In the last 30 days

• Open/close Tibetan singing bowls

  Denis Terwagne and John W M Bush 2011 Nonlinearity 24 R51 Tag this article

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  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.

• Open/close Nonlinear modelling of cancer: bridging the gap between cells and tumours

  J S Lowengrub et al 2010 Nonlinearity 23 R1 Tag this article

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  Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

• Open/close The nonlinear Schrödinger equation with a random potential: results and puzzles

  Shmuel Fishman et al 2012 Nonlinearity 25 R53 Tag this article

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  The nonlinear Schrödinger equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a random (Anderson like) potential has been done at various levels of control: numerical, analytical and rigorous. Yet this model equation presents us with a highly inconclusive and often contradictory picture. We will describe the main recent results obtained in this field and propose a list of specific problems to focus on, which we hope will enable to resolve these outstanding questions.

• Open/close Open mathematical problems regarding non-Newtonian fluids

  Helen J Wilson 2012 Nonlinearity 25 R45 Tag this article

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  We present three open problems in the mathematical modelling of the flow of non-Newtonian fluids. The first problem is rather long standing: a discontinuity in the dependence of the rise velocity of a gas bubble on its volume. This is very well characterized experimentally but not, so far, fully reproduced either numerically or analytically. The other two are both instabilities. The first is observed experimentally but never predicted analytically or numerically. In the second instability, numerical studies reproduce the experimental observations but there is as yet no analytical or semi-analytical prediction of the linear instability which must be present.

• Open/close Lower bounds for non-trivial travelling wave solutions of equations of KdV type

  C E Kenig et al 2012 Nonlinearity 25 1235 Tag this article

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  We prove that if a solution of an equation of KdV type is bounded above by a travelling wave with an amplitude that decays faster than a given linear exponential then it must be zero. We assume no restrictions neither on the size nor in the direction of the speed of the travelling wave.

• Open/close Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

  Giambattista Giacomin et al 2012 Nonlinearity 25 1247 Tag this article

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  We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long-term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Otherwise, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disc composed of radial trajectories connecting a saddle-point equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and coherent (or synchronized) equilibria. We prove in particular nonlinear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero.

• Open/close Nonlinear dynamics and statistical physics of DNA

  Michel Peyrard 2004 Nonlinearity 17 R1 Tag this article

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  DNA is not only an essential object of study for biologists—it also raises very interesting questions for physicists. This paper discuss its nonlinear dynamics, its statistical mechanics, and one of the experiments that one can now perform at the level of a single molecule and which leads to a non-equilibrium transition at the molecular scale.

  After a review of experimental facts about DNA, we introduce simple models of the molecule and show how they lead to nonlinear localization phenomena that could describe some of the experimental observations. In a second step we analyse the thermal denaturation of DNA, i.e. the separation of the two strands using standard statistical physics tools as well as an analysis based on the properties of a single nonlinear excitation of the model. The last part discusses the mechanical opening of the DNA double helix, performed in single molecule experiments. We show how transition state theory combined with the knowledge of the equilibrium statistical physics of the system can be used to analyse the results.

• Open/close Stabilization in a two-species chemotaxis system with a logistic source

  J I Tello and M Winkler 2012 Nonlinearity 25 1413 Tag this article

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  We study a system of three partial differential equations modelling the spatio-temporal behaviour of two competitive populations of biological species both of which are attracted chemotactically by the same signal substance. More precisely, we consider the initial-boundary value problem for
  
  
  
  under homogeneous Neumann boundary conditions in a bounded domain , n ≥ 1, with smooth boundary.

  When 0 ≤ a ₁ < 1 and 0 ≤ a ₂ < 1, this system possesses a uniquely determined spatially homogeneous positive equilibrium ( u , v ). We show that given any such a ₁ and a ₂ and any positive diffusivities d ₁ and d ₂ and cross-diffusivities χ ₁ and χ ₂, this steady state is globally asymptotically stable within a certain nonempty range of the logistic growth coefficients μ ₁ and μ ₂.

• Open/close A reaction–diffusion SIS epidemic model in a time-periodic environment

  Rui Peng and Xiao-Qiang Zhao 2012 Nonlinearity 25 1451 Tag this article

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  In this paper, we consider a susceptible–infected–susceptible (SIS) reaction–diffusion model, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous and temporally periodic and the total population number is constant. We introduce a basic reproduction number and establish threshold-type results on the global dynamics in terms of . In particular, we obtain the asymptotic properties of with respect to the diffusion rate d _I of the infected individuals, which exhibit the delicate influence of the time-periodic heterogeneous environment on the extinction and persistence of the infectious disease. Our analytical results suggest that the combination of spatial heterogeneity and temporal periodicity tends to enhance the persistence of the disease.

• Open/close Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations

  Song Jiang et al 2012 Nonlinearity 25 1351 Tag this article

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  The low Mach number limit for the multi-dimensional full magnetohydrodynamic (MHD) equations, in which the effect of thermal conduction is taken into account, is rigorously justified within the framework of classical solutions with small density and temperature variations. Moreover, we show that for a sufficiently small Mach number, the compressible MHD equations admit a smooth solution on the time interval where the smooth solution of the incompressible MHD equations exists. In addition, the low Mach number limit for the ideal MHD equations with small entropy variation is also investigated. The convergence rates are obtained in both cases.