Journal Article A reaction–diffusion–advection competition model with two free boundaries in heterogeneous time-periodic environment Get access Qiaoling Chen, Qiaoling Chen School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China Search for other works by this author on: Oxford Academic Google Scholar Fengquan Li, Fengquan Li School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China *Corresponding author: fqli@dlut.edu.cnqiaolingf@126.com Search for other works by this author on: Oxford Academic Google Scholar Feng Wang Feng Wang School of Mathematics and Statistics, Xidian University, Xi’an 710071, PR China e-mail: mathwangfeng@126.com Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Applied Mathematics, Volume 82, Issue 2, April 2017, Pages 445–470, https://doi.org/10.1093/imamat/hxw059 Published: 11 January 2017 Article history Received: 25 April 2016 Revision received: 23 October 2016 Accepted: 29 November 2016 Published: 11 January 2017

The construction of the second order solitary solutions to two Riccati differential equations coupled with a multiplicative term is discussed in this paper. The generalized differential operator method is used to derive necessary and sufficient conditions for the existence of these solutions in the space of system’s parameters and initial conditions. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Computational experiments illustrate the complexity of dynamical processes in nonlinear differential equations coupled with a multiplicative term.

A new model of fracture mechanics considered previously by Sendova and Walton \cite{SendovaWalton2010}, Zemlyanova \cite{Zemlyanova2013}, and Zemlyanova and Walton \cite{Zemlyanova2012} is further developed on the example of a mixed mode curvilinear interface fracture located on the boundary of a partially debonded thin elastic inclusion embedded in an infinite thin elastic matrix. The effect of the nano-structure of the material is incorporated into the model in the form of a curvature-depended surface tension acting on the boundary of the fracture. It is shown that the introduction of the surface tension allows to eliminate the classical oscillating and power singularities of the order $1/2$ present in the linear elastic fracture mechanics. The mathematical methods used to solve the problem are based on the Muskhelishvili's complex potentials and the Savruk's integral representations. The mechanical problem is reduced to the system of singular integro-differential equations which is further reduced to a system of weakly-singular integral equations. The numerical computations and comparison with known results are presented.

In 1952, Lighthill introduced the simplest possible model of a swimming micro-organism of finite size, intended as a model of a single-celled protozoan covered in beating cilia. The model consisted of a sphere, on the surface of which material points undergo small-amplitude oscillations. In 1971, Lighthill's student, John Blake, completed the calculations and applied the theory to various species of ciliate. Subsequently we, and many others, have used the even simpler (though less realistic) model of a steady squirmer, a sphere whose surface moves tangentially with time-independent velocity. In this article we survey: -low-Reynolds-number locomotion, nutrient uptake and optimisation of individual squirmers; -hydrodynamic interactions between pairs of steady squirmers and their influence on clustering, self-diffusion and rheology in suspensions of squirmers, including the effect of being bottom-heavy; -measurements and modelling of metachronal waves in Volvox, the only truly sphericalmulti-celled organism, culminating in predictions of the mean swimming speed and angular velocity of free-swimming Volvox. The predictions are compared with experiment.

The 120-year old so-called Painleve paradox involves the loss of determinism in models of planar rigid bodies in point contact with a rigid surface, subject to Coulomb-like dry friction. The phenomenon occurs due to coupling between normal and rotational degrees-of-freedom such that the effective normal force becomes attractive rather than repulsive. Despite a rich literature, the forward evolution problem remains unsolved other than in certain restricted cases in 2D with single contact points. Various practical consequences of the theory are revisited, including models for robotic manipulators, and the strange behaviour of chalk when pushed rather than dragged across a blackboard. Reviewing recent theory, a general formulation is proposed, including a Poisson or energetic impact law. The general problem in 2D with a single point of contact is discussed and cases or inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularisation are discussed from a dynamical systems point of view. By passing to the infinite stiffness limit and allowing impact without collision, inconsistent and indeterminate cases are shown to be resolvable for all open sets of conditions. However, two unavoidable ambiguities that can be reached in finite time are discussed in detail, so called dynamic jam and reverse chatter. A partial review is given of 2D cases with two points of contact showing how a greater complexity of inconsistency and indeterminacy can arise. Extension to fully three-dimensional analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the Painleve paradox is highlighted and open problems are discussed.

The properties of spatially localized structures in systems with a conservation law are reviewed and related to the properties of fronts in such systems. The theory is illustrated using the conserved Swift–Hohenberg equation, and the insights gained used to shed light on the process of crystallization from a melt and on two problems arising in hydrodynamics: convection in a rotating layer and convection in an imposed magnetic field. Spatially localized structures in systems with time-periodic forcing are also considered. Several open problems are highlighted.

In the sugar industry a major component of the extraction process is a device known as a diffuser. Crushed and shredded cane is repeatedly washed by warm water as it travels along a conveyor, leaching the sugar from the cane. An important part of the efficiency of the process is the cycling of the ‘used’ water through the bed from downstream to upstream so that the final juice has a higher concentration of sugar, thus requiring much less energy to burn off the water. In this paper, we propose a model that allows simulation of the process to investigate the effect of variations in the properties of the cane bed, the speed of the conveyor and how to initiate the flow. It is discovered that there are several controlling factors that need to be dealt with to lead to an orderly steady process.

In this paper, the concept of discontinuity geometry of rigid impact oscillators is extended to soft impact oscillators and the mechanisms of grazing bifurcations of an impact oscillator with one-sided elastic constraint are studied by the geometric and dynamical systems methods. The existence conditions of periodic solutions are given by the discontinuity geometry objects and then are used to derive the discontinuity curves. Several bifurcation scenarios near grazing bifurcation are studied to examine the evolution of the system dynamics from a geometrical point of view. The geometrical conditions are obtained for the existence of periodic orbits with one impact, grazing and saddle-node bifurcations. Some geometrical insight is gained into a question whether there is a discontinuous jump or a continuous transition from a non-impacting period-1 to an impacting period-1 attractor at a grazing bifurcation.

The inverse scattering transform for the focusing non-linear Schro&x0308;dinger (NLS) equation with non-zero boundary conditions at infinity and double zeros of the analytic scattering coefficients is presented. The direct problem is discussed, including a detailed analysis of the discrete spectrum in the presence of such double zeros. Such cases lead to double poles for the meromorphic matrices appearing in the inverse problem. Explicit formulae for the coefficients of the singular part of the Laurent expansions of the meromorphic functions at the discrete spectrum are also derived, and the inverse problem is reduced to a standard set of linear algebraic-integral equations. The general solution of the NLS equation in the case of an arbitrary finite number of double zeros is given, and an explicit formula for soliton solutions arising in the case of a single quartet of purely imaginary double zeros is presented. The long-time asymptotic behaviour of such a solution is also studied, and the solution is shown to describe the interaction of two solitons with same amplitude and velocity parameters, which diverge from each other logarithmically as in the case of zero boundary conditions.

A spherically symmetric isolated point defect in a 3D uniaxial nematic liquid crystal sample is often called a {\it radial hedgehog}. We use topological methods to describe local configurations of uniaxial and biaxial states into which a hedgehog naturally deforms under small perturbations: these include the biaxial torus and split core configurations studied in the literature using analytical and numerical methods. The topological results here take no account of the governing physical laws but provide a library of options from which the physics must make a choice.