扁球壳和扁锥壳的轴对称非线性热弹耦合振动

假设温度场与应变场相互耦合,研究了旋转扁薄球壳和锥壳的轴对称非线性热弹振动问题．基于von Krmn理论和热弹性理论,导出了本问题的全部控制方程及其简化形式．应用Galerkin技术进行时空变量分离后,得到了一个关于时间的非线性常微分方程组．根据方程的特点,分别用多尺度法和正则摄动法求得了壳体振动的频率与振幅间特征关系和振幅衰减规律的一次近似解析解,并讨论了壳体几何参数、热弹耦合参数以及边界条件等因素对其非线性热弹耦合振动特性的影响．     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique

The nonlinear modeling and subsequent dynamic analysis of cracked Timoshenko beams with functionally graded material (FGM) properties is investigated for the first time using harmonic balance method followed by an iterative technique. Crack is assumed to be open throughout. During modeling, nonlinear strain–displacement relation is considered. Rotational spring model is adopted in order to model the open cracks. Energy formulations are established using Timoshenko beam theory. Nonlinear governing differential equations of motion are derived using Lagrange's equation. In order to incorporate the influence of higher order harmonics, harmonic balance method is employed. This reduces the governing differential equations into nonlinear set of algebraic equations. These equations are solved using two different iterative techniques. Methodology is computationally easier and efficient as well. This is observed that although assumption of simple harmonic motion (SHM) simplifies the problem, it yields to erroneous results at higher amplitude of motion. However, accuracy of the solution is improved considerably when the contribution of higher order harmonic terms are considered in the analysis. Results are compared with the available results, which confirm the validity of the methodology. Subsequent to that a parametric study on influence of forcing term, material indices and crack parameters on large amplitude vibration of Timoshenko beams is performed for two different boundary conditions.     

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

A straightforward algorithm for the symbolic computation of generalized (higher‐order) symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations. With our Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Interacting Wave Fronts and Rarefaction Waves in a Second Order Model of Nonlinear Thermoviscous Fluids

A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation.     

Projectors in nonlinear evolution problem: Acoustic solitons of bubbly liquid

The method of one-dimensional disturbances splitting into components of rightward propagating, leftward propagating, and stationary components by projection technique is applied to compressible liquid with bubbles. By such projecting, the fundamental system of equations is transformed to three nonlinear equations of the interacting components. A small parameter is introduced which determines input of nonlinear and dispersive terms. The system is reduced to one of a Korteweg-de Vries type. It is shown that these three-mode evolution equations are approximately reduced to intergrable KdV-MKdV equation on a class of initial conditions specified by projecting, soliton solution is presented.     

A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

Persistence of bounded solutions to degenerate Sobolev-Galpern equations

In this paper the persistence of bounded solutions to degenerate evolution equations of Sobolev-Galpern type is discussed. In order to define the evolution operator well, we study the existence and uniqueness of solutions to its linear form. On this basis we discuss exponential dichotomies of the evolution operator and give the Fredholm alternative result for bounded solutions of nonhomogeneous linear degenerate equations. This result enables us to give a condition for the persistence of bounded solutions of a general degenerate nonlinear autonomous equation under a nonautonomous perturbation.     

Nonlinear Evolution Equations and the Braiding of Weakly Transporting Flows Over Gravel Beds

The flow of a river that transports sediment in the form of gravel as bedload is investigated for the case when the transport is small. The linear stability of such flows is discussed and used to formulate some strongly nonlinear investigations describing the interaction of bar instabilities that are known to occur. The key spatial scales in the asymptotic limit of small transport are identified, and highly nonlinear evolution equations derived for each case. A generalized KDV equation is found to govern the nonlinear evolution at small wavenumbers, while at O(1) wavenumbers an infinite set of "triad-like" amplitude equations describes the flow. The interactions demonstrate the natural tendency of rivers of width significantly higher than the critical width at which instability first occurs to form complex patterns that may be associated with braided rivers. The weak transport limit used in our anaysis makes our work directly relevant to rivers experiencing flood conditions where the onset of a flood causes transport to begin. The results shown suggest that in the highly nonlinear stages, bars take the form of slabs tilted in the flow direction with steep edges. In addition, it is found that there is no equilibrium state. These findings are consistent with observations.     

Existence Theorems and Estimates of Solutions for Equations of Principal Resonance

In this paper, we study systems of nonlinear, nonautonomous, ordinary differential equations that appear in the theory of averaging of nonlinear oscillations. We prove existence theorems for them and obtain conditions under which variables of the type of amplitude (or energy) are uniformly bounded with respect to time.     

Existence of solutions for gradient systems with application to diffusion problems involving nonconvex energies

In this paper, we establish the existence of solutions for gradient systems of evolution under some type (M) and semi-coerciveness conditions. The main result is applied in order to solve nonlinear diffusion equations involving nonconvex energies.     

Nonlinear stability and chaos in electrohydrodynamics

Using the method of multiple scales, the nonlinear instability problem of two superposed dielectric fluids is studied. The applied electric filed is taken into account under the influence of external modulations near a point of bifurcation. A time varying electric field is superimposed on the system. In addition, the viscosity and variable gravity force are considered. A generalized equation governing the evolution of the amplitude is derived in marginally unstable regions of parameter space. A bifurcation analysis of the amplitude equation is carried out when the dissipation due to viscosity and the control parameter are both assumed to be small. The solution of a nonlinear equation in which parametric and external excitations are obtained analytically and numerically. The method of generalized synchronization is applied to determine the equations that describe the modulation of the amplitude and phase. These equations are used to determine the steady state equations. Frequency response curves are presented graphically. The stability of the proposed solution is determined applying Liapunov's first method. Numerical solutions are presented graphically for the effects of the different equation parameters on the system stability, response and chaos.     

Traveling wavefronts for a reaction-diffusion-chemotaxis model with volume-filling effect

In this paper, we study the propagation of the pattern for a reaction-diffusionchemotaxis model. By using a weakly nonlinear analysis with multiple temporal and spatial scales, we establish the amplitude equations for the patterns, which show that a local perturbation at the constant steady state is spread over the whole domain in the form of a traveling wavefront. The simulations demonstrate that the amplitude equations capture the evolution of the exact patterns obtained by numerically solving the considered system.     

Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations

Cross-diffusion models the situation where the presence, absence or abundance of one species of population affects the movement of other species of population in the domain under consideration and vice versa. Inclusion of cross-diffusion terms makes the modeling approach more realistic and shows significant impact on the spatio-temporal pattern formation scenario. In this paper, cross-diffusion is considered in a prey-predator model with ratio-dependent functional response, in addition to self-diffusion. Weakly nonlinear analysis is used near the Turing bifurcation boundary to derive the amplitude equations. From the stability analysis of the amplitude equations, conditions for emergence of Turing patterns such as cold spot, hot spot, mixture of spots and stripes and labyrinthine are identified. The analytical results are then verified with the help of numerical simulations. Results are general in nature and can be used to study the effect of cross-diffusion on other prey predator models both analytically and numerically.     

Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field. Oblatum: 6-XI-1998 & 12-VI-1998 / Published online: 14 January 1999     

参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

The family index theorem and bifurcation of solutions of nonlinear elliptic BVP

We obtain some new bifurcation criteria for solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. The results are of different nature from the ones that can be obtained via the traditional Lyapunov–Schmidt reduction. Our sufficient conditions for bifurcation are derived from the Atiyah–Singer family index theorem and therefore they depend only on the coefficients of derivatives of leading order of the linearized differential operators. They are computed explicitly from the coefficients without the need of solving the linearized equations. Moreover, they are stable under lower order perturbations.     

Dynamical characteristics of decaying Lamb couples

A multiple-scale adiabatic asymptotic theory is developed to describe the dissipation of the solitary Lamb couple or modon solutions of the two-dimensional Navier-Stokes equations. The transport equations describing the evolution of the Lamb couple are obtained as solvability conditions for a direct asymptotic expansion assuming a relatively large but finite Reynolds number and are equivalent to globally-integrated leading-order enstrophy and energy balances. The asymptotic theory predicts that the spectral or spatial characteristics of the decaying Lamb couple are temporally invariant and that there is a simple exponential decay in the amplitude and translation speed. We compare the predictions of the theory with a high-resolution numerical simulation. The global and local predictions of the theory and the results of the numerical simulation are in very good agreement. As well, we present a time-series of vorticity-stream function scatter diagrams as derived from the numerical simulation to show that thenon-analytic linear vorticity-stream function relationship is being continuously maintained during the perturbed evolution of the Lamb couple.     

On stability for a class of semilinear stochastic evolution equations

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and using a limiting argument to pass on some properties of strong solution to our purposes. Several examples are studied to illustrate our theory. In particular, by means of the derived results we lose conditions of certain stochastic evolution systems from Haussmann (1978) to obtain the pathwise stability for mild solution with probability one.