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.     

Global Attractors for Mindlin–Timoshenko Plates and for Their Kirchhoff Limits

We consider dynamics of a class of Mindlin–Timoshenko plate models with nonlinear feedback forces.We prove the existence of a compact global attractor and study its limiting properties when the shear modulus tends to infinity. This limit corresponds to absence of transverse shear which is one of the Kirchhoff hypotheses in the plate theory. Lecture held by I. Chueshov in the Seminario Matematico e Fisico on April 18, 2005 Received: August 2005     

Attenuation and Diffraction of Bending Waves at Gaps in Fluid Loaded Plates

The scattering of a bending wave by a finite number of parallelrectilinear gaps in an infinite fluid-loaded plate is discussed.For the purpose of analysis, the widths of the gaps are assumedto be infinitesimal, but there is no physical contact betweenabutting edges of neighbouring sections of the plate. A sectionedge may be restrained by resilient supports or loadings, eitherindividually or jointly with the neighbouring edge. The theorydetermines the attenuation of the bending wave by the gaps andthe sound radiated into the ambient fluid during the interaction.Specific results are given for a steel plate which has a singlegap in air and in water, such that either (1) the abutting edgesare free to vibrate independently, (2) both edges are clamped,or (3) one edge is clamped and the other free. In each of thesecases the coupling between the two halves of the plate is providedsolely by the fluid loading: the bending wave would be totallyreflected at the gap in vacuo. The results are relevant to thecontrol and suppression of structure-borne sound     

Asymptotics of One-Dimensional Linear Standing Water Waves with Dispersion and Degeneracy on the Boundary

Mathematical Notes -     

连接1阶细鞍焦点的反转同宿环附近的动态及分支问题

在反转条件下研究了连接1阶细鞍焦点的对称同宿轨线附近包括具有各种不同缠绕数的周期轨线和同宿轨线在内的极其复杂的轨线结构,并且讨论了伴随Hopf分支的同宿轨线分支情况.     

Energy Spectrum Width and Nonequilibrium Energy of Rayleigh Waves Scattered on a Two-Dimensional Near-Surface Electron Layer

We obtain expressions for the energy spectrum widths of Rayleigh waves corresponding to their deformational coupling to Fermi and Boltzmann electrons in a two-dimensional layer near the surface of a semibounded solid. We evaluate the nonequilibrium energy of Rayleigh waves that depends on these widths and is caused by the same coupling to the corresponding hot electrons. We show that this energy is independent of the degeneracy degree of the electrons and is given by the mean energy of free Rayleigh waves heated up to temperature of the electrons. We find conditions under which the thermodynamics is determined by this nonequilibrium energy of Rayleigh waves in films of a certain thickness with Fermi electrons near the surface and by the equilibrium energy of bulk phonons in thicker samples. All the results are obtained using the Keldysh diagram technique applied to the case of semibounded media.