Robert Gardner

Existence and stability of travelling wave solutions of competition models: A degree theoretic approach

arise in various aspects of the scierices and engineering. In particular, they have been used to describe the evolution of interacting and diffusing species in mathematical ecology. This paper is concerned with competitive interactions; the purpose is to obtain large-amplitude, stable travelling wave solutions of (1) with arbitrary constant diffusion coefficients, and which are consistent with the principle of competitive exclusion. The existence and stability of large-amplitude solutions of systems is an important problem in nonlinear diffusion; see, for example, Fife [7]. We obtain such solutions for a robust class of nonlinear terms. This extends previous results of a similar nature which were proved with more restrictive hypotheses on f and g; see Gardner [ 11 ]. We also obtain the C O stability of such solutions. The method of proof employs topological degree. We construct a homotopy from a given field (f, g) to a new field, (f , ~) which is the gradient of a real-valued function. The results of [11] are then applied to obtain a travelling wave when the field is sufficiently near (f , ~). Next, we prove an a priori comparison theorem. In particular, we show that the components of the solution of (1) can be wedged between translates of the components of the travelling wave (modulo an exponentially decaying term in t), provided that the initial data are wavelike, (for a precise definition, see Theorem 2.3), and that a travelling front with monotone components exists. To this end, we

Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach

Abstract In this work, we analyze the linear stability of singular homoclinic stationary solutions and spatially periodic stationary solutions in the one-dimensional Gray-Scott model. This stability analysis has several implications for understanding the recently discovered phenomena of self-replicating pulses. For each solution constructed in A. Doelman et al. [Nonlinearity 10 (1997) 523–563], we analytically find a large open region in the space of the two scaled parameters in which it is stable. Specifically, for each value of the scaled inhibitor feed rate, there exists an interval, whose length and location depend on the solution type, of values of the activator (autocatalyst) decay rate for which the solution is stable. The upper boundary of each interval corresponds to a subcritical Hopf bifurcation point, and the lower boundary is explicitly determined by finding the parameter value where the solution ‘disappears’, i.e., below which it no longer exists as a solution of the steady state system. Explicit asymptotic formulae show that the one-pulse homoclinic solution gains stability first as the second parameter is decreased, and then successively, the spatially periodic solutions (with decreasing period) become stable. Moreover, the stability intervals for different solutions overlap. These stability results are derived via the reduction of a fourth-order slow-fast eigenvalue problem to a second-order nonlocal eigenvalue problem (NLEP). Explicit determination of these stability intervals plays a central role in understanding pulse self-replication. Numerical simulations confirm that the spatially periodic stationary solutions are attractors in the pulse-splitting regime; and, moreover, whenever, for a given solution, the value of the activator decay rate was taken to lie in the regime below that solution s stability interval, initial data close to that solution were observed to evolve toward a different spatially periodic stationary solution, one whose stability interval inclucded the parameter value. The main analytical technique used is that of matched asymptotic expansions.

Phase field models for hypercooled solidification

Abstract Properties of the solidification front in a hypercooled liquid, so called because the temperature of the resulting solid is below the melting temperature, are derived using a phase field (diffuse interface) model. Certain known properties for hypercooled fronts in specific materials are reflected within our theories, such as the presence of thin thermal layers and the trend towards smoother fronts (with less pronounced dendrites) when the undercooling is increased within the hypercooled regime. Both an asymptotic analysis, to derive the relevant free boundary problems, and a rigorous determination of the inner profile of the diffusive interface are given. Of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher order differential operators. These features necessitate a much richer mathematical analysis than previous theories. Anisotropic free boundary problems are derived from our models, the simplest of which involves determining the evolution of a set (a solid particle) whose boundary moves with velocity depending on its normal vector. Considerable attention is given to the identification of surface tension, to comparison with previous theories and to questions of stability.

Existence of multidimensional travelling wave solutions of an initial-boundary value problem

On considere le probleme suivant: u t =u xx +u yy +f(u), u(x,0,t)=u(x,L,t)=0, u(x,y,0)=u 0 (x,y), ou x∈R 1 , 0<y<L, et f(u)=u(u−1) (α-u) avec 0<α<1/2

The existence of travelling wave solutions of a generalized phase-field model

This paper establishes the existence and, in certain cases, the uniqueness of travelling wave solutions of both second-order and higher-order phase-eld systems. These solutions describe the propagation of planar solidication fronts into a hypercooled liquid. The equations are scaled in the usual way so that the relaxation time is 2 , where is a nondimensional measure of the interfacial thickness. The equations for the transition layer separating the two phases form a system identical to that for the travelling-wave problem, in which the temperature is strongly coupled with the order parameter. Thus there is no longer a well-dened temperature at the inteface, as is the case in the more frequently studied situation in which the liquid phase is undercooled but not hypercooled. For phase-eld systems of two second-order equations, we prove a general existence theorem based upon topological methods. A second, constructive proof based upon invariant-manifold methods is also given when the parameter is either suciently small or suciently large. In either regime, it is also proved that the wave and the wave velocity are globally unique. Analogous results are also obtained for generalized phase-eld systems in which the order pa- rameter solves a higher-order dierential equation. In this paper, the higher-order tems occur as a singular peturbation of the standard (isotropic) second-order equation. The higher-order terms are useful in modelling anisotropic interfacial motion.

Instability of oscillatory shock profile solutions of the generalized Burgers-KdV equation

Abstract Oscillatory shock profile solutions of the Burgers-KdV equation are studied in the limit as the viscosity e → 0. A rigorous proof of their instability is obtained by showing that the linearized operator about such a solution has many unstable eigenvalues for sufficiently small e. The result is obtained by applying a topological method introduced by Alexander, Gardner and Jones to extract spectral information about the perturbed wave with slightly positive viscosity from particular “pieces” of the underlying wave which are approximated by a solitary wave solution of the reduced equation with no viscous dissipation.

Global continuation of branches of nondegenerate solutions

On etudie les solutions des problemes aux valeurs limites de la forme: u t =u yd y+fα(u), 0<y<L, 0=l a (u,u y )| y=0 =r a (u,u y )| y=L ou l a et r a sont de la forme l a (u,u y )=au y −(1−a)u; r a (u,u y )=au y +(1−a)u; (0≤a≤1). Le terme non lineaire f est de la forme fα(u)=u(1−u) (u−α) (0<α<1/2)

An invariant-manifold analysis of electrophoretic traveling waves

A new, geometric proof of a theorem of Fife, Palusinski, and Su on electrophoretic traveling waves is presented. The proof is based upon the perturbation theory for invariant manifolds due to Fenichel. The results proved here reproduce the existence, uniqueness, and asymptotic approximation theorem proved by Fifeet al. The proof given here is substantially simpler, and in addition, it provides additional insight into the geometric structure of the phase space of the traveling wave equations for this system.

Solutions of Diffusion Equations in Channel Domains

We will discuss the existence and the qualitative properties of solutions of the problem n n