An Experimental Study of Inertial Waves in a Closed Cone

An experimental study of inertial waves in a closed cone is presented in which the inertial waves are excited by a slight periodic oscillation superimposed on the cone's basic rotation rate. The dynamic pressure field is measured along the cone's rotation axis; no standing modal structure is observed, confirming Greenspan's (1969) argument that the closed cone appears open to inertial oscillations and the inertial wave spectrum is continuous. Similar pressure measurements made in the frustum of a right circular cone show that removal of the singular apex of the cone leads to standing wave modes.     

Periodic Travelling Waves and Compactons in Granular Chains

We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime.     

On the lnviscid Theory of Rotating Fluids

Conservation theorems for “energy” and mean circulation in an inviscid rotating fluid are stated in their most general forms (which do not presume the existence of inertial modes). The geostrophic motion that contains all the mean circulation of an initial velocity field is identified as the time average of the flow. A preliminary study of the in viscid motion inside a closed cone reveals many singular features and indicates that the inertial wave spectrum may be continuous.     

Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

This Note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in Berestycki et al. (2009) [3] but their asymptotic behavior was still unclear. We use here a new numerical approximation of these traveling waves which shows that some traveling waves connect the two homogeneous steady states 0 and 1, which is a striking fact since 0 is dynamically unstable and 1 is unstable in the sense of Turing.     

The Maxey–Riley equation: Existence,uniqueness and regularity of solutions

The Maxey–Riley equation describes the motion of an inertial (i.e., finite-size) spherical particle in an ambient fluid flow. The equation is a second-order, implicit integro-differential equation with a singular kernel, and with a forcing term that blows up at the initial time. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving local existence and uniqueness of mild solutions. For certain initial velocities between the particle and the fluid, the results extend to strong solutions. We also prove continuous differentiability of the mild and strong solutions with respect to their initial conditions. This justifies the search for coherent structures in inertial flows using the Cauchy–Green strain tensor.     

Small Motions and Normal Oscillations of a Double Pendulum with Cavities Containing a Viscous Incompressible Liquid

The linear hydrodynamic problem involving the small motions and normal oscillations of a double pendulum with cavities completely filled with a liquid is examined. The problem is solved using the methods of functional analysis. An existence theorem is formulated for the solutions to the Cauchy problem and the properties of the normal oscillations are described.     

Stable odd solutions of some periodic equations modeling satellite motion

A new stability criterion is proved for second-order differential equations with symmetries in terms of the coefficients of the expansion of the nonlinearity up to the third order. Such a criterion provides solutions of twist type, which are Lyapunov-stable solutions with interesting dynamical properties. This result is connected with the existence of upper and lower solutions of a Dirichlet problem and applied to a known equation which model the planar oscillations of a satellite in an elliptic orbit, giving an explicit region of parameters for which there exists a Lyapunov-stable solution.     

Multiple solutions for semi-linear corner degenerate elliptic equations

The present paper is concerned with the existence of multiple solutions for semi-linear corner-degenerate elliptic equations with subcritical conditions. First, we introduce the corner type weighted p-Sobolev spaces and discuss the properties of continuous embedding, compactness and spectrum. Then, we prove the corner type Sobolev inequality and Poincaré inequality, which are important in the proof of the main result.     

A Canonical System of lntegrodifferential Equations Arising in Resonant Nonlinear Acoustics

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.     

Interaction of four rarefaction waves in the bi-symmetric class of the pressure-gradient system

The global existence and structure of solutions to multi-dimensional pressure-gradient system has some open problems. In this paper, we construct global classical solutions to the interaction of four planar rarefaction waves with two axes of symmetry for the pressure-gradient system in two space dimensions. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model

This is the first of a series of papers devoted to the initial value problem for the one‐dimensional Euler system of compressible fluids and augmented versions containing higher‐order terms. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity‐capillarity limit associated with the Navier‐Stokes‐Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. Our method of proof combines energy and effective energy estimates, a nonlinear Sobolev inequality, high‐integrability properties for the mass density and for the velocity, and compactness properties based on entropies.© 2015 Wiley Periodicals, Inc.     

Solvability and stability of von Karman model without rotational inertia with nonlinear forces

We investigate the nonlinear oscillations of plates described by von Karman evolution equations with mixed–hinged/simply supported nonlinear boundary conditions. The von Karman evolution system is of “hyperbolic type” with nonlinear transverse and boundary forces. We prove the existence of global solution (on given interval) constructing new variational approach to that system. We do not prove the Hadamard well-posedness, however a kind of new dependence of solutions on initial data is considered.     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

On the Structure of Spectra of Modulated Travelling Waves

Modulated travelling waves are solutions to reaction-diffusion equations that are time-periodic in an appropriate moving coordinate frame. They may arise through Hopf bifurcations or essential instabilities from pulses o fronts. In this article, a framework for the stability analysis of such solutions is presented: the reaction-diffusion equation is cast as an ill-posed elliptic dynamical system in the spatial variable acting upon time-periodic functions. Using this formulation, points in the esolvent set, the point spectrum, and the essential spectrum of the linearization about a modulated travelling wave are related to the existence of exponential dichotomies on appopriate intervals for the associated spatial elliptic eigenvalue problem. Fredholm properties of the linearized operator are characterized by a relative Morse-Floe index of the elliptic equation. These results are proved without assumptions on the asymptotic shape of the wave. Analogous results are true for the spectra of travelling waves to parabolic equations on unbounded cylinders. As an application, we study the existence and stability of modulated spatially-periodic patterns with long-wavelength that accompany modulated pulses.     

Two dimensional solitary waves in shear flows

In this paper we study existence and asymptotic behavior of solitary-wave solutions for the generalized Shrira equation, a two-dimensional model appearing in shear flows. The method used to show the existence of such special solutions is based on the mountain pass theorem. One of the main difficulties consists in showing the compact embedding of the energy space in the Lebesgue spaces; this is dealt with interpolation theory. Regularity and decay properties of the solitary waves are also established.     

The harmonic oscillations of a viscoelastic rod of triangular cross-section

Two exact solutions of the plane strain problem of the harmonic oscillations of a viscoelastic rod, the cross-section of which is a right triangle, are proposed. Either the normal displacement and the shear stress or the shear displacement and the normal stress of the side surface of the rod are given. Six dimensionless parameters which affect the dynamic deformation process are derived. Two parameters characterize the contribution of the viscous properties with respect to the elastic properties, two others define the logarithmic decrement of the longitudinal and shear harmonic waves, and two other parameters affect the wavelength of the corresponding wave and the velocity of motion of the wave front of these waves. The velocities of both types of waves and their wavelengths turn out to be greater than the velocities and wavelengths of the corresponding elastic waves. It is shown that, for certain values of the viscosity and the oscillation frequency, pseudo-resonance frequencies are possible which are higher than the resonance frequencies for an elastic medium.     

Structure and properties of four-kink multisolitons of the sine-Gordon equation

The dynamics of nonlinear waves of the sine-Gordon equation with a spatially modulated periodic potential are studied using analytical and numerical methods. The structure and properties of four-kink multisolitons excited on two identical attracting impurities are determined. For small-amplitude oscillations, an analytical spectrum of the oscillations is obtained, which is in qualitatively agreement with the numerical results.     

Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation

This paper is concerned with the traveling waves and entire solutions for a delayed nonlocal dispersal equation with convolution- type crossing-monostable nonlinearity. We first establish the existence of non-monotone traveling waves. By Ikehara’s Tauberian theorem, we further prove the asymptotic behavior of traveling waves, including monotone and non-monotone ones. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. Finally, the entire solutions are considered. By introducing two auxiliary monostable equations and establishing some comparison arguments for the three equations, some new types of entire solutions are constructed via the traveling wavefronts and spatially independent solutions of the auxiliary equations.