Highly Downloaded Collection 2011

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.

The nonlinear dynamics of biochemical reactions in a small-sized system on the order of a cell are stochastic. Assuming spatial homogeneity, the populations of n molecular species follow a multi-dimensional birth-and-death process on . We introduce the Delbrück–Gillespie process, a continuous-time Markov jump process, whose Kolmogorov forward equation has been known as the chemical master equation, and whose stochastic trajectories can be computed via the Gillespie algorithm. Using simple models, we illustrate that a system of nonlinear ordinary differential equations on emerges in the infinite system size limit. For finite system size, transitions among multiple attractors of the nonlinear dynamical system are rare events with exponentially long transit times. There is a separation of time scales between the deterministic ODEs and the stochastic Markov jumps between attractors. No diffusion process can provide a global representation that is accurate on both short and long time scales for the nonlinear, stochastic population dynamics. On the short time scale and near deterministic stable fixed points, Ornstein–Uhlenbeck Gaussian processes give linear stochastic dynamics that exhibit time-irreversible circular motion for open, driven chemical systems. Extending this individual stochastic behaviour-based nonlinear population theory of molecular species to other biological systems is discussed.

The formation of rogue waves is studied in the framework of nonlinear hyperbolic systems with an application to nonlinear shallow-water waves. It is shown that the nonlinearity in the random Riemann (travelling) wave, which manifests in the steeping of the face-front of the wave, does not lead to extreme wave formation. At the same time, the strongly nonlinear Riemann wave cannot be described by the Gaussian statistics for all components of the wave field. It is shown that rogue waves can appear in nonlinear hyperbolic systems only in the result of nonlinear wave–wave or/and wave–bottom interaction. Two special cases of wave interaction with a vertical wall (interaction of two Riemann waves propagating in opposite directions) and wave transformation in the basin of variable depth are studied in detail. Open problems of the rogue wave occurrence in nonlinear hyperbolic systems are discussed.

Co-limitation of marine phytoplankton growth by light and nutrient, both of which are essential for phytoplankton, leads to complex dynamic behaviour and a wide array of coherent patterns. The building blocks of this array can be considered to be deep chlorophyll maxima, or DCMs, which are structures localized in a finite depth interior to the water column. From an ecological point of view, DCMs are evocative of a balance between the inflow of light from the water surface and of nutrients from the sediment. From a (linear) bifurcational point of view, they appear through a transcritical bifurcation in which the trivial, no-plankton steady state is destabilized. This paper is devoted to the analytic investigation of the weakly nonlinear dynamics of these DCM patterns, and it has two overarching themes. The first of these concerns the fate of the destabilizing stationary DCM mode beyond the centre manifold regime. Exploiting the natural singularly perturbed nature of the model, we derive an explicit reduced model of asymptotically high dimension which fully captures these dynamics. Our subsequent and fully detailed study of this model—which involves a subtle asymptotic analysis necessarily transgressing the boundaries of a local centre manifold reduction—establishes that a stable DCM pattern indeed appears from a transcritical bifurcation. However, we also deduce that asymptotically close to the original destabilization, the DCM loses its stability in a secondary bifurcation of Hopf type. This is in agreement with indications from numerical simulations available in the literature. Employing the same methods, we also identify a much larger DCM pattern. The development of the method underpinning this work—which, we expect, shall prove useful for a larger class of models—forms the second theme of this paper.

We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schrödinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both Lyapunoff's method and virial identities. We find that in the one-dimensional case, i.e. for n = 1, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension n ⩾ 2 and singular kernel ∼1/r^α, no collapse takes place if α < 2, whereas collapse is possible if α ⩾ 2. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of ∼1/r² kernels for n = 3. Moreover, different evolution scenarios for the three-dimensional physically relevant case of Bose–Einstein condensates are studied numerically for both the ground state soliton and higher order toroidal states with, and without, an additional local repulsive nonlinear interaction. In particular, we show that the presence of local repulsive nonlinearity can prevent collapse in those cases.

We present an overview of mathematical results and methods relevant for the spectral study of semiclassical Schrödinger (or wave) operators of scattering systems, in cases where the corresponding classical dynamics is chaotic; more precisely, we assume that in some energy range, the classical Hamiltonian flow admits a fractal set of trapped trajectories, which hosts chaotic (hyperbolic) dynamics. The aim is then to connect the information on this trapped set with the distribution of quantum resonances in the semiclassical limit.

Our study encompasses several models sharing these dynamical characteristics: free motion outside a union of convex hard obstacles, scattering by certain families of compactly supported potentials, geometric scattering on manifolds with (constant or variable) negative curvature. We also consider the toy model of open quantum maps, and sketch the construction of quantum monodromy operators associated with a Poincaré section for a scattering flow.

The semiclassical density of long-living resonances exhibits a fractal Weyl law, related to the fact that the corresponding metastable states are 'supported' by the fractal trapped set (and its outgoing tail). We also describe a classical condition for the presence of a gap in the resonance spectrum, equivalently a uniform lower bound on the quantum decay rates, and present a proof of this gap in a rather general situation, using quantum monodromy operators.

Discrete breathers, originally introduced in the context of biopolymers and coupled nonlinear oscillators, are also localized modes of excitation of Bose–Einstein condensates (BEC) in periodic potentials such as those generated by counter-propagating laser beams in an optical lattice. Static and dynamical properties of breather states are analysed in the discrete nonlinear Schrödinger equation that is derived in the limit of deep potential wells, tight-binding and the superfluid regime of the condensate. Static and mobile breathers can be formed by progressive re-shaping of initial Gaussian wave-packets or by transporting atomic density towards dissipative boundaries of the lattice. Static breathers generated via boundary dissipations are determined via a transfer-matrix approach and discussed in the two analytic limits of highly localized and very broad profiles. Mobile breathers that move across the lattice are well approximated by modified analytical expressions derived from integrable models with two independent parameters: the core-phase gradient and the peak amplitude. Finally, possible experimental realizations of discrete breathers in BEC in optical lattices are discussed in the presence of residual harmonic trapping and in interferometry configurations suitable to investigate discrete breathers' interactions.

We study elliptic lower dimensional invariant tori of Hamiltonian systems via parametrizations. The method is based on solving iteratively the functional equations that stand for invariance and reducibility. In contrast with classical methods, we do not assume that the system is close to an integrable one nor that it is written in action-angle variables. We only require an approximation of an invariant torus with a fixed vector of basic frequencies and a basis along the torus that approximately reduces the normal variational equations to constant coefficients. We want to highlight that this approach presents many advantages compared with methods which are built in terms of canonical transformations, e.g., it produces simpler and more constructive proofs that lead to more efficient numerical algorithms for the computation of these objects. Such numerical algorithms are suitable to be adapted in order to perform computer assisted proofs.

We investigate a model for the dynamics of a solid object, which moves over a randomly vibrating solid surface and is subject to a constant external force. The dry friction between the two solids is modelled phenomenologically as being proportional to the sign of the object's velocity relative to the surface, and therefore shows a discontinuity at zero velocity. Using a path integral approach, we derive analytical expressions for the transition probability of the object's velocity and the stationary distribution of the work done on the object due to the external force. From the latter distribution, we also derive a fluctuation relation for the mechanical work fluctuations, which incorporates the effect of the dry friction.

We establish the existence of travelling wave solutions for delayed cooperative recursions that are allowed to have more than two equilibria. We define an important extended real number that is used to determine the speeds of travelling wave solutions. The results can be applied to a large class of delayed cooperative reaction–diffusion models. We show that for a delayed Lotka–Volterra reaction–diffusion competition model, there exists a finite positive number that can be characterized as the slowest speed of travelling wave solutions connecting two mono-culture equilibria or connecting a mono-culture with the coexistence equilibrium.

We investigate a reaction–diffusion–advection equation that models the dynamics of a single phytoplankton species in a eutrophic vertical water column. First, we extend the results of Du and Hsu (2010 SIAM J. Math. Anal.42 1305–33) to show that even with variable diffusion and sinking rates, the global dynamics of the model is completely determined by its unique steady-state solution. This implies that the bistable behaviour observed through numerical simulation in Ryabov et al (2010 J. Theor. Biol. 263 120–33) for the phytoplankton dynamics can only occur when one assumes limitation of nutrients in the model. Second, we examine the asymptotic profiles of the positive steady-state solution for small diffusion, large diffusion and deep water column, respectively. Our results reveal that for small diffusion, the phytoplankton population concentrates at the bottom of the water column, while for large diffusion, the population tends to distribute evenly in the water column, and when all the other factors are the same, in a water column with positive background turbidity, the total biomass is bigger in the large diffusion case than in the small diffusion case, and in a water column with zero (or negligible) background turbidity, the total biomass tends to the same limit in both cases; when the water column depth goes to infinity, the population distribution approaches that obtained in Ishii and Takagi (1982 J. Math. Biol. 16 1–24) with infinite water depth, and it reaches a unique maximum at a certain finite water level. We also give a complete answer to a question left open in Hsu and Lou (2010 SIAM J. Appl. Math. 70 245–54) regarding the behaviour of the critical death rate for deep water column, which plays a key role in determining whether a critical water depth exists.

Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction–drift–diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman–Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.

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 Nonlinearity 19 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 review recent progress in modelling the probability distribution of wave heights in the deep ocean as a function of a small number of parameters describing the local sea state. Both linear and nonlinear mechanisms of rogue wave formation are considered. First, we show that when the average wave steepness is small and nonlinear wave effects are subleading, the wave height distribution is well explained by a single 'freak index' parameter, which describes the strength of (linear) wave scattering by random currents relative to the angular spread of the incoming random sea. When the average steepness is large, the wave height distribution takes a very similar functional form, but the key variables determining the probability distribution are the steepness, and the angular and frequency spread of the incoming waves. Finally, even greater probability of extreme wave formation is predicted when linear and nonlinear effects are acting together.

The Poisson–Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. In this paper we study a new Poisson–Boltzmann type (PB_n) equation with a small dielectric parameter ² and non-local nonlinearity which takes into consideration the preservation of the total amount of each individual ion. This equation can be derived from the original Poisson–Nernst–Planck system. Under Robin-type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviours of one-dimensional solutions of PB_n equations as the parameter approaches zero. In particular, we show that in case of electroneutrality, i.e. α = β, solutions of 1D PB_n equations have a similar asymptotic behaviour as those of 1D PB equations. However, as α ≠ β (non-electroneutrality), solutions of 1D PB_n equations may have blow-up behaviour which cannot be found in 1D PB equations. Such a difference between 1D PB and PB_n equations can also be verified by numerical simulations.

We present a detailed analysis of invariant phase space structures near higher-rank saddles of Hamiltonian systems. Using the theory of pseudo-hyperbolic invariant surfaces, we show the existence of codimension-one normally hyperbolic invariant manifolds that govern transport near the higher-rank saddle points. Such saddles occur in a number of problems in celestial mechanics, chemical reactions, and atomic physics. As an example, we consider the problem of double ionization of helium in an external electric field, a basis of many modern ionization experiments. In this example, we illustrate our main results on the geometry and transport properties near a rank-two saddle.

This work considers the Keller–Segel system of parabolic–parabolic type in for n ⩾ 2. We prove existence results in a new framework and with initial data in . This initial data class is larger than the previous ones, e.g., Kozono–Sugiyama (2008 Indiana Univ. Math. J. 57 1467–500) and Biler (1998 Adv. Math. Sci. Appl. 8 715–43), and covers physical cases of initial aggregation at points (Diracs) and on filaments. Self-similar solutions are obtained for initial data with the correct homogeneity and a certain value of parameter γ. We also show an asymptotic behaviour result, which provides a basin of attraction around each self-similar solution.

Within the abstract framework of dynamical system theory we describe a general approach to the transient (or Evans–Searles) and steady state (or Gallavotti–Cohen) fluctuation theorems of non-equilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. In addition to its conceptual simplicity, another advantage of our approach is its natural extension to quantum statistical mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.

We consider the aggregation equation ρ_t − ∇ · (ρ∇K * ρ) = 0 in , where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric density that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which is the steady-state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results.

A sufficient and necessary condition for the existence of monotone travelling waves in the nonlocal Fisher–KPP equation is established, and the uniqueness of travelling wavefronts (up to translation) is also proved.