On the third- and fourth-order constants of incompressible isotropic elasticity

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress-strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incompressible elasticity μ, ā, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.     

Quantum problems nonlinear in the classical limit

We discuss the solutions of quantum problems which are nonlinear in the classical limit. It is shown that in this case it is necessary to solve the corresponding nonlinear classical problem and study its bifurcation properties.Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 104–107, July, 1992.     

The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit

A short-time existence theorem is proven for the Euler equations for nonisentropic compressible fluid flow in a bounded domain, and solutions with low Mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow.Partially supported by Department of Energy contracts DEAC0276ER03077-III, V, Current address: Mathematics Department, Princeton University, Princeton, NJ 08544, USA     

Wave operators and the incompressible limit of the compressible Euler equation

In an exterior domain inR ⁿ (n2), the solution of the compressible Euler equation is shown to converge to that of the incompressible Euler equation when the Mach number tends to 0. The initial layer appears.     

�����Բ���ѹ����������巽����ľ�ȷ��

得到了不可压缩理想霍尔磁流体方程组的平面波精确解,这些平面波是斜传播的左旋圆极化波或右旋圆极化波,并且涨落速度和涨落磁场通过波数联系起来。讨论了这些平面波解的叠加性质。当波的传播方向平行时,任意两支圆极化平面波都可以叠加;而当波的传播方向不平行时,只有极化方向相同,波数相同,且各自的旋度与自身都是同方向(或都是反方向)的两支圆极化平面波才可以叠加。     

非线性不可压缩霍尔磁流体方程组的精确解

得到了不可压缩理想霍尔磁流体方程组的平面波精确解,这些平面波是斜传播的左旋圆极化波或右旋圆极化波,并且涨落速度和涨落磁场通过波数联系起来。讨论了这些平面波解的叠加性质。当波的传播方向平行时,任意两支圆极化平面波都可以叠加;而当波的传播方向不平行时,只有极化方向相同,波数相同,且各自的旋度与自身都是同方向(或都是反方向)的两支圆极化平面波才可以叠加。     

The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation

We consider the flow of a gas in a channel whose walls are kept at fixed (different) temperatures. There is a constant external force parallel to the boundaries which may themselves also be moving. The system is described by the stationary Boltzmann equation to which are added Maxwellian boundary conditions with unit accommodation coefficient. We prove that when the temperature gap, the relative velocity of the planes, and the force are all sufficiently small, there is a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip voundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.     

Tracking noisy limit cycle oscillation with nonlinear filters

For engineering systems, the dynamic state estimates provide valuable information for the detection and prediction of failure due to noise and vibration. From this perspective, nonlinear filtering techniques are applied to the problem of state estimation of dynamical systems undergoing noisy limit cycle oscillation. Specifically, the extended Kalman filter, ensemble Kalman filter and particle filter are used to track the limit cycle oscillations of a Duffing oscillator using noisy observational data. The noisy limit cycle oscillations feature highly non-Gaussian trends. The efficiency and limitations of the extended Kalman filter, ensemble Kalman filter and particle filter in tracking limit cycle oscillations are examined with respect to the model and measurement noise and sparsity of measurement data. For the limit cycle oscillations considered here, it is demonstrated that the ensemble Kalman filter and particle filter outperform the extended Kalman filter in the presence of sparse observational data or strong measurement noise. For moderate measurement noise and frequent measurement data, the ensemble Kalman filter and particle filter perform equally well in comparison to the extended Kalman filter.     

Determination of limit cycles for strongly nonlinear oscillators

An innovative approach to finding limit cycles is proposed and illustrated on the van der Pol equation. The technique developed in this Letter is similar to the Ritz's method in variational theory. The present theory can be applied to not only weakly nonlinear equations, but also strongly nonlinear ones, and the obtained results are valid for the whole solution domain.     

Self-synchronization of populations of nonlinear oscillators in the thermodynamic limit

A population of identical nonlinear oscillators, subject to random forces and coupled via a mean-field interaction, is studied in the thermodynamic limit. The model presents a nonequilibrium phase transition from a stationary to a time-periodic probability density. Below the transition line, the population of oscillators is in a quiescent state with order parameter equal to zero. Above the transition line, there is a state of collective rhythmicity characterized by a time-periodic behavior of the order parameter and all moments of the probability distribution. The information entropy of the ensemble is a constant both below and above the critical line. Analytical and numerical analyses of the model are provided.     

The physical origin of sigmoidal respiratory pressure-volume curves: Alveolar recruitment and nonlinear elasticity

An important unsolved problem in medical science concerns the physical origin of the sigmoidal shape of pressure-volume curves of healthy (and some unhealthy) lungs. Conventional wisdom holds that linear response, i.e., Hooke’s law, together with alveolar overdistention play a dominant role in respiration, but such assumptions cannot explain the crucial empirical sigmoidal shape of the curves. Here, we propose a theory of alveolar recruitment together with nonlinear elasticity of the alveoli. The proposed model surprisingly and correctly predicts the observed sigmoidal pressure-volume curves. We discuss the importance of this result and its implications for medical practice.     

Dynamics of nonlinear mass-spring chains near the continuum limit

We study the dynamics of mass-spring chains with arbitrary interparticle and substrate potentials and describe a systematic approach to derive the equations of motion near the continuum limit. Our method, which applies both to strictly one-dimensional problems and to one-dimensional chains free to move in three dimensions, correctly captures all terms to given order in discreteness and allows us to formulate well-behaved nonlinear partial differential equations for these systems. We re-examine the familiar cases of the Fermi-Pasta-Ulam problem and the Frenkel-Kontorova model and obtain new insights into these problems.     

Lie algebras and dynamic nonlinear systems containing limit cycles

Certain Lie algebras, represented as linear partial differential operators of first order, are used to derive autonomous systems of differential equations which involve limit cycles. To illustrate the approach an example is given.     

New applications of the calculus of variations in the large to nonlinear elasticity

Global methods of the calculus of variations and the infinite dimensional critical point theories of Morse and Ljusternik are applied to investigate the structure of the equilibrium states of thin flexible elastic plates under general body forces. The arguments used are equally applicable to broad classes of physical systems governed by nonlinear elliptic partial differential equations.