Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.     

Four-component gas/water/oil displacements in one dimension: Part I. structure of the conservation law

This paper presents the first analysis of the mathematical structure of a system of conservation laws modeling compositional flow of four components in three phases. The phase behavior that results from assuming the equilibrium volume ratios of the components in the phases are fixed (constant K-values) when up to three phases may form, is studied. We parameterize the equations in the three-phase region and show that within the three-phase region two of the characteristic curves can be found using three-phase immiscible flow theory. The third eigenvalue can also be found analytically when the K-values are constant. We show that the eigenvalue problem given by the conservation law has a discontinuity at the boundary of the two- and three-phase regions. Finally, the loss of strict hyperbolicity is discussed.     

The Kinetic Limit of a System of Coagulating Brownian Particles

We consider a random model of diffusion and coagulation. A large number of small particles is randomly scattered in at an initial time. Each particle has some integer mass and moves as a Brownian motion whose rate of diffusion is determined by that mass. When any two particles are close, they are liable to combine into a single particle that bears the mass of each of them. The range of interaction between pairs of particles is chosen so that a typical particle is liable to interact with a unit order of other particles in a unit of time. We determine the macroscopic evolution of the system, in any dimension d ≧ 3. The density of particles evolves according to the Smoluchowski system of partial differential equations, indexed by the mass parameter, in which the interaction term is a sum of products of densities. Central to the proof is the task of establishing the so-called Stosszahlansatz, which asserts that, at any given time, the presence of particles of two given masses at any given point in macroscopic space is asymptotically independent, as the initial number of particles is taken to be high. Nonetheless, there is, in a microscopic region about each particle, a reduced probability of finding another particle. Determining this deficit precisely is necessary in computing the coefficients appearing in the interaction terms of the Smoluchowski partial differential equation.     

On the Chaotic Behaviour of Discontinuous Systems

We follow a functional analytic approach to study the problem of chaotic behaviour in time-perturbed discontinuous systems whose unperturbed part has a piecewise C ¹ homoclinic solution that crosses transversally the discontinuity manifold. We show that if a certain Melnikov function has a simple zero at some point, then the system has solutions that behave chaotically. Application of this result to quasi periodic systems are also given.     

Large Time Well-posedness of the Three-Dimensional Capillary-Gravity Waves in the Long Wave Regime

In the regime of weakly transverse long waves, given long-wave initial data, we prove that a non-dimensionalized water wave system in an infinite strip under the influence of gravity and surface tension on the upper free interface has a unique solution on [0, T/ e{0, T/ \varepsilon} ] for some e{\varepsilon} independent of constant T. We shall prove in the subsequent paper (Ming et al., The long wave approximation to the three-dimensional capillary gravity waves, 2011) that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev–Petviashvili (KP) equations.     

Analyticity and Gevrey-Class Regularity for the Second-Grade Fluid Equations

We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of the solutions that does not vanish as t → ∞, and which is independent of the Rivlin–Ericksen material parameter α. Applications to the damped incompressible Euler equations are also given.     

Scattering of a plane monochromatic wave by a system of strips

Summary The diffraction of a plane wave by a general system of strips is treated by means of separation of variables. A new addition theorem for Mathieu functions is used to satisfy the boundary conditions on the strips. Numerical calculations are performed for the case of two strips lying in the same plane, the boundary conditions being grad _n =0, while the angle of incidence of the plane wave is arbitrary. The transmission coefficient for a system of two slits in a plane, perfectly conducting screen is calculated for a range of values of the parameters. An approximate expression relating the transmission coefficient for the system of two slits to the transmission coefficient for a system of a single slit is given. As the distance between the two slits is increased, the transmission coefficient for the system of two slitsvery rapidly becomes nearly identical with the transmission coefficient for a single slit.     

An electric point charge moving along the poling direction of a transversely isotropic piezoelectric solid

The problem of an electric point charge moving constantly along the poling direction of a transversely isotropic piezoelectric solid is considered in a moving coordinate system, which moves together with the electric point charge. A general solution in the moving coordinate system is given, and all the field components, such as displacements, electric potential, stresses and electric displacements, can be concisely expressed in terms of four quasi-harmonic functions. We also present two examples to demonstrate the effect of the moving velocity on the values of _i. Once the general solution is given, the axisymmetric problem of a moving electric point charge can be easily solved. The explicit expressions of all the field components caused by the moving electric charge are presented, and the effect of the moving velocity on these field components is numerically investigated.     

Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

The evolution of heat in crystalline solids is described at low temperatures by the Boltzmann-Peierls equation, which is a kinetic equation for the phase density of phonons.In this study, we solve initial value problems for the Boltzmann-Peierls equation in relation to the following issues: In thermodynamics, a given kinetic equation is usually replaced by a truncated moment system, which in turn is supplemented by a closure principle so that a system of PDEs results for some moments as thermodynamic variables. A very popular closure principle is the maximum entropy principle, which yields a symmetric hyperbolic system. In recent times, this strategy has led to serious studies on two problems that might arise: 1. Do solutions of the maximum entropy principle exist? 2. Is the physics that is embodied by the kinetic equation more or less equivalently displayed by the truncated moment system? It was Junk who proved for the BOLTZMANN equation of gases that maximum entropy solutions do not exist. The same failure appears for the Fokker-Planck equation, which was proved by means of explicit solutions by Dreyer, Junk, and Kunik.This study has two main objectives:1. We give a positive existence result for the maximum entropy principle if the underlying kinetic equation is the Boltzmann-Peierls equation. In other words we show that the maximum entropy principle can be used here to establish a closed hyperbolic moment system of PDEs. However, the intent of the paper is by no means a general justification of the maximum entropy principle.2. We develop an approximative method that allows the solutions of the kinetic equations to be compared with the solutions of the hyperbolic moment systems. To this end we introduce kinetic schemes that consists of free flight periods and two classes of update rules. The first class of rules is the same for the kinetic equation as well as for the maximum entropy system, while the second class of update rules contains additional rules for the maximum entropy system. It is shown that if a sufficient number of moments are taken into account, the two solutions converge to each other. However, in terms of numerical effort, the presented solver for the kinetic equation clearly outperforms the one for the maximum entropy principle.Received: 15 August 2003, Accepted: 8 November 2003, Published online: 11 February 2004PACS: 02.30.Jr, 02.60.Cb, 05.30.Jp, 44.10. + i, 63.20.-e, 66.70. + f, 65.40.Gr Correspondence to: M. Herrmann     

Statistical mechanics of temporary polymer networks I. The equilibrium theory

In part I of this work (the present article) the equilibrium state of temporary polymer networks is treated in the framework of thermodynamics and statistical mechanics. The network is described as an open system. Thereby we use a modified spring-bead model in which the beads represent junctions that decay and reform thus adding a viscous component to the assumed elastic behaviour of the permanent network. The relevant statistical equation — analogous to Liouville's equation — is solved. The grand-canonical probability density function and two of three equations of state are derived. Explicit formulae are given for several relevant probabilities. For instance the probabilityw (z)dz that a network chain connecting two junctions has a contour length betweenz andz +dz is given by the Wien type formulaw(z) =A z ³ exp {–B z} whereA andB do not depend onz.     

Strong Solutions for a Phase Field Navier–Stokes Vesicle–Fluid Interaction Model

In this paper we study a mathematical model for the dynamics of vesicle membranes in a 3D incompressible viscous fluid. The system is in the Eulerian formulation, involving the coupling of the incompressible Navier–Stokes system with a phase field equation. This equation models the vesicle deformations under external flow fields. We prove the local in time existence and uniqueness of strong solutions. Moreover, we show that, given T > 0, for initial data which are small (in terms of T), these solutions are defined on [0, T] (almost global existence).     

Asymptotic Resonance, Interaction of Modes and Subharmonic Bifurcation

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam β-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.     

Bifurcation Analysis of a Predator–Prey System with Generalised Holling Type III Functional Response

We consider a generalised Gause predator–prey system with a generalised Holling response function of type III: . We study the cases where b is positive or negative. We make a complete study of the bifurcation of the singular points including: the Hopf bifurcation of codimensions 1 and 2, the Bogdanov–Takens bifurcation of codimensions 2 and 3. Numerical simulations are given to calculate the homoclinic orbit of the system. Based on the results obtained, a bifurcation diagram is conjectured and a biological interpretation is given.      

On the transient response of forced nonlinear oscillators

We consider the transient response of a prototypical nonlinear oscillator modeled by the Duffing equation subjected to near resonant harmonic excitation. Of interest here is the overshoot problem that arises when the system is undergoing free motion and is suddenly subjected to harmonic excitation with a near resonant frequency, which leads to a beating type of transient response during the transition to steady state. In some design applications, it is valuable to know the peak value of this response and the manner in which it depends on system parameters, input parameters, and initial conditions. This nonlinear overshoot problem is addressed by considering the well-known averaged equations that describe the slowly varying amplitude and phase for both transient and steady state responses. For the undamped system, we show how the problem can be reduced to a single parameter χ that combines the frequency detuning, force amplitude, and strength of nonlinearity. We derive an explicit expression for the overshoot in terms of χ, describe how one can estimate corrections for light damping, and verify the results by simulations. For zero damping, the overshoot approximation is given by a root of a quartic equation that depends solely on χ, yielding a simple bound for the overshoot of lightly damped systems.     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

Designing Freeform Lenses for Intensity and Phase Control of Coherent Light with Help from Geometry and Mass Transport

We study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position, shape and intensity. Such interfaces can be refracting surfaces of two different lenses or of one lens. In geometrical optics approximation, the analytic formulation of this problem in both cases requires construction of maps with controlled Jacobian. Though this Jacobian can be expressed as a second order partial differential equation of Monge-Ampère type for a scalar function defining one of the refracting surfaces, its analysis is not straightforward. In this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions. Existence and uniqueness of weak solutions in Lipschitz classes for both cases are established by variational methods. Our results show, in particular, that two types of interfaces exist in each case for the same data: one of these types always consists of two interfaces, one of which is concave or convex and the second convex or concave, while the interfaces of the second type may be neither convex nor concave. The availability of a design with convex/concave lenses is particularly important for fabrication. The truly geometric nature of this problem permits its statement and investigation in \mathbb R^N+1, N \geqq 1{\mathbb {R}^{N+1},\, N \geqq 1} .     

MHD rotating flow of a viscous fluid over a shrinking surface

This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows.     

On Minimality of Nonautonomous Dynamical Systems

The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a sequence of continuous self-maps of X is studied. A sufficient condition for the nonminimality of such a system is formulated. Special attention is given to the particular case where X is a real compact interval I. A sequence of continuous self-maps of I forming a minimal nonautonomous system may converge uniformly. For example, the limit may be any topologically transitive map. However, if all maps in the sequence are surjective, then the limit is necessarily monotone. An example where the limit is the identity is given. As an application, in a simple way we construct a triangular map in the square I ² with the property that every point except those in the leftmost fiber has an orbit whose -limit set coincides with the leftmost fiber.     

Change-of-type behavior in viscoelastic slender jet models

In this paper a slender jet model of viscoelastic fluids which is asympotically derived from the full free surface boundary-value problem. The model system consists of four coupled quasi-linear differential equations in one space dimensions, where the nonlinear characteristics are given in closed form. Two characteristics are always real, two others may be real or complex, leaving open the possibility for change-of-type from hyperbolic to mixed elliptic/hyperbolic type. We proceed to exhibit exact solutions (constant, steady time dependent and space-time dependent) along which this model system undergoes a variety of change-of-type phenomena. Viewed purely as a one-dimensional (1-D) model for change-of-type, we explain the significance of type for the stability of these solutions and describe the numerical implications for each type. We also explain the physical significance of these model phenomena with respect to the original 3-D system, since these asymptotic equations are no longer valid once small-scale instabilities develop. Remarkably, these special solutions of the model system that exhibit change-of-type correspond to exact solutions of the 3-D Maxwell model with cylindrical free surface. The 1-D model equations, however, are not an invariant reduced system of the full 3-D free surface Maxwell model, so that the change-of-type exhibited here in the 1-D model is not directly responsible for a 3-D free surface change-of-type. Regularizations of this model as a catastrophic change-of-type develops are suggested.Research support is gratefully acknowledged from the Air Force Office of Scientific Research, Grant No. 88-0164.     

An exact analytic method applied to nonhomogeneous ring- and stringer-stiffened cylindrical shells

In this paper, the nonlinear axial symmetric deformation problem of nonhomogeneous ring- and stringer-stiffened shells is first solved by the exact analytic method. An analytic expression of displacements and stress resultants is obtained and its convergence is proved. Displacements and stress resultants converge to exact solution uniformly. Finally, it is only necessary to solve a system of linear algebraic equations with two unknowns. Four numerical examples are given at the end of the paper which indicate that satisfactory results can be obtained by the exact analytic method.