arXiv subjects

John R. King

Publications and source records attributed to John R. King.

18 recordsLinked to original sources

arXiv2026

Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. II: The two-dimensional space case

The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.

math-ph↗

arXiv2026

On the independence of the slow and fast scales in multiple-scale expansions, with application to Van der Pol's equation

When implementing the method of multiple scales, one is traditionally instructed to treat the slow and fast time scales as if they were independent. Despite the intuitive motivation and the effectiveness of this perturbation method, one cannot failt to notice that these two scales relate to the same unique variable, so independence can only be formal. How sensible is it, then, to split a variable asymptotically into two (or more) independent ones? In this paper, we elucidate this issue with Van der Pol's equation, one of the simplest weakly nonlinear oscillators, as well as a simple example of a Hopf bifurcation. The discussion involves carrying the multiple-scale analysis up to arbitrarily large order and dealing with the divergent character of the resulting asymptotic series. Using the technique of optimal truncation, we re-connect the two scales. Specifically, we show that an initial translation of the fast coordinate leads to a non-trivial, exponentially small, phase shift that depends on the slow coordinate. This phase shift breaks the independence of the slow and fast scales and is found to result from the nonlinearity. Numerical simulations confirm its existence, as well as the predicted scaling. The calculation is carried out in sufficient detail to provide confidence in the generality of our result, both in its essence and in its form. In particular, we find strong indications that a Hopf bifurcation with a quadratic nonlinearity would lead to the same phenomenon, but with a larger magnitude.

math.DS↗

arXiv2026

Small-time asymptotics and the emergence of complex singularities for the KdV equation

While real-valued solutions of the Korteweg--de Vries (KdV) equation have been studied extensively over the past 50 years, much less attention has been devoted to solution behaviour in the complex plane. Here we consider the analytic continuation of real solutions of KdV and investigate the role that complex-plane singularities play in early-time solutions on the real line. We apply techniques of exponential asymptotics to derive the small-time behaviour for dispersive waves that propagate in one direction, and demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of double-pole singularities of the initial condition in the complex plane. Using matched asymptotic expansions in the limit $t\rightarrow 0^+$, we show how complex singularities of the time-dependent solution of the KdV equation emerge from these double-pole singularities. Generically, their speed as they move from their initial position is of $\mathcal{O}(t^{-2/3})$, while the direction in which these singularities propagate initially is dictated by a Painlevé II (P$_{\mathrm{II}}$) problem with decreasing tritronquée solutions. The well-known $N$-soliton solutions of KdV correspond to rational solutions of P$_{\mathrm{II}}$ with a finite number of singularities; otherwise, we postulate that infinitely many complex-plane singularities of KdV solutions are born at each double-pole singularity of the initial condition. We also provide asymptotic results for some non-generic cases in which singularities propagate more slowly than in the generic case. Our study makes progress towards the goal of providing a complete description of KdV solutions in the complex plane and, in turn, of relating this behaviour to the solution on the real line.

nlin.SI↗

arXiv2025

Stokes' phenomenon in continuous limits of discrete Painlevé I

We use exponential asymptotic analysis to identify the relevance of Stokes' phenomenon to integrability in discrete systems. We study Stokes' phenomenon in two discrete problems with the same (leading-order) continuous limit, a finite-difference discretisation of the first continuous Painlevé equation and the first discrete Painlevé equation, as well as a family of differential equation associated with each discrete problem. This analysis reveals two important observations. Firstly, the orderly behaviour that characterises Stokes' phenomenon in discrete equations emerges naturally from corresponding continuous differential equations as the order of the latter increases, although this is not apparent at low orders. Secondly, Stokes' phenomenon vanishes in the continuum limit of the integrable discrete equation, but not the non-integrable discrete equation. This means that subdominant exponentials do not appear in the integrable equation, and therefore do not cause moveable singularities to form in the solution. The results are clarified further by consideration of one-parameter family of difference equations that interpolates between the two considered in detail.

nlin.SI↗

arXiv2024

Smoothing of the higher-order Stokes phenomenon

For nearly a century and a half the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989 Berry demonstrated how it is possible to smooth out this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution that is being switched on/off taking the universal form of an error function. Following pioneering work of Berk {\it et al.} \cite{BNR82} and the Japanese school of formally exact asymptotics \cite{Aokietal1994,AKT01}, the concept of the higher-order Stokes phenomenon was introduced in \cite{HLO04} and \cite{CM05}, whereby the ability for the exponentially small terms to cause a Stokes phenomenon may change, depending on the values of parameters in the problem, corresponding to the associated singularities in the Borel plane transitioning between different Riemann sheets. Until now, the higher-order Stokes phenomenon has also been treated as a discontinuous event. In this paper we show how the higher-order Stokes phenomenon is, in fact, also smooth and occurs universally with a prefactor that takes the form of a new special function, based on a Gaussian convolution of an error function that gives rise to a rich structure. We provide a rigorous derivation of the result, with examples spanning the gamma function, a second-order nonlinear ODE and the telegraph equation, giving rise to a ghost-like smooth contribution that is present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation and example of the effect of the smoothed higher-order Stokes phenomenon on the individual terms in the asymptotic series, where the additional contributions appear prefactored by an error function.

math.CA↗

arXiv2024

The Shigesada-Kawasaki-Teramoto model: conditional symmetries, exact solutions and their properties

We study a simplification of the well-known Shigesada-Kawasaki-Teramoto model, which consists of two nonlinear reaction-diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerisation process is suggested and exact solutions are found in this case as well.

math-ph↗

arXiv2023

Conservation laws that depend on functions and PDE reduction: extending Noether $1\tfrac{1}{2}$

This paper develops methods for simplifying systems of partial differential equations that have families of conservation laws which depend on functions of the independent or dependent variables. In some cases, such methods can be combined with reduction using families of symmetries, giving a multiple reduction that is analogous to the double reduction of order for ordinary differential equations with variational symmetries. Applications are given, including a widely-used class of pseudoparabolic equations and several mean curvature equations.

math.AP↗

arXiv2023

Burgers' equation in the complex plane

Burgers' equation is a well-studied model in applied mathematics with connections to the Navier-Stokes equations in one spatial direction and traffic flow, for example. Following on from previous work, we analyse solutions to Burgers' equation in the complex plane, concentrating on the dynamics of the complex singularities and their relationship to the solution on the real line. For an initial condition with a simple pole in each of the upper- and lower-half planes, we apply formal asymptotics in the small- and large-time limits in order to characterise the initial and later motion of the singularities. The small-time limit highlights how infinitely many singularities are born at $t=0$ and how they orientate themselves to lie increasingly close to anti-Stokes lines in the far-field of the inner problem. This inner problem also reveals whether or not the closest singularity to the real axis moves toward the axis or away. For intermediate times, we use the exact solution, apply method of steepest descents, and implement the AAA approximation to track the complex singularities. Connections are made between the motion of the closest singularity to the real axis and the steepness of the solution on the real line. While Burgers' equation has an exact solution, we deliberately apply a mix of techniques in our analysis in an attempt to develop methodology that can be applied to other nonlinear partial differential equations that do not.

math.CV↗

arXiv2022

Blow up in a periodic semilinear heat equation

Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition. Novel results include various asymptotic approximations that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Preliminary results on continuing a numerical solution beyond the singularity are also presented.

math.AP↗

arXiv2021

Multiscale analysis of nutrient uptake by plant roots with sparse distribution of root hairs: Nonstandard scaling

In this paper we undertake a multiscale analysis of nutrient uptake by plant root hairs, considering different scale relations between the radius of hairs and the distance between them. We combine the method of formal asymptotic expansions and rigorous derivation of macroscopic equations. The former prompt us to study a distinguished limit (which yields a distinct effective equation), allow us to determine higher order correctors and provide motivation for the construction of correctors essential for rigorous derivation of macroscopic equations. In the final section, we validate the results of our asymptotic analysis by direct comparison with full-geometry numerical simulations.

math.AP↗

arXiv2020

Asymptotic Analysis of a Two-Phase Model of Solid Tumour Growth

We investigate avascular tumour growth as a two-phase process consisting of cells and liquid. Based on the one-dimensional continuum moving-boundary model formulated by (Byrne, King, McElwain, Preziosi, Applied Mathematics Letters, 2003, 16, 567-573), we defined boundary conditions for the analogous model of tumour growth in two dimensions. We investigate linear stability of one dimensional time-dependent solution profiles in the moving-boundary formulation of a limit case (with negligible nutrient consumption and cell drag). For this, we obtain an asymptotic limit of the two-dimensional perturbations for large time (in the case where the tumour is growing) by using the method of matched asymptotic approximations. Having characterised an asymptotic limit of the perturbations, we compare it to the time-dependent solution profile in order to analytically obtain a condition for instability. Numerical simulations are mentioned.

math.AP↗

arXiv2019

Termination points and homoclinic glueing for a class of inhomogeneous nonlinear ordinary differential equations

Solutions $u(x)$ to the class of inhomogeneous nonlinear ordinary differential equations taking the form \[u'' + u^2 = αf(x) \] for parameter $α$ are studied. The problem is defined on the $x$ line with decay of both the solution $u(x)$ and the imposed forcing $f(x)$ as $|x| \to \infty $. The rate of decay of $f(x)$ is important and has a strong influence on the structure of the solution space. Three particular forcings are examined primarily: a rectilinear top-hat, a Gaussian, and a Lorentzian, the latter two exhibiting exponential and algebraic decay, respectively, for large $x$. The problem for the top hat can be solved exactly, but for the Gaussian and the Lorentzian it must be computed numerically in general. Calculations suggest that an infinite number of solution branches exist in each case. For the top-hat and the Gaussian the solution branches terminate at a discrete set of $α$ values starting from zero. A general asymptotic description of the solutions near to a termination point is constructed that also provides information on the existence of local fold behaviour. The solution branches for the Lorentzian forcing do not terminate in general. For large $α$ the asymptotic analysis of Keeler, Binder $\&$ Blyth (2018 "On the critical free-surface flow over localised topography", J. Fluid Mech., 832, 73-96) is extended to describe the behaviour on any given solution branch using a method for glueing homoclinic connections.

math.CA↗

arXiv2019

Moving boundary problems for quasi-steady conduction limited melting

The problem of melting a crystal dendrite is modelled as a quasi-steady Stefan problem. By employing the Baiocchi transform, asymptotic results are derived in the limit that the crystal melts completely, extending previous results that hold for a special class of initial and boundary conditions. These new results, together with predictions for whether the crystal pinches off and breaks into two, are supported by numerical calculations using the level set method. The effects of surface tension are subsequently considered, leading to a canonical problem for near-complete-melting which is studied in linear stability terms and then solved numerically. Our study is motivated in part by experiments undertaken as part of the Isothermal Dendritic Growth Experiment, in which dendritic crystals of pivalic acid were melted in a microgravity environment: these crystals were found to be prolate spheroidal in shape, with an aspect ratio initially increasing with time then rather abruptly decreasing to unity. By including a kinetic undercooling-type boundary condition in addition to surface tension, our model suggests the aspect ratio of a melting crystal can reproduce the same non-monotonic behaviour as that which was observed experimentally.

physics.flu-dyn↗

arXiv2017

Finite indentation of highly curved elastic shells

Experimentally measuring the elastic properties of thin biological surfaces is non-trivial, particularly when they are curved. One technique that may be used is the indentation of a thin sheet of material by a rigid indenter, whilst measuring the applied force and displacement. This gives immediate information on the fracture strength of the material (from the force required to puncture), but it is also theoretically possible to determine the elastic properties by comparing the resulting force-displacement curves with a mathematical model. Existing mathematical studies generally assume that the elastic surface is initially flat, which is often not the case for biological membranes. We previously outlined a theory for the indentation of curved isotropic, incompressible, hyperelastic membranes (with no bending stiffness) which breaks down for highly curved surfaces, as the entire membrane becomes wrinkled. Here we introduce the effect of bending stiffness, ensuring that energy is required to change the shell shape without stretching, and find that commonly neglected terms in the shell equilibrium equation must be included. The theory presented here allows for the estimation of shape- and size-independent elastic properties of highly curved surfaces via indentation experiments, and is particularly relevant for biological surfaces.

cond-mat.soft↗

arXiv2017

Bifurcations of self-similar solutions for reversing interfaces in the slow diffusion equation with strong absorption

Bifurcations of self-similar solutions for reversing interfaces are studied in the slow diffusion equation with strong absorption. The self-similar solutions bifurcate from the time-independent solutions for standing interfaces. We show that such bifurcations occur at the bifurcation points, at which the confluent hypergeometric functions satisfying Kummer's differential equation is truncated into a finite polynomial. A two-scale asymptotic method is employed to obtain the asymptotic dependencies of the self-similar reversing interfaces near the bifurcation points. The asymptotic results are shown to be in excellent agreement with numerical computations.

nlin.PS↗

arXiv2017

Lie symmetries of nonlinear parabolic-elliptic systems and their application to a tumour growth model

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other not. A complete Lie symmetry classification, including a number of the cases characterised being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications tumour growth). Graphical representations of the solutions are provided and biological interpretation addressed briefly. The results are generalised on multi-dimensional case under assumption of radially symmetrical shape of the tumour.

math-ph↗

arXiv2013

Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a continuum of algebraic rates of convergence to a self--similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631--680, 2010) and provides answers to some open problems.

math.AP↗

arXiv2007

Travelling-wave analysis of a model describing tissue degradation by bacteria

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the `degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.

math.AP↗