光学双稳性的强迫振动模型

本文提出一个新的光学双稳性模型.基于平均场近似,慢变振幅近似和绝热近似的思想,把光学双稳系统看成一个“黑箱”,类比非线性振动理论,指出不同的光学双稳过程(包括不同的工作物质和不同的腔(F-P腔或环腔)),能用一个恰当的强迫振动方程统一地描述.方程中所包含的都是可测量的宏观参量,使得可能直接用实验拟合,与理论结果进行定量对照.用振动理论的方法,把方程演变成自治方程组,从而方便地得到光学双稳性稳态和动态解的一般形式.并用若干已报道的具体例子验证了这个模型的普适性.     

Nonlinear elastic waves in a monatomic chain with nonlinear interaction

1　ＩＮＴＲＯＤＵＣＴＩＯＮＢｅｃａｕｓｅｓｏｍｅｉｍｐｏｒｔａｎｔ ｐｒｏｐｅｒｔｉｅｓｏｆｃｒｙｓｔａｌ(ｓｕｃｈａｓｓｐｅｃｉｆｉｃｈｅａｔａｔｈｉｇｈｔｅｍｐｅｒａｔｕｒｅ ,ｍｅｌｔｉｎｇ ,ｔｈｅｒｍａｌｅｘｐａｎｓｉｏｎ ,ｔｅｍｐｅｒａｔｕｒｅｄｅ     

Long-time relaxation processes in the nonlinear Schrödinger equation

The nonlinear Schrödinger equation, known in low-temperature physics as the Gross-Pitaevskii equation, has a large family of excitations of different kinds. They include sound excitations, vortices, and solitons. The dynamics of vortices strictly depends on the separation between them. For large separations, some kind of adiabatic approximation can be used. We consider the case where an adiabatic approximation can be used (large separation between vortices) and the opposite case of a decay of the initial state, which is close to the double vortex solution. In the last problem, no adiabatic parameter exists (the interaction is strong). Nevertheless, a small numerical parameter arises in the problem of the decay rate, connected with an existence of a large centrifugal potential, which leads to a small value of the increment. The properties of the nonlinear wave equation are briefly considered in the Appendix A.     

Nonlinear wave processes in media containing cracks partially filled with a viscous liquid

A nonlinear (in the cubic approximation) relaxation equation of state is derived for a rod containing cracks partially filled with an incompressible viscous liquid. The nonlinear effects of the self-action and interaction of low-and high-frequency longitudinal elastic waves propagating in such a rod are studied for the cases of identical and size-varied cracks. Linear and nonlinear acoustic parameters characterizing the self-action and interaction of elastic waves in a cracked rod are determined.     

Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.     

Demkov-Kunike model in cold molecule formation: the fast resonance sweep regime of the strong interaction limit

We study a quadratic-nonlinear version of the level-crossing Demkov-Kunike problem relevant to coherent molecule formation via photo- or Feshbach-association of ultracold atoms. Using an exact third-order nonlinear differential equation for the molecular state probability, we construct an approximate solution to the problem in the fast sweep regime of the strong interaction limit. The constructed approximation, defined as a solution of a first-order nonlinear equation, contains a fitting parameter which we determine through a variational procedure. The suggested approach can be interpreted as a modification of the adiabatic approximation and can further be used for the treatment of analogous nonlinear two-state problems.     

Localized optical structures in the scheme of a nonlinear layer with a feedback mirror

An analysis is made of the dynamics of the transverse structure of the field in the scheme of a layer of a medium with nonlinear amplitude-phase transmittance and a feedback mirror separated from it by a linear spacing. For the case of small field changes in a single passage, an approximate equation is derived, which is close in form to the equation used in the average-field approximation for nonlinear interferometers excited with external emission. For the nonlinearity of threshold type, an analytical form is presented for the field distribution corresponding to localized dissipative structures (dissitons).     

A coupled arbitrary Lagrangian–Eulerian and Lagrangian method for computation of free surface flows with insoluble surfactants

A finite element scheme to compute the dynamics of insoluble surfactant on a deforming free surface is presented. The free surface is tracked by the arbitrary Lagrangian–Eulerian (ALE) approach, whereas the surfactant concentration transport equation is approximated in a Lagrangian manner. Since boundary resolved moving meshes are used in the ALE approach, the surface tension, which may be a linear or nonlinear function of surfactant concentration (equation of state), and the Marangoni forces can be incorporated directly into the numerical scheme. Further, the Laplace–Beltrami operator technique, which reduces one order of differentiation associated with the curvature, is used to handle the curvature approximation. A number of 3D-axisymmetric computations are performed to validate the proposed numerical scheme. An excellent surfactant mass conservation without any additional mass correction scheme is obtained. The differences in using a linear and a nonlinear equation of state, respectively, on the flow dynamics of a freely oscillating droplet are demonstrated.     

A Theoretical Solution for a Nonlinear Master Equation

Based on the master equation describing the interaction of a single-mode bosonic state with a heat bath at finite temperature in the Born-Markov approximation, we constructed a new nonlinear master equation and derived the infinite operator sum representation of quasi-Kraus operators for the density operator     

A meshless method for the nonlinear generalized regularized long wave equation

This paper presents a meshless method for the nonlinear generalized regularized long wave(GRLW) equation based on the moving least-squares approximation.The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method.A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm.Compared with numerical methods based on mesh,the meshless method for the GRLW equation only requires the scattered nodes instead of meshing the domain of the problem.Some examples,such as the propagation of single soliton and the interaction of two solitary waves,are given to show the effectiveness of the meshless method.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential: Excited states

The quasilinearization method (QLM) is used to approximate analytically, both the ground state and the excited state solutions of the Schrödinger equation for arbitrary potentials. The procedure of approximation was demonstrated on examples of a few often used physical potentials such as the quartic anharmonic oscillator, the Yukawa and the spiked harmonic oscillator potentials. The accurate analytic expressions for the ground and excited state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first QLM iteration. In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist. The choice of zero iteration is based on general features of solutions near the boundaries. In order to estimate the accuracy of the QLM solutions, the exact numerical solutions were found as well. The first QLM iterate given by analytic expression allows to estimate analytically the role of different parameters and the influence of their variation on different characteristics of the relevant quantum systems.     

Intrinsic optical bistability in Kerr ferroelectric materials

In this paper we investigate the intrinsic optical bistability (IOB) in a ferroelectric (FE) single layer using an alternative analysis. The FE material is considered to have an intensity-dependent refractive index where the third order nonlinear susceptibility χ⁽³⁾ acts like Kerr coefficient. The nonlinear response of the FE medium is modeled using the Landau-Khalatnikov (LK) dynamical equation with the nonlinear anharmonic potential obtained from the Landau-Devonshire free energy expressed in terms of polarization. Within a single frequency approximation, the electromagnetic wave equation is written in terms of the polarization P rather than the electric field E as the dependent variable. With the application of the nonlinear boundary conditions we have derived expressions for both reflectance and transmittance as a function of the electric field incident amplitude, polarization and other material parameters. The formalism proves to be more suitable for FE materials since most of these materials have highly linear and nonlinear coefficients especially when the operating frequency is in the resonance region. The effects of thickness, operating frequency and temperature on BaTiO₃ single film are investigated theoretically. The results presented here agree in principle with the recent experimental observations of intrinsic OB in BaTiO₃ monocrystal and other FE photorefractive materials.     

Wide-Angle Parabolic Approximation for Modeling High-Intensity Fields from Strongly Focused Ultrasound Transducers

A novel numerical algorithm based on the wide-angle parabolic approximation is developed for modeling linear and nonlinear fields generated by axially symmetric ultrasound transducers. An example of a strongly focused single-element transducer is used to compare the results of ultrasound field simulations based on the Westervelt equation, Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation with differently modified boundary condition, and nonlinear wide-angle parabolic equation. It is demonstrated that having a computational speed comparable to modeling the KZK equation, the use of wide-angle parabolic approximation makes it possible to obtain solutions for highly focused ultrasound beams that are closer in accuracy to solutions based on the Westervelt equation.     

Nonlinear amplitude approximation for bilinear systems

An efficient method to predict vibration amplitudes at the resonant frequencies of dynamical systems with piecewise-linear nonlinearity is developed. This technique is referred to as bilinear amplitude approximation (BAA). BAA constructs a single vibration cycle at each resonant frequency to approximate the periodic steady-state response of the system. It is postulated that the steady-state response is piece-wise linear and can be approximated by analyzing the response over two time intervals during which the system behaves linearly. Overall the dynamics is nonlinear, but the system is in a distinct linear state during each of the two time intervals. Thus, the approximated vibration cycle is constructed using linear analyses. The equation of motion for analyzing the vibration of each state is projected along the overlapping space spanned by the linear mode shapes active in each of the states. This overlapping space is where the vibratory energy is transferred from one state to the other when the system switches from one state to the other. The overlapping space can be obtained using singular value decomposition. The space where the energy is transferred is used together with transition conditions of displacement and velocity compatibility to construct a single vibration cycle and to compute the amplitude of the dynamics. Since the BAA method does not require numerical integration of nonlinear models, computational costs are very low. In this paper, the BAA method is first applied to a single-degree-of-freedom system. Then, a three-degree-of-freedom system is introduced to demonstrate a more general application of BAA. Finally, the BAA method is applied to a full bladed disk with a crack. Results comparing numerical solutions from full-order nonlinear analysis and results obtained using BAA are presented for all systems.     

Nonlinear Excitation in Antiferromagnet RbFeCl3

Introducing the Dyson-Maleev transformation and the coherent state ansatz, we obtain two partial differential equations of motion in antiferromagnet RbFeCl₃. Employing the method of multiple scales and the long-wavelength approximation, we reduce these equations in to a nonlinear Schrödinger equation. By making use of inverse-scattering transformation, we obtained the nonlinear excitations in an tiferromagnet RbFeCl₃.     

玻色-爱因斯坦凝聚体中的超流现象

在玻色－爱因斯坦凝聚（BEC）的超流现象的研究中，人们通常采用平均场近似下求解Gross-Pitaveskii方程的方法，我们采用更严格的准确对角化的方法对弱排斥相互作用下两维旋转N－Boson体系的凝聚状态进行了研究，研究表明，弱相互作用下的基态并不是人们通常认为的单一凝聚态，而是一个碎裂凝聚态，通过碎裂态能谱与平均场方法给出的能谱之间的比较以及条件几率分布函数的计算，我们指出这种碎裂凝聚态有着内在的不稳定性，很容易破缺到一个单一凝聚状态；计算给出的条件几率分布可以用来揭示破缺石的状态，其分布图案与平均场近似下得到的涡旋图形相类似，我们进一步注意到过去研究工作主要集中在弱相互作用极限下和强相互作用Thomas-Fermi近似极限下这两种极端情况，为考察两种极限间的中间过渡区域，我们研究了中等相互作用强度下体系的基态性质。     

Quantum noise generated by four-wave mixing process within a fiber ring resonator

We have analyzed the dissipative effect of the entangled photons generated by a nonlinear optical ring resonator from a non-degenerate four-wave mixing (FWM) process. The system and reservoir Hamiltonian are established in the interaction picture. To eliminate the reservoir operators, the Markov approximation is used and result them in a Linblad form in the master equation. Consequently, the positive P representation can recast this equation to the Fokker-Planck equation, and then the stochastic differential equations i.e., the entangled photon state equation of motion for photons propagating in the fiber, are obtained and easy to analyze numerically. Results obtained have shown that the entangled strength measurement depends on three main factors; first the nonlinear susceptibility of the third harmonic generation, second the damping rate that represents loss of energy from the system to the reservoir, and final the diffusion of fluctuations in the reservoir into the entangled photon modes.     

Weak coupling regime of the Landau-Zener transition for association of an atomic Bose-Einstein condensate

In the framework of a basic semiclassical time-dependent nonlinear two-state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo- and magneto-association of an atomic Bose-Einstein condensate. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom-molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Landau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al. [Phys. Rev. A 69, 043612 (2004); J. Phys. A 38, 3505 (2005)] by order of magnitude.     

Stable bubble oscillations beyond Blake’s critical threshold

The equilibrium radius of a single spherical bubble containing both non-condensable gas and vapor is determined by the mechanical balance at the bubble interface. This expression highlights the fact that decreasing the ambient pressure below the so called Blake’s critical threshold, the bubble has no equilibrium state at all. In the last decade many authors have tried to find evidence for the existence of stable bubble oscillation under harmonic forcing in this regime, that is, they have tried to stabilize the bubble motion applying ultrasonic radiation on the bubble. The available numerical results provide only partial proof for the existence as they are usually based on linearized or weakly nonlinear (higher order approximation) bubble models. Here, based on numerical techniques of the modern nonlinear and bifurcation theory, the existence of stable bubble motion has been proven without any restrictions in nonlinearities. Although the model, applied in this paper, is the rather simple Rayleigh–Plesset equation, the presented technique can be extended to more complex bubble models easily.