Table of contents

We present the results of an experimental investigation of the acoustics and fluid dynamics of Tibetan singing bowls. Their acoustic behaviour is rationalized in terms of the related dynamics of standing bells and wine glasses. Striking or rubbing a fluid-filled bowl excites wall vibrations, and concomitant waves at the fluid surface. Acoustic excitation of the bowl's natural vibrational modes allows for a controlled study in which the evolution of the surface waves with increasing forcing amplitude is detailed. Particular attention is given to rationalizing the observed criteria for the onset of edge-induced Faraday waves and droplet generation via surface fracture. Our study indicates that drops may be levitated on the fluid surface, induced to bounce on or skip across the vibrating fluid surface.

In this paper we consider the dynamics in the one-dimensional piston problem. We give a description of the dynamics of the system in the second approximation of the perturbation theory. The first approximation of the perturbation theory was studied previously (Gorelyshev and Neishtadt 2006 J. Appl. Math. Mech.70 4–17, Sinai 1999 Theor. Math. Phys.125 1351–7, Wright 2006 Nonlinearity19 2365–89). Given the small parameter of the system ε, the first approximation prescribes non-damping oscillations of the piston in time intervals of order ε⁻¹. We prove that in time intervals of order ε⁻² the oscillations of the piston are not damped either.

We consider the propagation of wave packets for a one-dimensional nonlinear Schrödinger equation with a matrix-valued potential, in the semi-classical limit. For an initial coherent state polarized along an eigenvector, we prove that the nonlinear evolution preserves the separation of modes, in a scaling such that nonlinear effects are critical (the envelope equation is nonlinear). The proof relies on a fine geometric analysis of the role of spectral projectors, which is compatible with the treatment of nonlinearities. We also prove a nonlinear superposition principle for these adiabatic wave packets.

In this paper we develop a general approach of studying the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker–Planck operator and linearized Boltzmann operator are considered when the spatial domain takes the whole space or torus and when there is a confining force or not. The key part of the developed approach is to construct some equivalent temporal energy functionals for obtaining time rates of the solution trending towards equilibrium in some Hilbert spaces. The result in the case of the linear Boltzmann equation with confining forces is new. The proof mainly makes use of the macro–micro decomposition combined with Kawashima's argument on dissipation of the hyperbolic–parabolic system. At the end, a Korn-type inequality with probability measure is provided to deal with dissipation of momentum components.

This paper deals with the well-posedness of the Cauchy problem for the magnetic Zakharov type system. This system describes the pondermotive force and magnetic field generation effects resulting from the nonlinear interaction between plasma-wave and particles. Using the Fourier restriction norm method, we obtain low regularity for the magnetic Zakharov system in the case of d = 2, 3.

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the d-dimensional sphere to itself for 3 ⩽ d ⩽ 6. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times at which there occurs the type I blow-up at one of the poles of the sphere. We give evidence that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time T_i, and eventually the solution comes to rest at the zero energy constant map.

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a 'higher order analogue of the discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient–quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed s-periodic initial value problems, for almost all . From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.

Gu and Zhu (2008 Commun. Anal. Geom.16 467–94) have shown that type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on . In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.

This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the behaviour of the flow and in which situations the dead-water effect appears.

This paper is devoted to a study of dimensional entropy, especially entropy on the fibres of factor maps. We show that dimensional entropy and topological entropy of sets are not usually equal, while dimensional entropy over fibres always corresponds to entropy of the factor. Then we provide an estimate of the dimensional entropy of the image of a set under a factor map. Finally we solve a few questions stated recently by Dai and Jiang on distance entropy (which is a modification of dimensional entropy).

The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness ε undergo a slow-time coarsening dynamics. With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when ε → 0.

The approach is inspired by the paper by Mielke and Zelik (2009 Mem. Am. Math. Soc.198 1–97), where the centre manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear degenerate parabolic PDEs such as lubrication equations.

While it has previously been shown by Glasner and Witelski (2003 Phys. Rev. E 67 016302), that the system of ODEs governing the coarsening dynamics can be obtained via formal asymptotic methods, the centre manifold reduction approach presented here pursues the rigorous justification of this asymptotic limit.

We obtain bounds on fluctuations of two entropy estimators for a class of one-dimensional Gibbs measures on the full shift. They are the consequence of a general exponential inequality for Lipschitz functions of n variables. The first estimator is based on empirical frequencies of blocks scaling logarithmically with the sample length. The second one is based on the first appearance of blocks within typical samples.