Stability of solitary waves in dispersive media described by a fifth-order evolution equation

The problem of the existence and dynamical stability of solitary wave solutions to a fifth-order evolution equation, generalizing the well-known Korteweg-de Vries equation, is treated. The theoretical framework of the paper is largely based on a recently developed version of positive operator theory in Fréchet spaces (which is used for the existence proof) and the theory of orbital stability for Hamiltonian systems with translationally invariant Hamiltonians. The validity of sufficient conditions for stability are established. The shape of solitary waves under analysis are determined by a numerical solution of the boundary-value problem followed by a correction using the Picard method of 4–12 orders of accuracy.     

An invariant-manifold analysis of electrophoretic traveling waves

A new, geometric proof of a theorem of Fife, Palusinski, and Su on electrophoretic traveling waves is presented. The proof is based upon the perturbation theory for invariant manifolds due to Fenichel. The results proved here reproduce the existence, uniqueness, and asymptotic approximation theorem proved by Fifeet al. The proof given here is substantially simpler, and in addition, it provides additional insight into the geometric structure of the phase space of the traveling wave equations for this system.     

Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit

We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier–Stokes equations with an axisymmetric background straining flow. It is numerically known that the Burgers vortex obtains better stabilities as the circulation number (or the vortex Reynolds number) is increasing. Although the global stability of the axisymmetric Burgers vortex is already proved rigorously, mathematical understanding of this numerical observation has been lacking. In this paper we study a linearized operator that includes the circulation number as a parameter, and prove that if the operator is restricted on a suitable invariant subspace, then its spectrum moves to the left as the circulation number goes to infinity. As an application, we show that the axisymmetric Burgers vortex with a high rotation has a better stability, in the sense that the non-radially symmetric part of solutions to the associated evolution equation decays faster in time if the circulation number is sufficiently large.     

The Linearized Crocco Equation

In this paper, we study the existence and uniqueness of a degenerate parabolic equation, with nonhomogeneous boundary conditions, coming from the linearization of the Crocco equation [12]. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations with the so-called Crocco transformation. The linearized Crocco equation plays a major role in stabilization problems of fluid flows described by the Prandtl equations [5]. To study the infinitesimal generator associated with the adjoint linearized Crocco equation – with homogeneous boundary conditions – we first study degenerate parabolic equations in which the x-variable plays the role of a time variable. This equation is doubly degenerate: the coefficient in front of ∂_x vanishes on a part of the boundary, and the coefficient of the elliptic operator vanishes in another part of the boundary. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operator is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument (see Th. 4.9). This kind of argument is also used to prove estimates, which cannot be obtained in a classical way.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

The Influence of Moduli Slope of a Linearly Graded Matrix on the Bulk Moduli of Some Particle- and Fiber-Reinforced Composites

The stored energy functional of a homogeneous isotropic elastic body is invariant with respect to translation and rotation of a reference configuration. One can use Noether's Theorem to derive the conservation laws corresponding to these invariant transformations. These conservation laws provide an alternative way of formulating the system of equations governing equilibrium of a homogeneous isotropic body. The resulting system is mathematically identical to the system of equilibrium equations and constitutive relations, generally, of another material. This implies that each solution of the system of equilibrium equations gives rise to another solution, which describes the reciprocal deformation and solves the system of equilibrium equations of another material. In this paper we derive conservation laws and prove the theorem on conjugate solutions for two models of elastic homogeneous isotropic bodies – the model of a simple material and the model of a material with couple stress (Cosserat continuum). This revised version was published online in August 2006 with corrections to the Cover Date.     

A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem

A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.     

Stability of the Front under a Vlasov–Fokker–Planck Dynamics

We consider a kinetic model for a system of two species of particles interacting through a long range repulsive potential and a reservoir at given temperature. The model is described by a set of two coupled Vlasov–Fokker–Plank equations. The important front solution, which represents the phase boundary, is a stationary solution on the real line with given asymptotic values at infinity. We prove the asymptotic stability of the front for small symmetric perturbations.     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, a rather general model is obtained by assuming that the real noise is an output of a linear filter system-a zeromean stationary Gaussian diffusion process which satisfies detailed balance condition. By means of the asymptotic analysis approach given by L. Arnold and the expression of the eigenvalue spectrum of Fokker-Planck operator, the asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are obtained. Foundation item: the National Natural Science Foundation of China (19602016)     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

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A Simplified Proof of a Liouville Theorem for Nonnegative Solution of a Subcritical Semilinear Heat Equations

We give a new proof of the Liouville theorem proved by Merle and Zaag for nonnegative solutions of the semilinear heat equation with power nonlinearity. Our proof has a pedagogical interest and is based on Kaplan’s blow-up criterion.      

Variation transforming analysis (I)

This paper is concerned with operator variation from the transforming point of view, and presents some new concepts and new relations. Related problems and concepts include: convex operator, reciprocity set, reciprocity principles, unilateral variation principles, (H, H ₁,H ₂) — generalized solution and operator-differential equation, etc..Variation principle and variation method is a sort of basic concept and method in the analysis of many problems concerning mathematics, mechanics, physics and control theory. The change and transformation of variation or stationary value relations is the most fundamental form and process of these princeples and methods. This paper analyzes some properties of operator variation and stationary value from the transforming point of view and presents some new concepts, unifiedly treats and extends the reciprocity theorem of classical variation, variation theorem of quadratic functional and unilateral variation principles. Furthermore, we investigate the calculus of normed ring and the solution of a sort of operator differential equation.     

Nonmonotonicity of Kneading Invariants in a Family of Kinked Maps

We study monotonicity properties of the kneading invariant for one-parameter families of piecewise-linear unimodal maps and prove a theorem on the violation of monotonicity of the kneading invariant for maps that are symmetric and convex and consist of four linear pieces. The fact that such maps cannot be approximated by smooth mappings with negative Schwarzian is proved using a dynamics argument. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 159–164, April–June, 2005.     

Theorem on perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications

We study the problem of perturbations of quasiperiodic motions on coisotropic invariant tori in a class of locally Hamiltonian systems. We prove a general KAM-theorem on the perturbation of coisotropic invariant tori for locally Hamiltonian systems. As applications of this theorem, we consider the motion of an electron on a two-dimensional torus under the action of an electromagnetic field and extend results concerning the bifurcation of a Cantor set of coisotropic invariant tori to the case of locally Hamiltonian systems. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 490–515, October–December, 2005.     

一类随机分叉系统概率1分叉研究

对于三维中心流形上实噪声参数的一类余维2分叉系统,使用Arnold的渐近方法以及Fokker-Planck算子的特征谱展式,求解了不变测度以及最大的Lyapunov指数的渐进展式。     

Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.     

Non-Ergodicity of the Nosé–Hoover Thermostatted Harmonic Oscillator

The Nosé–Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nosé–Hoover dynamics being ergodic. Hoover presented numerical experiments to show that the Nosé–Hoover dynamics are non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nosé–Hoover thermostat does not give an ergodynamical system for the one- dimensional harmonic oscillator when the “mass” of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nosé–Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass.     

Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions

For finite-dimensional maps and periodic systems, Palmer rigorously proved Smale horseshoe theorem using shadowing lemma in 1988 [20]. For infinite-dimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990 [30], and Henry in 1994 [9]. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996 [17]. For infinite-dimensional autonomous systems, the current article offers such a proof. First we prove an Inclination Lemma to set up a coordinate system around a pseudo-orbit. Then we utilize graph transform and the concept of persistence of invariant manifold, to prove the existence of a shadowing orbit.     

A historical perspective of Menabrea’s theorem in elasticity

This paper presents the theorem proposed by Luigi Federico Menabrea to study linear elastic redundant systems. Some of Menabrea’s papers on the subject are examined, as well as the criticism and the corrections brought to his first proof. We consider Menabrea’s work in the frame of the studies of his contemporaries; we try to provide a historical and epistemological background for Menabrea’s theorem and for its consequences in modern mechanics.     

Invariant Solutions of the Thermal-Diffusion Equations for a Binary Mixture in the Case of Plane Motion

The group properties of the thermal-diffusion equations for a binary mixture in plane flow are studied. Optimal systems of first-and second-order subalgebras are constructed for the admissible Lie operator algebra, which is infinite-dimensional. Examples of the exact invariant solutions are given, which are found by solving ordinary differential equations. Exact solutions are found that describe thermal diffusion in an inclined layer with a free boundary and in a vertical layer in the presence of longitudinal temperature and concentration gradients. The effect of the thermal-diffusion parameter on the flow regime is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 95–108, January–February, 2006.