We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb {R}^3$ , in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including ‘Zeno breaking’, or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the ‘Contact’ effect, in which in the discontinuous limit, trajectories can pass through a ‘saddle point’ without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

We consider slow–fast delayed systems and discuss pulsating periodic solutions, which are characterised by specific properties that (a) the period of the periodic solution is close to the delay, and (b) these solutions are formed close to a bifurcation threshold. Such solutions were previously found in models of mode-locked lasers. Through a case study of population models, this work demonstrates the existence of similar solutions for a rather wide class of delayed systems. The periodic dynamics originates from the Hopf bifurcation on the positive equilibrium. We show that the continuous transformation of the periodic orbit to the pulsating regime is simultaneous with multiple secondary almost resonant Hopf bifurcations, which the equilibrium undergoes over a short interval of parameter values. We derive asymptotic approximations for the pulsating periodic solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realisation of the bifurcation scenario is highlighted.

The high Reynolds-number structure of the laminar, chaotic and turbulent attractors is investigated in a two-dimensional Kolmogorov flow. The laminar attractors include the families of multi-phased travelling waves and quasi-periodic standing waves both of which form the backbone of the transition to a turbulent flow. At leading order, each laminar attractor under study is obtained by solving the Euler equations on a manifold subject to the appropriate periodicity and symmetry conditions. The manifold is determined by a finite number of vorticity equations, these being required to suppress the secular terms at the next order. Our results show that, for the multi-phased travelling waves, the first phase velocity can be determined by an integral conservation law for kinetic energy and the subsequent phase velocities can be evaluated by a non-linear eigenvalue problem. The results also reveal that whereas viscosity determines the smallest scales and controls the amplitude of the flow, the inertial terms govern the shape and form of the flow. The comparison of our analytical predictions for evaluating the stable single-phased travelling wave with the direct numerical simulation of the Navier–Stokes equations has been undertaken, the agreement being excellent. For sufficiently high Reynolds number, after the bifurcation to chaotic flow, all of the multi-phased travelling waves and quasi-periodic standing waves become unstable non-wandering sets. Based on the above new findings for these unstable non-wandering sets and other travelling and standing waves of this kind in phase space, necessary conditions for the invariant manifolds of the chaotic and turbulent attractors are obtained, these necessary conditions being conjectured to be also sufficient.

In this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundary-value problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli–Kohn–Nirenberg exponent.

The sine-Gordon (SG) partial differential equation (PDE) with an arbitrary perturbation is initially considered. Using the method of Kuzmak–Luke, we investigate the conditions, which the perturbation must satisfy, for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion. The motivation for this study is that the mathematical modelling of physical systems often leads to the discrete SG system of ordinary differential equations, which are then approximated in the long wavelength limit by the continuous SG PDE. Such limits typically produce fourth-order spatial derivatives as correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi-continuum SG equation and not the damped SG system.

In the limit of an asymptotically large diffusivity ratio of order $\mathcal{O}$(ϵ−2) ≫ 1, steady-state spatially periodic patterns of localized spots, where the spots are centred at lattice points of a Bravais lattice, are well-known to exist for certain two-component reaction–diffusion systems (RD) in $\mathbb{R}$2. For the Schnakenberg RD model, such a localized periodic spot pattern is linearly unstable when the diffusivity ratio exceeds a certain critical threshold. However, since this critical threshold has an infinite-order logarithmic series in powers of the logarithmic gauge ν ≡ −1/log ϵ, a low-order truncation of this series is expected to be in rather poor agreement with the true stability threshold unless ϵ is very small. To overcome this difficulty, a hybrid asymptotic-numerical method is formulated and implemented that has the effect of summing this infinite-order logarithmic expansion for the stability threshold. The numerical implementation of this hybrid method relies critically on obtaining a rapidly converging infinite series representation of the regular part of the Bloch Green's function for the reduced-wave operator. Numerical results from the hybrid method for the stability threshold associated with a periodic spot pattern on a regular hexagonal lattice are compared with the two-term asymptotic results of [10] (Iron et al. J. Nonlinear Science, 2014). As expected, the difference between the two-term and hybrid results is rather large when ϵ is only moderately small. A related hybrid method is devised for accurately approximating the stability threshold associated with a periodic pattern of localized spots for the Gray-Scott RD system in $\mathbb{R}$2.

In the low-Froude number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Previous studies have considered linearized steady flow past a submerged source in infinite-depth fluids, in which exponential asymptotics were used to determine the behaviour of downstream longitudinal and transverse free-surface gravity waves. Here, unsteady flow past a submerged source in an infinite-depth fluid is investigated, with the free surface taken to be initially waveless. The source is taken to be weak, and the flow is linearized about the undisturbed solution. Exponential asymptotics are applied to determine the wave behaviour on the free surface in terms of the two-dimensional plan-view, in order to show how the free surface waves evolve over time and eventually tend to the steady solution.

In this note we prove compactness for the Cahn–Hilliard functional without assumingcoercivity of the multi-well potential.

In this paper, a fitted Numerov method is constructed for a class of singularly perturbed one-dimensional parabolic partial differential equations with a small negative shift in the temporal variable. Similar boundary value problems are associated with a furnace used to process a metal sheet in control theory. Here, the study focuses on the effect of shift on the boundary layer behavior of the solution via finite difference approach. When the shift parameter is smaller than the perturbation parameter, the shifted term is expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable. When the shift parameter is larger than the perturbation parameter, a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted scheme is developed. This scheme is also unconditionally stable. The applicability of the proposed methods is demonstrated by means of two examples.

In a one-dimensional domain, the stability of localized spike patterns is analysed for two closely related singularly perturbed reaction–diffusion (RD) systems with Brusselator kinetics. For the first system, where there is no influx of the inhibitor on the domain boundary, asymptotic analysis is used to derive a non-local eigenvalue problem (NLEP), whose spectrum determines the linear stability of a multi-spike steady-state solution. Similar to previous NLEP stability analyses of spike patterns for other RD systems, such as the Gierer–Meinhardt and Gray–Scott models, a multi-spike steady-state solution can become unstable to either a competition or an oscillatory instability depending on the parameter regime. An explicit result for the threshold value for the initiation of a competition instability, which triggers the annihilation of spikes in a multi-spike pattern, is derived. Alternatively, in the parameter regime when a Hopf bifurcation occurs, it is shown from a numerical study of the NLEP that an asynchronous, rather than synchronous, oscillatory instability of the spike amplitudes can be the dominant instability. The existence of robust asynchronous temporal oscillations of the spike amplitudes has not been predicted from NLEP stability studies of other RD systems. For the second system, where there is an influx of inhibitor from the domain boundaries, an NLEP stability analysis of a quasi-steady-state two-spike pattern reveals the possibility of dynamic bifurcations leading to either a competition or an oscillatory instability of the spike amplitudes depending on the parameter regime. It is shown that the novel asynchronous oscillatory instability mode can again be the dominant instability. For both Brusselator systems, the detailed stability results from NLEP theory are confirmed by rather extensive numerical computations of the full partial differential equations system.

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multi-scale theory for such systems and apply it to the Kuramoto–Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing. For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset so that there are two unstable modes, we observe numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacsoperators. We establish such an estimate for the parabolic Cauchy problem in the wholespace  [0, +∞) × ℝn and, under some periodicity and eitherellipticity or controllability assumptions, we deduce a similar estimate for the ergodicconstant associated to the operator. An interesting byproduct of the latter result will bethe local uniform convergence for some classes of singular perturbation problems.

We analyze the problem of shear-induced electrokinetic lift on a particle freely suspended near a solid wall, subject to a homogeneous (simple) shear. To this end, we apply the large-Péclet-number generic scheme recently developed by Yariv et al. (J. Fluid Mech., Vol. 685, 2011, p. 306). For a force- and torque-free particle, the driving flow comprises three components, respectively describing (i) a particle translating parallel to the wall; (ii) a particle rotating with an angular velocity vector normal to the plane of shear; and (iiii) a stationary particle in a shear flow. Symmetry arguments reveal that the electro-viscous lift, normal to the wall, is contributed by Maxwell stresses accompanying the induced electric field, while electro-viscous drag and torque corrections, parallel to the wall, are contributed by the Newtonian stresses accompanying the induced flow. We focus upon the near-contact limit, where all electro-viscous contributions are dominated by the intense electric field in the narrow gap between the particle and the wall. This field is determined by the gap-region pressure distributions associated with the translational and rotational components of the driving Stokes flow, with the shear-component contribution directly affecting only higher-order terms. Owing to the similarity of the corresponding pressure distributions, the induced electric field for equal particle–wall zeta potentials is proportional to the sum of translation and rotation speeds. The electro-viscous loads result in induced particle velocities, normal and tangential to the wall, inversely proportional to the second power of particle–wall separation.

Recent attention has focused on deriving localised pulse solutions to various systems of reaction–diffusion equations. In this paper, we consider the evolution of localised pulses in the Brusselator activator–inhibitor model, long considered a paradigm for the study of non-linear equations, in a finite one-dimensional domain with feed of the inhibitor through the boundary and global feed of the activator. We employ the method of matched asymptotic expansions in the limit of small activator diffusivity and small activator and inhibitor feeds. The disparity of diffusion lengths between the activator and inhibitor leads to pulse-type solutions in which the activator is localised while the inhibitor varies on an O(1) length scale. In the asymptotic limit considered, the pulses become spikes described by Dirac delta functions and evolve slowly in time until equilibrium is reached. Such quasi-equilibrium solutions with N activator pulses are constructed and a differential-algebraic system of equations (DAE) is derived, characterising the slow evolution of the locations and the amplitudes of the pulses. We find excellent agreement for the pulse evolution between the asymptotic theory and the results of numerical computations. An algebraic system for the equilibrium pulse amplitudes and locations is derived from the equilibrium points of the DAE system. Both symmetric equilibria, corresponding to a common pulse amplitude, and asymmetric pulse equilibria, for which the pulse amplitudes are different, are constructed. We find that for a positive boundary feed rate, pulse spacing of symmetric equilibria is no longer uniform, and that for sufficiently large boundary flux, pulses at the edges of the pattern may collide with and remain fixed at the boundary. Lastly, stability of the equilibrium solutions is analysed through linearisation of the DAE, which, in contrast to previous approaches, provides a quick way to calculate the small eigenvalues governing weak translation-type instabilities of equilibrium pulse patterns.

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris,SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732],we developeda class of iterative algorithmswithin the contextof equation-free methodsto approximatelow-dimensional,attracting,slow manifoldsin systemsof differential equationswith multiple time scales.For user-specified valuesof a finite numberof the observables,the mth memberof the classof algorithms( $m = 0, 1, \ldots$ )finds iterativelyan approximationof the appropriate zeroof the (m+1)st time derivativeof the remaining variablesanduses this rootto approximate the locationof the pointon the slow manifoldcorresponding to these valuesof the observables.This articleis the firstof two articlesin whichthe accuracy and convergenceof the iterative algorithmsare analyzed.Here,we work directlywith fast-slow systems,in which there isan explicit small parameter, ε,measuring the separationof time scales.We show that,for each $m = 0, 1, \ldots$ ,the fixed pointof the iterative algorithmapproximates the slow manifoldup to and includingterms of ${\mathcal O}(\varepsilon^m)$ .Moreover,for each m,we identify explicitlythe conditionsunder whichthe mth iterative algorithmconverges to this fixed point.Finally,we show thatwhenthe iterationis unstable(orconverges slowly)it may be stabilized(orits convergencemay be accelerated)by applicationof the Recursive Projection Method.Alternatively,the Newton-KrylovGeneralized Minimal Residual Methodmay be used.In the subsequent article,we will considerthe accuracy and convergenceof the iterative algorithmsfor a broader classof systems – in whichthere need not bean explicitsmall parameter – to whichthe algorithms also apply.

We consider the following nonlinear elliptic equations

\begin{gather*} \begin{cases} \Delta u+u_{+}^{N/(N-2)}=0\amp\quad\text{in }\sOm, \\ u=\mu\amp\quad\text{on }\partial\sOm\quad(\mu\text{ is an unknown constant}), \\ \dsty\int_{\partial\sOm}\biggl(-\dsty\frac{\partial u}{\partial n}\biggr)=M, \end{cases} \end{gather*}

where $u_{+}=\max(u,0)$, $M$ is a prescribed constant, and $\sOm$ is a bounded and smooth domain in $R^N$, $N\geq3$. It is known that for $M=M_{*}^{(N)}$, $\sOm=B_R(0)$, the above problem has a continuum of solutions. The case when $M>M_{*}^{(N)}$ is referred to as supercritical in the literature. We show that for $M$ near $KM_{*}^{(N)}$, $K>1$, there exist solutions with multiple condensations in $\sOm$. These concentration points are non-degenerate critical points of a function related to the Green's function.

AMS 2000 Mathematics subject classification: Primary 35B40; 35B45. Secondary 35J40