of coupled integro-differential Bogolyubov equations for distribution functions have been used to determine conditions, both in temperature and in concentration, under which a spatially uniform distribution of excitons transforms spontaneously into a periodic state with a small amplitude. The analysis rests on the concept assuming the existence of a biexciton interaction potential. The criterion for the appearance of a weak periodic component in the exciton distribution is derived from the condition of branching of solutions of Hammerstein-type equations at the point where the parameters of this equation reach critical values.     

Natural curvilinear coordinates for ideal MHD equations. Non-stationary flows with constant total pressure

Equations of magnetohydrodynamics (MHD) in the natural curvilinear system of coordinates where trajectories and magnetic lines play a role of coordinate curves are reduced to the non-linear vector wave equation coupled with the incompressibility condition in the form of the generalized Cauchy integral. The symmetry group of obtained equation, equivalence transformation, and group classification with respect to the constitutive equation are calculated. New exact solutions with functional arbitrariness describing non-stationary incompressible flows with constant total pressure are given by explicit formulae. The corresponding magnetic surfaces have the shape of deformed nested cylinders, tori, or knotted tubes.