ON AXISYMMETRIC TRAVELING WAVES AND RADIAL …

NATURAL RESOURCE MODELINGVolume 13, Number 3, Fall 2000

ON AXISYMMETRIC TRAVELING WAVESAND RADIAL SOLUTIONS OF

SEMI-LINEAR ELLIPTIC EQUATIONS

THOMAS P. WITELSKIDepartment of Mathematics

Duke UniversityDurham, NC 27708-0320

E-mail: [email protected]

KINYA ONODepartment of Mathematics

Boston UniversityBoston, MA 02215

E-mail: [email protected]

TASSO J. KAPERDept. of Mathematics and Center for BioDynamics

Boston University111 Cummington Street

Boston, MA 02215E-mail: [email protected]

ABSTRACT. Combining analytical techniques from per-turbation methods and dynamical systems theory, we presentan elementary approach to the detailed construction of ax-isymmetric diffusive interfaces in semi-linear elliptic equa-tions. Solutions of the resulting non-autonomous radial differ-ential equations can be expressed in terms of a slowly varyingphase plane system. Special analytical results for the phaseplane system are used to produce closed-form solutions for theasymptotic forms of the curved front solutions. These axisym-metric solutions are fundamental examples of more generalcurved fronts that arise in a wide variety of scientific fields,and we extensively discuss a number of them, with a par-ticular emphasis on connections to geometric models for themotion of interfaces. Related classical results for travelingwaves in one-dimensional problems are also reviewed briefly.Many of the results contained in this article are known, andin presenting known results, it is intended that this article beexpository in nature, providing elementary demonstrations ofsome of the central dynamical phenomena and mathematicaltechniques. It is hoped that the article serves as one possi-ble avenue of entree to the literature on radially symmetricsolutions of semilinear elliptic problems, especially to thosearticles in which more advanced mathematical theory is de-veloped.

Copyright c©2000 Rocky Mountain Mathematics Consortium

339

340 T.P. WITELSKI, K. ONO AND T.J. KAPER

1. Introduction. In a wide variety of scientific fields, includingchemical physics (Borgis and Moreau [1988], Witelski et al. [1998]),combustion theory (Fife [1988], Volpert et al. [1994]), reaction-diffusionproblems (Grindrod [1996]), fluid mechanics (Lamb [1993], Majda andBertozzi [1999]), population dynamics (Murray [1990]), and mathemat-ical biology (Keener and Sneyd [1998], Murray [1990]), to name but afew, problems describing the formation of spatial structure involve theanalysis of localized transition layers or diffusive interfaces. These in-terfaces are boundaries that separate regions of spatially uniform homo-geneous solutions of the nonlinear systems. The form of the governingequations in these problems determines the details of the local structureof the transition layers, as well as their motion and their stability. Thelong-time asymptotic state of the system is ultimately determined bythe motion and interaction of these localized structures, and possibleoutcomes include complex patterns (Goldstein et al. [1996], Grindrod[1996], Murray [1990]) and rotating spiral structures (Bernoff [1991],Tyson and Keener [1988]). There are extensive bodies of mathemati-cal literature examining existence, uniqueness, stability and dynamicsof such problems in applications in reaction-diffusion systems, fluiddynamics, reactive-transport phenomena, and other areas of appliedphysics; as an introduction to this field, we cite the following incom-plete list of some of the fundamental earlier works and some of the morerecent reviews (Aronson and Weinberger [1978], Berestycki and Lions[1980], Brezis and Lieb [1984], Fife [1988], Gidas et al. [1979], Jones[1983b], Kaper and Kwong [1988], Kerner and Osipov [1994], Kuzinand Pohozaev [1997], McLeod and Serrin [1968] and [1987], Ni [1988],Powell [1997], Terman [1987], Vasileva et al. [1995]).

In this article we review, from an elementary point of view, someof the results for problems in the study of nonlinear reaction-diffusionproblems and their equilibria, corresponding to solutions of nonlinearelliptic problems. Combining analytical techniques from dynamical sys-tems theory (Guckenheimer and Holmes [1983]), perturbation methods(Kevorkian and Cole [1996]), and from the study of motion of diffusiveinterfaces (Fife [1988]), we present a detailed construction of radiallysymmetric curved fronts, see Figure 1. Asymptotic and explicit formu-lae are given for the positions of the fronts and their wavespeeds. Wealso discuss how these simple solutions can serve as the fundamentalbuilding blocks for problems in more complicated geometries that are

AXISYMMETRIC TRAVELING WAVES 341

u(x, y)

U(r)

r

0

1

0

FIGURE 1. A two-dimensional localized axisymmetric solution of the semi-linear elliptic equation (1.2) and its radial profile, u = U(r).

often studied using more advanced techniques.

We will make use of the simplest model that qualitatively describesthe behavior of localized transition layers. The equation we will analyzeis the nonlinear second order differential equation:

(1.1)d2U

dz2+ c

dU

dz+ F (U ;ω) = 0,

where c is a damping parameter describing the level of dissipation in thesystem and ω is a control parameter modifying the form of the nonlinearfunction F (U ;ω). F (U ;ω) is a cubic-like, or bi-stable, function of U ,see Figure 2a. Bistability implies that there is a well-defined range ofvalues, ωmin < ω < ωmax, such that F (U ;ω) has three real roots, seeFigure 2a, and for ω outside of this range, F has only one real root. Forevery ω ∈ (ωmin, ωmax), equation (1.1) has three equilibrium solutions,and we label these UL(ω) < UM (ω) < UR(ω), corresponding to thethree roots of F (U ;ω).


Motivated by problems from the disciplines cited above and by earlieranalytical studies, we examine problems in which the bistable functionF (U ;ω) is the sum of a term with linear dependence on a controlparameter ω and a fixed nonmonotone function f(U),

(1.2) F (U ;ω) = f(U)− ωU,

where f(U) = o(U) as U → 0. For example, in the study of firstorder phase transitions, f(U) is called the free energy density and ωis a measure of the temperature in the system (Chan [1977]). Withthis choice for the form of F (U ;ω), the trivial state UL = 0 is ahomogeneous solution for all values of ω.

As an ordinary differential equation, the system (1.1) may be inter-preted as a damped anharmonic oscillator (Goldstein [1980]), with thedamping parameter c describing the level of dissipation of the motionand the force F being derived from a nonconvex inverted double-wellpotential energy function V(U), see Figure 2b,

(1.3) F (U ;ω) =∂V∂U

.

To uniquely specify the potential, we define its value at the pointU = UL as V(UL) = 0. In terms of the inverted double well potentialV(U), UL and UR are maxima while UM is a local minimum. The valueof ω determines the relative heights of the maxima. We will show thatfor an appropriate level of damping c, dissipation can exactly balancethe difference in potential heights to yield a unique bounded solutionconnecting UL and UR. This solution is called a heteroclinic orbit,and it is the solution of (1.1) that describes the structure of diffusiveinterfaces. We now review some applications that depend upon thismodel.

Equation (1.1) arises directly from the description of planar travel-ing wave solutions of reaction-diffusion equations. For example, con-sider the FitzHugh-Nagumo equations for the propagation of electricalimpulses in nerve cells from mathematical biology (Grindrod [1996],Murray [1990])

(1.4)∂u

∂t=

∂2u

∂x2+ f(u)− ωu,

∂ω

∂t= εu.


ωminU

ωmaxU

UR

UM

UL

ωU

f(U)

U

0

0

(a)

−V(U)

UM

UR

UL

U

0

0

(b)

FIGURE 2. (a) Construction of the zeroes of F (U ;ω) = f(U)− ωU . (b) Theinverted double well potential V(U).


In the limit that ε→ 0, this problem is a singular perturbation problemfor a relaxation oscillator. For ε = 0, the two equations decouple, andone can seek traveling wave solutions in the form u(x, t) = U(x − ct),where U(z) is given by the solution of (1.1), see McKean [1970].More generally, in the study, in several dimensions, of scalar reaction-diffusion equations, also called Alan-Cahn models (Taylor and Cahn[1994]),

(1.5)∂u

∂t= ∇2u + F (u;ω),

one also finds one-dimensional planar fronts of the form u(x, t) = U(k ·x−ct). This model appears in many classical studies of the propagationof phase transitions in time-dependent Ginzburg-Landau problems(Dixon et al. [1991]). For different forms of the nonlinear functionF (u;ω), (1.1) has been used in population genetics to study the spreadof a favored trait, originally given by Fisher’s equation (Murray [1990]).In all of these problems, solutions of the partial differential equationexist whose spatial structure is given by the solution of (1.1) and thespeed of propagation c is determined by a balance of nonlinear anddissipative effects in the ordinary differential equation. Analyticaldetermination of the speed of propagation in closed form has beenstudied extensively for classes of polynomial nonlinearities F (U ;ω)(Benguria and Depassier [1994], Dixon et al. [1991], Powell and Tabor[1992]). However, some open questions still remain regarding thecalculation of c for more general problems (Goriely [1995], Cisternasand Depassier [1996]).

Nontrivial equilibrium solutions of (1.5) have also been the focus ofmuch study (Ni [1988]). The bistable reaction-diffusion equation (1.5)is also the basis for nucleation theory, describing the formation andgrowth of spatial inhomogeneities in chemical reactions. In this setting,small initial perturbations in a uniform reactive medium can grow tobecome nucleation centers, the sources of larger scale pattern formation.Finite-size equilibrium nuclei satisfy the steady-state equation

(1.6) ∇2u + F (u;ω) = 0,

sometimes called the scalar field equation (Ni [1988]).

Another application where semilinear elliptic equations of the form(1.6) occur is in the study of incompressible fluid dynamics. The two-dimensional vorticity-streamfunction form of the Euler equations for


incompressible inviscid flow can be written as (Lamb [1993], Majdaand Bertozzi [1999])

(1.7)∂Ω∂t

+ k · (∇Ψ×∇(∇2Ψ)) = 0,

where Ψ = Ψ(x, y) is a streamfunction. If the Laplacian of thestreamfunction is only a function of Ψ, ∇2Ψ = −F (Ψ), then the crossproduct vanishes, ∇Ψ × F ′(Ψ)∇Ψ = 0, and Ψ is a steady solution ofthe Euler equation (1.7) for any function F (Ψ). Localized axisymmetricsolutions of this problem represent vortex flows with zero total vorticityand are commonly called eddies.

Similarly, in chemical physics, nonlinear elliptic equations occur asmean field models to describe density functions. In Witelski et al.[1998], a semilinear elliptic equation (1.2) was derived for the densityfunction of the spherically symmetric globule state of a collapse polymerchain. In this context, the problem is stated as an eigenvalue problemfor a nonlinear elliptic problem:

(1.8) H[ψ] = ωψ, ψ(x→∞)→ 0, H[ψ] ≡ ∇2ψ + f(ψ),

where ω gives a measure of the potential energy of the solution.

It is well known that positive solutions of (1.6) must be radiallysymmetric (Gidas et al. [1979]). Therefore, we write u(x) = U(r) andwrite the reduced form of the radial Laplacian operator in n dimensions,

(1.9) ∇2 ≡ 1rn−1

d

dr

(rn−1 d

dr

)=

d2

dr2+n− 1r

d

dr;

consequently, radially symmetric solutions of (1.6) satisfy the ordinarydifferential equation

(1.10)d2U

dr2+n− 1r

dU

dr+ F (U ;ω) = 0.

We note that equation (1.10) has the same qualitative structure as(1.1); it is a second order differential equation with linear damping andnonlinearity F (U ;ω). However, there are some important technicaldifferences between the two equations. While (1.1) is an autonomous


equation defined on −∞ < z <∞, (1.10) is a nonautonomous equationdefined on 0 ≤ r <∞.

We will go on to demonstrate how the solution of the nonautonomousequation (1.10) can be related to the solution of (1.1) using perturbationmethods. In Section 2 we discuss the basic properties of equation(1.1) in the phase plane. In Section 3 we will outline the propertiesof the nonautonomous radial equation (1.10). In Section 4 we use anautonomous approximation to reduce the radial problem to the form(1.1). In Section 5 we justify the use of (1.1) for the solution of (1.10)through a perturbation expansion that reduces the radial problem toa series of phase plane problems with slowly varying coefficients. InSection 6 we study a threshold effect by studying the instability of theseradial solutions for time dependent reaction-diffusion equations. InSection 7 we review transversality arguments from dynamical systemstheory for the analysis of the heteroclinic interface solution. Finally, inSection 8 we summarize the scope of our examination of this topic, andwe describe briefly how the results have been extended, as well as howthey can be used to construct more general curved front solutions.

2. The autonomous problem. In this section we review theclassical phase plane analysis of the autonomous differential equation:

(2.1)d2U

dz2+ c

dU

dz+ F (U ;ω) = 0,

and we review some of its exact solutions. As described above, thisequation has been studied extensively in the literature in connectionwith the structure of planar front traveling wave solutions.

2.1. Phase plane analysis. Equation (2.1) can be written as asystem of first-order differential equations in the form

(2.2)dU

dz= V,

dV

dz= −F (U ;ω)− cV.

To determine the local behavior near the equilibria, (UE , 0)|UE =UL, UM , UR, we compute the Jacobian of vector field (2.2),

(2.3) J =(

0 1ω − f ′(U) −c

)


as part of a local stability analysis (Guckenheimer and Holmes [1983]).The eigenvalues of the Jacobian matrix at the equilibria (UE , 0) are:

(2.4) λE± =

−c±√c2 + 4ω − 4f ′(UE)

2, UE = UR, UM , UL.

Hence for our bistable function F (U ;ω), (UL, 0) and (UR, 0) are saddlepoints, while (UM , 0) is a spiral sink for c > 0 (and is a center for c = 0).

The heteroclinic orbit is the only monotone solution U = U(z) thatis bounded for all −∞ < z < ∞. The heteroclinic orbit connectingthe two maxima in (2.3) is not a structurally stable property of thesystem. Existence of the heteroclinic solution requires a special balancebetween the nonlinear and dissipative terms, see Fife and McLeod[1981] for a uniqueness result. This result implies the existence of aunique functional relationship between the parameters necessary for aheteroclinic connection, c = c(ω). For every value ω ∈ (ωmin, ωmax)we can define the local stable and unstable manifolds (Guckenheimerand Holmes [1983]) of the saddle points (UL, 0) and (UR, 0) that areinvolved in the construction of the heteroclinic orbit. The local stablemanifold of (UL, 0) is the set of initial conditions in a neighborhoodof (UL, 0) that stay in the neighborhood for all z > z0, for some z0,and that approach the equilibrium, U(z → ∞) → UL, as z → ∞.Linearized analysis shows that solutions on the local stable manifoldapproach (UL, 0) tangent to the eigenvector corresponding to the stableeigenvalue λL

− < 0. Similarly, the local unstable manifold of (UR, 0) isthe set of all initial conditions in a neighborhood of that point thatstay in the neighborhood for all z < z0, for some z0, and that approach(UR, 0) exponentially in backward time, i.e., U(z → −∞) → UR asz → −∞. Solutions on this manifold approach the equilibrium (UR, 0)tangent to the eigenvector corresponding to the unstable λR

+ > 0eigenvalue. Figure 3 shows the three possible relations between thesetwo manifolds for different values of the parameters c, ω; (a) the stablemanifold of (UL, 0) lies above the unstable manifold of (UR, 0), (b) thetwo manifolds coincide, and the orbits lying in them are heteroclinicconnections from (UR, 0) to (UL, 0), or (c) the stable manifold of(UL, 0) lies below the unstable manifold of (UR, 0). In case (c), theunstable manifold of (UR, 0) approaches the spiral sink (UM , 0) asz →∞, see Figure 3c.


(a)

U

URUMUL

0

(b)

(c)

FIGURE 3. Phase plane for the nonlinear damped equation (1.1) with c = c(ω)chosen for the existence of a heteroclinic connection (b middle). Breaking ofthe saddle-saddle connection for perturbed parameters c = c(ω): c < c(ω) (atop), c > c(ω) (c bottom).


0

(a)

U

URUMUL

0

(b)

0

(c)

FIGURE 4. Phase plane for the nonlinear undamped equation (1.1) withparameters ω = ω0 chosen for the existence of a heteroclinic connection (bmiddle). Breaking of the saddle-saddle connection for perturbed parametersω = ω0: ω < ω0 (a top), ω > ω0 (c bottom).


For the special case of c = 0 in which there is no damping in thesystem, (2.2) is Hamiltonian with the conserved quantity,

(2.5) H = 12V

2 + V(U).

In the phase plane (UL, 0) and (UR, 0) remain hyperbolic saddle points,while (UM , 0) becomes a center, see Figure 4. A heteroclinic connectionwill exist only if the two maxima have the same value of the potentialV(UL) = V(UR). If this is the case, then from the symmetry of thephase plane system (2.2) for V → −V , both monotone increasing anddecreasing connections exist, see Figure 4b. This will occur at a valueω = ω0 corresponding to a root of c(ω) = 0. For other values of ω atc = 0, either (UL, 0), see Figure 4a, or (UR, 0), see Figure 4c, will havea homoclinic orbit, but no heteroclinic connection will exist.

In Section 7 we will review a constructive proof for the existence anduniqueness of the heteroclinic connection in terms of the transverseintersection of the stable and unstable manifolds in extended parameterspace. The proof of existence will include persistence under smallperturbations of the vector field, including the introduction of weaklynonautonomous, or slowly varying, coefficients resulting for the radialproblem that we consider in Section 3.

2.2. Exact solutions. For completeness, we review some of theknown exact traveling wave heteroclinic solutions of equation (1.1).Numerous studies have discovered classes of closed-form analytical so-lutions for polynomial nonlinearities F (U ;ω) (Dixon et al. [1991], Pow-ell and Tabor [1992], Goriely [1995], Benguria and Depassier [1994]).Powell and Tabor [1992] describe a set of exact solutions for F (U ;ω)of the form (1.2) when f(U) has two terms with a special relation,

(2.6) F (U ;ω) = −U(ω − βUk + (k + 1)U2k),

where k = 1, 2, 3, . . . is a positive integer and β, ω > 0 are parameters.In (2.6) we have factored F (U ;ω) to show that it is the product of Utimes a quadratic polynomial in Uk, hence the zeroes of F (U ;ω) canbe obtained analytical as

(2.7)UL = 0, UM =

(β −√

β2 − 4(k + 1)ω2(k + 1)

)1/k

,

UR =(β +

√β2 − 4(k + 1)ω2(k + 1)

)1/k

.


From these solutions it is clear that the range of ω where F has threereal roots is

(2.8) 0 < ω <β2

4(k + 1).

For each value of ω in this range, ωmin < ω < ωmax, the integral curvefor the heteroclinic solution of (2.3) is given by

(2.9) V (U) ≡ dU

dz= U [Uk − (UR)k];

this equation can be integrated to yield the explicit heteroclinic solu-tion,

(2.10) U(z) =UR

[1 + (UR)k exp k(UR)k(z − z0)]1/k,

where z0 is a constant of integration corresponding to translationinvariance of the solution. Substituting ansatz (2.9) into (2.2) andbalancing O(U) terms yields an explicit formula for the dampingrelation, c = c(ω),

(2.11) c(ω) = UR(ω)k − ω

UR(ω)k,

where we have explicitly indicated the dependence of the root UR on ω,(2.7). The question of obtaining the wavespeed relation c = c(ω) for thegeneral form of the damped equation (2.2) has attracted some recentattention (Cisternas and Depassier [1996], Goriely [1995], Witelski, etal. [1998]). We will make use of this exact solution in a later section topresent a closed-form solution of an example problem.

3. Stationary axisymmetric front solutions of semilinear el-liptic equations. We now focus on obtaining radially symmetric,localized, positive, monotone decreasing front solutions of the semilin-ear elliptic equation (1.6). As was described in the introduction, thesesolutions are of great interest for many problems in applied physics,fluid dynamics, and mathematical biology. There is a large body ofliterature connected with the existence and uniqueness of nonnegative


solutions for this problem (Aronson and Weinberger [1978], Atkinsonand Peletier [1986], Berestycki and Lions [1980], Brezis and Lieb [1984],Gidas et al. [1979], Jones [1983a] and [1983b], Kaper and Kwong [1988],Kuzin and Pohozaev [1997], McLeod and Serrin [1968] and [1987], Ni[1988], Terman [1987] and [1988]).

For the positive radially symmetric solutions that we seek, the ellipticequation (1.6) reduces to the nonlinear boundary value problem:

d2U

dr2+n− 1r

dU

dr+ F (U ;ω) = 0, 0 ≤ r <∞,(3.1a)

dU

dr

∣∣∣∣r=0

= 0, U(r →∞)→ 0,(3.1b)

where F (U ;ω) is given by (1.2). This is a singular two-point boundaryvalue problem for a nonautonomous nonlinear second order ordinarydifferential equation.

3.1. Asymptotics of solutions of the radial boundary valueproblem. In the far-field, i.e., as r → ∞, equation (3.1a) can belinearized about U = 0:

(3.2)d2U

dr2+n− 1r

dU

dr− ωU = 0.

This equation is a modified Bessel equation of order (n/2)−1 for ω > 0.The solution is given in terms of a modified Bessel function of the secondkind, U(r) = Cr1−n/2K(n/2)−1(

√ωr) (Abramowitz and Stegun [1965]).

Using the asymptotics of the Bessel functions, this reduces to

(3.3) U(r →∞) ∼ C

√π

2√ωr−(n−1)/2e−

√ωr.

The value of the solution at the origin, U(0) ≡ U0 > 0, is unknown andcan be used to obtain the solution of boundary value problem (3.1) via aone-parameter shooting method (Kuzin and Pohozaev [1997], McLeodand Serrin [1987]) with U0 as the parameter. This method exploits theproperty of continuous dependence of solutions on initial conditions ina clear way. We now show that U0 must lie in the interval (UM , UR]for monotonically decreasing radial solutions to exist. The analysis


proceeds in two steps; first, we analyze the system linearized aboutU = U0, then, a comparison argument is made to establish the desiredresult.

Near the origin, solutions of (3.1) are given by a power series expan-sion in r2 as r → 0,

(3.4) U(r → 0) ∼ U0 − F (U0;ω)2n

r2

(1− F ′(U0;ω)

4(n + 2)r2 + O(r4)

).

Linearizing problem (3.1) about U = U0 yields another Bessel equation,similar to (3.2), and hence the local solution can also be expressed inthe form

(3.5) U(r → 0) ∼ U0 − F (U0;ω)F ′(U0;ω)

·(

1− Γ(n2 )( 1

2

√|F ′(U0;ω)| r)1−(n/2)B(n/2)−1(

√|F ′(U0;ω)| r)

),

where Bν(z) represents the regular or modified Bessel functions of thefirst kind according to

Bν(z) =Jν(z) F ′(U0;ω) > 0,Iν(z) F ′(U0;ω) < 0.

Recall that, for all real ν, Jν(z) is a decaying oscillatory function, whileIν(z) is monotone increasing. When the value of U0 passes through theroot of F ′(U0;ω) = 0, the linearized behavior of the solution changesfrom oscillatory (Chen and Lu [1997]), suggesting the influence of thespiral sink UM , to monotone growth, suggesting the influence of thesaddle point UR. When U0 exactly satisfies F ′(U0;ω) = 0, the localsolution degenerates to the quadratic polynomial given by (3.4).

We now use the results obtained from the linearized analysis to carryout a comparison argument for the desired solution of the nonlinearproblem. Solutions with U0 > UR are monotone increasing and can beuniformly bounded from below by the modified Bessel function solution(3.5), see Figure 5. Therefore, we conclude that, if a monotone decreas-ing radial solution exists, it must satisfy the condition UM < U0 ≤ UR,see Figure 5ab. Note from (3.4) that if U0 < UM then the solu-tion is not monotone decreasing in a neighborhood of r = 0. In the


U0

UR

UM

UL= 0

(a)

U0

U

UR

UM

UL= 0

0

(b)

UR

UM

UL= 0

(c)

FIGURE 5. Plots of the trajectories obtained by numerically integrating the non-autonomous equation (3.1a) in the (U, V ) plane. Solutions are shown for a series ofuniformly spaced initial conditions chosen along the U-axis over a range of valuesof U(0) = U0. the plots obtained using three distinct values of ω are shown. Ifa solution of the boundary value problem (3.1) exists, it is shown as a darkenedcurved. In (a top) the solution is given by the stable manifold of UL with someU0 ∈ (UM , UR). In (b middle) the solution is approximately a heteroclinicconnection with U0 ∼ UR. In (c bottom) there is no positive localized radialsolution.


limiting case that U0 → UR, the solution approaches the heteroclinicconnection between the saddle points at UL = 0 and UR, see Figure 5b.When U0 UR, we are no longer seeking a heteroclinic connection,but just the portion of the stable manifold of the point UL = 0 for0 ≤ r < ∞. Finally, for values of ω when the unstable manifold ofUR approaches the spiral sink at UM for r → ∞, then no positivemonotone decreasing solution exists, see Figure 5c. Note the closeparallels between the structure of Figure 5 and the phase plane for theautonomous problem shown in Figure 3.

3.2. Integral formulation of the radial problem. To gain moreinsight into the nature of the solution of (3.1), we rewrite (3.1a) usingan integral formulation. Integrating equation (3.1a) over the entiredomain with a weight function of r2(n−1)U ′(r),

(3.7)∫ ∞

0

[1

rn−1

d

dr

(rn−1 dU

dr

)+ F (U ;ω)

]dU

drr2(n−1) dr = 0,

yields

(3.8) −12

(rn−1 dU

dr

)2∣∣∣∞0

=∫ ∞

0

r2(n−1)F (U ;ω)dU

drdr.

From the boundary condition (3.1b) at r = 0 and the far-field asymp-totics for r →∞ (3.3), we conclude that the boundary terms on the left,and hence also the integral on the right side of this equation, vanish.Since we seek monotone decreasing solutions U = U(r), these solutionshave well-defined inverse functions r = r(U) ≥ 0 for 0 ≤ U ≤ U0.Hence, we can write the integral on the right as

(3.9)∫ U0

0

F (U ;ω)r(U)2(n−1) dU = I− + I+ = 0,

where

(3.10)I− =

∫ UM

0

F (U ;ω)r(U)2(n−1) dU < 0,

I+ =∫ U0

UM

F (U ;ω)r(U)2(n−1) dU > 0.


F (U ;!)

F(U ;!)

U0

U

UR

UM0

0

FIGURE 6. Comparison of F (U ;ω) and the equal-area rule for the weighted

nonlinear function F(U ;ω) = F (U ;ω)r(U)2(n−1).

Note that r(U) is positive for all U , 0 ≤ U < U0, while F (U ;ω)is negative for 0 ≤ U ≤ UM and positive for UM < U ≤ U0.Consequently, equation (3.9) expresses a weighted balance betweenpositive and negative contributions of the nonlinearity F (U ;ω). Recallthat F (U ;ω) has zeros at UL = 0, UM , and UR. From the far fieldexponential behavior of U(r → ∞) → 0, (3.3), we see that r(U) mustdiverge logarithmically r(U → 0)→∞. From the boundary conditionat the origin, we see that r′(U → U0)→ −∞ is an integrable singularityas r → 0. Consequently, the integrand F(U ;ω) = F (U ;ω)r(U)2(n−1)

is a generalization of the nonlinear function F (U ;ω), with the zeros at0, UM , and U0. Equation (3.9) can be interpreted as an equal-area rulefor F(U ;ω) (Rubinstein and Sternberg [1992], Witelski [1996], Witelskiet al. [1998]), see Figure 6. For n = 1, (3.9) trivially reduces to an equal-area rule for F (U ;ω). We will now examine under what conditions thisreduction occurs for n > 1.

3.3. Generalized equal area rule for sharp interfaces solutions.To understand the implications of this integral relation for the structureof monotone front solutions, we consider a special limiting case. Assumethat the front connects UL to U0, with intermediate values, U ∈(0, U0), being covered in a transition layer whose width is narrowcompared to its radial distance from the origin. This is often calleda sharp interface approximation (Caginalp [1991], Taylor and Cahn


r

U

1510R50

U0

UM

0

FIGURE 7. The axisymmetric sharp-interface approximate solution (4.7),(4.9), and the exact numerical solution U(r) of (3.1) showing the positionof the transition front, r = R.

[1994]). Let r = R denote the position of this transition layer, and letr(U) be the relative displacement from R, see Figure 7,

(3.11) r(U) = R + r(U).

A natural definition for the position of the front is the value of r whereU = UM . This choice implies F = 0 at r = 0:

(3.12) UM = U(R)←→ R = r(UM ).

This value also marks the boundary between the positive and negativecontributions to the integral relation for F(U ;ω), (3.10).

For narrow transition layers |r(U)| R for all values of U except insmall neighborhoods of U = UL and U = U0. Examining the integralrelation (3.9), one sees that F(U ;ω) vanishes in those limits, and toleading order in O(r) one obtains

(3.13)∫ U0

0

F(U ;ω) dU ∼ R2(n−1)

∫ U0

0

F (U ;ω) dU = 0.

Hence the equal area rule for F(U ;ω) reduces to one for F (U ;ω), andin this limit, the root U0 of F(U ;ω) must approach the root UR ofF (U ;ω) for a particular value of ω0 where F (U ;ω) has a heteroclinic


solution. We will see that the width of the front is inversely related tothe magnitude of the eigenvalues of the two saddle points, and if weassume that those eigenvalues remain bounded, then the front will havean O(1) width. Therefore, in the limiting case of a “narrow transitionlayer,” |r| R implies that the radial position of the front is going toinfinity, i.e., R→∞. As will be discussed further, this behavior couldhave been expected, since for large radii, the local curvature of the frontis small, and the results known for one-dimensional planar fronts willbe recovered (Tyson and Keener [1988]). In the following sections, westudy how weak curvature effects have been traditionally incorporatedin perturbation expansions and how leading order curvature effects canbe used to obtain approximate front solutions of the radial differentialequation.

4.1. The autonomous approximation of the radial solution.Motivated by the results of the previous section, we now present aheuristic construction of an approximate radial solution with a sharpinterface. These localized solutions have a finite mass with exponentialdecay to U(r → ∞) → 0. In Sections 5 and 6, we will comparethe accuracy of this approximate solution with a formal asymptoticexpansion obtained from a regular perturbation series.

Writing the independent variable as the sum, r = R+ r, of the positionof the transition front, R, and the relative displacement from the front,r, we put (3.1a) in the form

(4.1)d2U

dr2+

n− 1R + r

dU

dr+ F (U ;ω) = 0.

If the front is a sharp transition layer occurring at a large distancefrom the origin, large relative to the width of the front, R r, thenthe coefficient of the damping term can be expanded as

(4.2)n− 1R + r

= c− (n− 1)rR2

+(n− 1)r2

R3+ O(R−4), c =

n− 1R

,

for R → ∞. Retaining the leading order term from this expansionreduces (4.1) to

(4.3)d2U

dr2+ c

dU

dr+ F (U ;ω) = 0.


This is just the autonomous damped equation of the form (1.1) thatwas solved in Section 2 to yield bounded heteroclinic connections forall ω ∈ (ωmin, ωmax) and c = c(ω).

Solutions of (4.3) are translation invariant under r → r + r0, leavingthe position of the front unspecified. However, using (3.12) as thedefinition of the front position, R, we can select a unique solution fromthe continuous one-parameter translation invariant family. We nowpresent an example of this approximation that yields a closed-formradial solution.

4.1. An example. Consider the problem of finding the uniquelocalized solution of the radial differential equation

(4.4)d2U

dr2+n− 1r

dU

dr− ωU + 3U2 − 2U3 = 0.

This problem is analogous to the nonlinear eigenvalue problem forspherical polymer globules solved in (Witelski et al. [1998]) with n = 3and ω > 0. Using the large-radius approximation described above, wewrite the corresponding autonomous equation (4.3) as

(4.5)d2U

dr2+ c

dU

dr− ωU + 3U2 − 2U3 = 0.

We recognize (4.5) as an exactly solvable problem of the form (2.6)with k = 1, β = 3, and use the results obtained in Section 2.2. Theequilibrium solutions of (4.4) and (4.5) are

(4.6) UL = 0, UM =3−√9− 8ω

4, UR =

3 +√

9− 8ω4

,

for the range 0 ≤ ω < 9/8. The explicit heteroclinic solution of (4.5) is

(4.7) U(r) =URUM

UM + (UR − UM ) exp [URr],

where the phase condition U(r = 0) = UM , (3.12), has been satisfied.The relation for c(ω) may now be obtained by plugging UR into (2.11):

(4.8) c(ω) =6(1− ω)

1 +√

9− 8ω.


Finally, substituting (4.8) into (4.2) yields an approximation for thecore radius, R:

(4.9) R = (n− 1)1 +√

9− 8ω6(1− ω)

.

Combining (4.7) and (4.9) gives us the approximate solution of (4.4) asU(r) ≈ U(r − R). A search of the scientific literature reveals that thisapproximation has been used in an ad hoc manner earlier in chemicalphysics (Chan [1977]), among other places. In later subsections, weanalyze the limits of validity of this solution in comparison with anasymptotic expansion of the solution of the full problem.

4.2. Properties of the localized solution. Before carrying out thiscomparison, it is useful to describe the properties of the solution thatare of primary interest in applications. We tie this description to theproblem of finding the structure of the collapsed polymer globule statefrom Witelski et al. [1998], though it is equally relevant to all of theother problems described in the introduction. Our primary focus hasbeen the structure of the sharp interface, or transition layer, formingthe boundary between the high density inner core of the solution, whereU ≈ U0 ≈ UR, and the surrounding empty space, U = 0, see Figures 1,7. There are well-developed experimental and theoretical works givinga variety of results for problems with these phase transition interfaces,see for example Bates and Fife [1993], Lifschitz and Freed [1993], Pego[1989], Puri and Binder [1991], Witelski [1998]). First, we observe that,by (4.7), the dimensionality of the solution, n = 1 for planar interfaces,n = 2 cylindrical, or n = 3 spherical solutions, does not affect the localdensity profile structure of the interface. Instead, the dimensionalityenters through the core radius R (4.9). Second, equation (4.5) supportsheteroclinic solutions for a range of ω, corresponding to a range ofsystem temperatures or levels of kinetic energy, namely 0 ≤ ω < 9/8.However, for equation (4.9) to describe a physically meaningful finitepositive core radius, only a reduced range of ω, namely 0 ≤ ω < 1, canbe used. In the limit ω → 1, we approach the infinite radius solutionR → ∞. Since the curvature is inversely proportional to R, this limitreturns the planar front solution. Note that excellent agreement withthe exact solution is achieved even for moderate radii, as shown inFigure 7 for n = 2, ω = 0.95 and R ≈ 7.22. Finally, for the range


1 < ω < 9/8, no localized positive radial solutions of (4.4) are expectedto exist.

For the many problems in which the transition is narrow compared tothe radius of the inner core, the sharp interface approximation is useful,and one can directly compute the mass, or integral, of the radiallysymmetric solution:

(4.10) M =2πn/2

Γ(n/2)

∫ ∞

0

U(r)rn−1 dr.

To leading order, in the sharp interface approximation, we can replacethe smoothed exponential solution (4.7) by a limiting step function:

(4.11) U(r) ≈ U(−R)H(R− r),

where H(z) is the Heaviside step function, and U0 ≈ U(−R) is used asan approximation of the uniform core density. This yields the followingclosed-form estimate of the mass:

(4.12) M =2πn/2Rn

nΓ(n/2)U(−R).

We now turn to compare this solution with asymptotic expansions forthe solution of (4.1).

5. Asymptotic analysis of the radial problem. In this sectionwe present a regular perturbation series solution to the radial problem(3.1). The solution is expanded about a stationary planar interfacesolution, and it incorporates curvature of the front in higher ordercorrections.

Using the same change of variables employed in Section 4, namelyr = R + r, we start with (4.1):

(5.1)d2U

dr2+

n− 1R + r

dU

dr+ F (U ;ω) = 0.

The primary weaknesses of the approximate solution given in Section 4are the lack of information about the limits of its validity and theabsence of a clear method for estimating the error and improving the


approximation. These problems can be corrected with a more carefuljustification of the expansion of the damping coefficient (4.2). Formula(4.2) is an asymptotic expansion in the limit R→∞. From the integralformulation of problem (3.1) presented in Sections 3.2 and 3.3, werecognize that this limit occurs only for a special value of ω, ω = ω0,where F (U ;ω0) satisfies the equal area rule,

(5.2)∫ UR(ω0)

UL(ω0)

F (U ;ω0) dU = 0.

Hence we will write ω = ω0 − ε, where 0 ≤ ε 1 is a detuningparameter (Kevorkian and Cole [1996]) that will serve as the basis ofour perturbation expansion. The problem we solve can then be writtenas

(5.3a)d2U

dr2+

n− 1R + r

dU

dr+ F (U ;ω0 − ε) = 0, −∞ < r <∞,

with boundary conditions (3.1b) for |r| → ∞ and the phase condition(3.12):

(5.3b) U ′(r → −∞)→ 0, U(r →∞)→ 0, U(r = 0) = UM .

5.1. Regular perturbation expansions. Our solution will be givenas a regular perturbation expansion for U(r),

(5.4a) U(r) = U0(r) + εU1(r) + ε2U2(r) + · · · ,along with a strained parameter expansion for the position of the front:

(5.4b) R =1εR−1 + R0 + εR1 + ε2R2 + · · · .

Moreover, since UM = UM (ω), (2.7), we expand the phase conditionas:

(5.5) UM = UM0 + εUM

1 + ε2UM2 + · · · .

To leading order, we obtain a nonlinear autonomous equation for U0(r):

(5.6) N [U0] ≡ d2U0

dr2+ F (U0;ω0) = 0.


This equation is an undamped Hamiltonian equation that possesses aunique monotone decreasing heteroclinic solution satisfying the condi-tion U0(0) = UM

0 . Its solution describes a stationary one-dimensionalplanar front, and it does not include any curvature effects or determineany finite radius contributions to the speed of the front. A representa-tion for its integral curve may be found implicitly in terms of V = V0(U)as follows. Since V0(U) = dU0/dr, a straight-forward calculation givesU ′′

0 (r) = dV0/dr = V0(U)V ′0(U) (Bender and Orszag [1978]). Hence,

equation (5.5) is transformed into the first order differential equation

(5.6) V0dV0

dU+ F (U ;ω0) = 0.

Direct integration then yields the integral curve V0(U) for the hetero-clinic orbit,

(5.7) V0(U) = −√ω0U2 − 2

∫ U

0

f(u) du.

At higher orders, the expansions yield inhomogeneous linear problemsfor Uk(r), k = 1, 2, 3, . . . ,

(5.8a) LU1 = R1(U0, R−1) ≡ −U0 − n− 1R−1

dU0

dr,

(5.8b)LU2 = R2(r, U0, U1, R−1, R0)

≡ −U1 − 12f

′′(U0)U21 −

n− 1R−1

dU1

dr+

(n− 1)(R0 + r)(R−1)2

dU0

dr,

(5.8c) LUk = Rk(r, . . . , Uk−1, . . . Rk−2), k = 2, 3, . . . .

Here, the linear operator is the functional derivative of N , which isgiven by (5.5),

(5.9) LU ≡(δNδU

∣∣∣U0

)U =

d2U

dr2+ [f ′(U0)− ω0]U.


To solve for the functions U1, U2, . . . in (5.4a), we need to considerthe properties of the linear operator L given by (5.9). If any of theterms in (5.4a) are not bounded for all r, then the expansion may notbe uniformly well-ordered in an asymptotic sense. To suppress termsthat would yield unbounded growth in Uk(r) as |r| → ∞, we apply theFredholm alternative (Friedman [1990]) to obtain a solvability conditionat each order in the expansion:

(5.10)∫ ∞

−∞Rk(r, . . . , Rk−2)W (r) dr = 0,

where W (r) is an element of the null space of the adjoint operator.This function W (r) is a solution of the problem

(5.11) L†W = 0, W ′(r → −∞)→ 0, W (r →∞)→ 0.

Since the operator is self-adjoint (Friedman [1990]), L† = L, theadjoint solution is W (r) = U ′

0(r). Consequently, at each order, i.e.,for each k = 1, 2, . . . , equation (5.10) yields a solvability condition interms of the lower order solutions U0, . . . , Uk−1 and R−1, . . . , Rk−3

that uniquely determines the value of the next correction to the frontposition Rk−2 so that secular growth is eliminated from the solutionUk(r).

To calculate the leading order term, R−1, in the expansion of thefront position, (5.4b), it suffices to evaluate the function R1 for thefunction U ′

0(r) = V0(U) given by (5.7), and then to substitute theresult into the solvability condition (5.10) with k = 1. Integrating thefirst term, observing that only the lower limit of integration yields anonzero contribution, and transforming the variable of integration inthe second term from r to U , we arrive at:

(5.12) R−1 = − 2(n− 1)UR(ω0)2

∫ UR(ω0)

0

V0(U) dU > 0,

see also Witelski et al. [1998].

5.2. An example. To illustrate the solution procedure discussed inSection 5.1, we return to the example given by equation (4.4), withF (U ;ω) = −2U3 + 3U2 − ωU and n = 2. The equal-area condition


(5.2) is satisfied for ω0 = 1 on the interval (UL = 0) ≤ U ≤ (UR = 1).Therefore, we write ω = 1−ε. Using the perturbation expansions givenabove, we obtain the solution, U ∼ Uε(r):

(5.13) Uε(r) =1

1 + er+ ε

(5 + r)er − 1(1 + er)2

+ O(ε2),

and the corresponding front position is given by

(5.14) R =13ε

+23

+143ε + O(ε2).

The structure of the solution U(r) ∼ Uε(r−R) matches the numerically-obtained exact solutions very well for all moderate and large values ofR, as shown in Figure 7. However, for small R, i.e., for the case ofsolutions for which the front position is very close to the origin, theexpansion becomes much less accurate.

From Figure 8, we see that the expansion of the front position (5.14)approximates the exact front position, as calculated from a directnumerical solution of (4.4), quite well. Moreover, it is asymptoticallyaccurate as ε → 0, i.e., as ω → 1. In fact, the expansion carriedout to O(ε2) has a wider range of validity than the expression for theapproximate front position R, given by (4.9), in Section 4.1. In thenext subsection, we will make precise the relation of the approximatesolution (4.7), (4.9) to the asymptotic solution (5.13), (5.14) anddiscuss the advantages of the approximate solution over the asymptoticexpansion given above.

We note that the O(ε2) term in the expansion of Uε(r), U2(r), is givenby a cumbersome expression, involving the dilogarithm special function(Lewin [1981]),

(5.15) Li2(z) = −∫ z

0

ln(1− s)s

ds,

which arises from integrals of the inhomogeneous terms in equation(5.8b). Higher order terms in the expansion will contain more compli-cated polylogarithms, which are generalizations of (5.15).

5.3. Analysis of the autonomous approximate solution (4.7).In order to compare the approximate solution, (4.7) and (4.9), found in


R

R

~R

!

R

10.90.80.7

100

10

1

FIGURE 8. Comparison of the numerically obtained exact front position, orcore radius, R, for problem (4.4), with the results of the asymptotic expansion,

Rε from (5.14), and the approximate front position, R from (4.9) for a rangeof different values of the parameter ω. See Section 8 for a discussion ofω > ω0 = 1.

Section 4.1 with the asymptotic expansion determined above, we writeω as ω = 1− ε in equations (4.6), (4.7) and (4.9). Then, by expanding(4.7) in the limit ε→ 0, we obtain:(5.16)

U(r) =1

1 + er+ ε

(5 + r)er − 1(1 + er)2

+ ε2(r2 + 14r + 56)e2r − (r2 + 10r + 20)er − 4

2(1 + er)3+ O(ε3),

and the front position (4.9) is approximated by:

(5.17) R =13ε

+23− 4

3ε + O(ε2).

The first two terms in the expansion for U(r) in (5.16) exactly matchthose for Uε(r) found above in (5.13), and similarly for the first twoterms in the respective expansions for R and R. However, the secondorder correction terms differ. The O(ε2) term in (5.16) is a rationalpolynomial function of r and er, and it does not contain the dilogarithmterms (5.15) found in U2(r), as was described above. Also, R1 = −4/3in equation (5.17) does not match the correct asymptotic coefficient


R1 = 14/3 that was found above in (5.14). Therefore, in this examplewe have seen that the approximate solution (5.16) captures both thecorrect leading order solution and the first order correction terms inthe asymptotic expansion of the interface structure and the radial frontposition. In the next subsection, we demonstrate that this is true forthe problem for equation (5.1) with general bi-stable F (U ;ω) as well.

5.4. Analysis of the general autonomous approximate solu-tion. As a preliminary step, we make some general observations aboutthe nature of the large front-radius limit, R → ∞. Recall from Sec-tion 3 that the linearized behavior of the radial solution for r → ∞ isU ∼ Cr−(n−1)/2e−

√ωr, (3.3). In contrast, the asymptotic behavior of

the heteroclinic solution is U ∼ CeλL−r, where the exponential decay

rate is given by the eigenvalue (2.4), with UE = UL. Using f ′(UL) = 0from (1.2) and c ∼ (n − 1)/(R + r) from (4.2), we get the followingexpansion:

(5.18) λL− = −√ω − n− 1

2(R + r)+ O(R−2),

as R→∞. Note that the leading term of (5.18) gives the correct rateof exponential decay matching (3.3), sometimes called the controllingfactor (Bender and Orszag [1978]). Furthermore, if λL

− is interpretedas the derivative of the phase, λL

− = S′(r), in a WKB approximation(Bender and Orszag [1978]) of U(r) ∼ eS(r) as r →∞, then we recoverthe full leading order behavior,

(5.19) U(r →∞) ∼ C(R + r)−(n−1)/2e−√

ωr.

Hence, the approximate solution is consistent with the far-field asymp-totics of the exact solution.

We now establish the limitations on the order of accuracy of theapproximate solution constructed in Section 4 by comparing it to theasymptotic expansion of the exact solution, constructed in Section 5.1.By comparing the terms in the respective series expansions for ε → 0we will show that the first discrepancies occur at O(ε2) for U(r) andat O(ε) for R. Recall that the approximation made in reducing thenonautonomous equation (4.1) to the damped autonomous equation(4.3) was in retaining only the leading term in the geometric series


expansion for the non-autonomous damping coefficient as R → ∞,(4.2). The full series is given by (4.2). Consequently, the first correctionterm is

(5.20) − (n− 1)rR2

dU

dr∼ − (n− 1)r

(R−1)2dU0

drε2 = O(ε2).

This is the first explicitly nonautonomous term that occurs in theperturbation expansion of equation (5.3a), and it comes in at O(ε2)in equation (5.8b). This term is uniformly bounded by O(ε2) as ε→ 0for the heteroclinic solution, and it is exponentially small outside of aneighborhood of the front at r = 0, since dU0/dr decays exponentiallyfor large r. While the nonautonomous coefficient in this term growsalgebraically away from the front, the derivative of the leading ordersolution decays exponentially.

To see the effect of this term in the perturbation expansion, weneed to distinguish autonomous terms from explicitly r-dependentinhomogeneous terms in (5.8). Consider rewriting the asymptoticexpansions (5.4) in terms of the autonomous approximate solution andan expansion of the correction to that approximation:

(5.21) U(r) = U(r, ε) + U(r, ε), U(r, ε) =∞∑

k=0

εkUk(r),

and

(5.22) R = R(ε) + R, R =∞∑

k=−1

εkRk.

Using this ansatz in equation (5.3a) and neglecting nonautonomouscontributions to the damping coefficient, to leading order equationwe get (4.3). We could expand this equation as a series of regularperturbation problems for ε→ 0 for the autonomous equation:

(5.23) N [U0] = 0,

andLUk = Rk(. . . , Uk−1, . . . , Rk−2), k = 1, 2, 3, . . . ,


where Rk involves only autonomous terms, with no explicit depen-dence on r. However, we know that U(r, ε) is an exact solution ofthe autonomous equation (4.3) for all values of ε in the acceptablerange of ω. Hence, we do not need any correction to the approximatesolution until non-autonomous terms enter the expansion (5.8). Theautonomous problem (5.23) exactly matches (5.5) and (5.8a), there-fore the autonomous approximate solution correctly reproduces U0(r),U1(r) and R−1, and therefore U0 = U1 = R−1 = 0.

At O(ε2) the nonautonomous term (5.20) enters in equation (5.8b),

LU2 + LU2 = R2(U0, U1, R−1, R0) + R2(r, . . . ),

R2 =(n− 1)r(R−1)2

dU0

dr,

where LU2 = R2. Here, since R2 is present, we get a nonautonomouscorrection U2(r) and therefore U2(r) = U2(r) as observed above. Atthis order, the R0 coefficient in the expansion of the front position iscalculated from the solvability condition (5.10) for R2 = R2 + R2. Wenow determine the condition needed so that the nonautonomous termR2 does not yield a correction to the autonomous front coefficient R0.Since W (r) = U ′

0(r), the integral for the nonautonomous contributionbecomes

(5.25)∫R2W dr =

∫ ∞

−∞r

(dU0

dr

)2

dr.

If this term vanishes, then the autonomous solvability condition,∫ R2W dr = 0, is equivalent to the full solvability condition (5.10),and we find that R0 = R0 and R0 = 0. We observed this behavior inthe example above. However, the integral (5.25) does not genericallyvanish for all F (U ;ω). It will disappear trivially if U ′

0(r) is an evenfunction. From the implicit form of the solution U0(r),

(5.26) r(U) =∫ UM

0

U

du√−2V(u),

it can be shown that V(U) must be even with respect to U = UM0 for

the autonomous solvability condition to hold, where V(U) is negative


over the entire interval of integration, UL0 < U < UR

0 , see Figure 2b.For the exact solutions given in Section 2.1, we observe that F (U ;ω0),(2.7), is odd with respect to UM for ω0 = 2β2/(9[k+1]), and hence thecorresponding potential V(U) is symmetric. Satisfying the autonomousfront condition (5.25) is yet another special and very useful property ofthese solutions. For problems with nonsymmetric potential functionsV(U), it is still possible to obtain second order accuracy for the frontposition, but the definition of the front position (3.12) must be replacedby the less convenient, non-local integral condition (5.25).

The autonomous approximate solution of equation (5.1) provides anasymptotic solution in a more compact form and involving fewer calcu-lations than the corresponding result from the regular perturbation ex-pansion. Our construction using the solution of the single autonomousequation (4.3) eliminates the need to solve equations (5.5) and (5.8a)and the solvability integral conditions for R−1 and R0. Furthermore,extending the accuracy of the approximate solution is exactly equiva-lent to solving for the higher order asymptotic corrections in the regularperturbation expansion. Consequently, it is clear that the autonomousapproximate solution is advantageous since it provides a very conve-nient form of a second order accurate solution of (5.1).

6. Instability of the equilibrium solution. We now apply ourresults to provide an elementary demonstration of the fact that thelocalized radial solution of the elliptic equation ∇2u + F (u;ω) = 0 isan unstable equilibrium of the reaction-diffusion equation

(6.1)∂u

∂t= ∇2u + F (u;ω),

in Rn. If F (u;ω) is a bi-stable nonlinearity then it is well known thatthe solution will develop well-defined fronts connecting smooth, slowlyvarying outer regions (Fife [1988], Rubinstein, et al. [1989]). Supposethat we search for axisymmetric traveling wave front solutions of (6.1)of the form u(r, t) = U(r−ct) where c is the wavespeed. These solutionssatisfy the differential equation

(6.2)d2U

dz2+

(c +

n− 1r

)dU

dz+ F (U ;ω) = 0,

where the traveling wave phase variable is z = r − ct. Equation (6.2)shows that, for c = 0, equation (6.1) is not separable in terms of the


variables z and t for the steady-profile traveling wave ansatz u = U(z)and hence such traveling waves are not exact solutions. More properly,the solution should be expressed in terms of a multi-scale expansion(Kevorkian and Cole [1996]) to allow the solution to slowly changeshape as it evolves in r and t. However, we will show that in thelarge radius limit, these traveling waves give very accurate informationabout the behavior of problem (6.1). In fact, a multi-scale perturbationexpansion of the full solution would return U(r−ct) as the leading orderterm.

We begin by expressing (6.2) as an ordinary differential equation interms of the phase variable z alone. Note that the variable r appearingin the coefficient of the first derivative above can be written as r = z+ct.As was done for the equilibrium problem in (3.11), we can write thephase z in terms of the phase Z corresponding to the motion of thefront and the relative displacement, z, from the front:

(6.3) z = Z + z, Z = R − ct,

where R = R(t) is the time-dependent radial position of the front. Asin (3.12), we define the position of the front by the UM level set,

(6.4) u(R(t), t) = UM .

Combining the changes of variables (6.3), we express r as

(6.5) r = z + ct = R + z,

reducing (6.2) to

(6.6)d2U

dz2+

(c +

n− 1R + z

)dU

dz+ F (U ;ω) = 0.

We treat the factor of R appearing in (6.6) as if it were a large constant;we will justify this approximation a posteriori by showing that it is infact a slowly varying function of time, see Kevorkian and Cole [1996] forgeneral theory underlying this type of quasistationary approximation.Having reduced the problem to a nonautonomous differential equationin z, we now use an expansion of the damping coefficient for large frontradius, R→∞,

(6.7) c +n− 1R + z

= c− (n− 1)zR2

+(n− 1)z2

R3+ O(R−4),


where

c =(c +

n− 1R

),

to obtain the autonomous approximation for the equation of the frontstructure,

(6.8)d2U

dz2+ c

dU

dz+ F (U ;ω) = 0.

Since (6.8) is of the form (1.1), in order to obtain a bounded monotonedecreasing front solution, the damping coefficient must be determinedby the relation c = c(ω). Hence the damping coefficient c is a time-independent constant value determined by the form of the nonlinearityF and we have a fixed relation between the front position and the wavespeed,

(6.9) c = c(ω)− n− 1R

.

Identifying the wavespeed c as the speed of the front, we see thatc = dR/dt, and that (6.9) is a differential equation for the motionof the front,

(6.10)dR

dt= c(ω)− n− 1

R.

Note that while R is large i.e., the front is bounded away well fromthe origin, the front speed is finite, and R(t) is slowly varying. Thisjustifies our earlier approximation of R as a slowly varying parameterin the large radius regime.

Equation (6.10) has an equilibrium solution with radius R = (n −1)/c(ω). This value is the front radius (4.9) in the solution of theequilibrium elliptic problem (4.1) since from (6.9) we have c = 0.This radius is called the critical nucleation radius in the context ofchemical reactions and nucleation theory (Borgis and Moreau [1988],Chan [1977]). It is an unstable equilibrium that separates solutionsthat grow outward indefinitely from those that collapse inward towardthe origin. If we have a solution with a front radius larger than theequilibrium value, R > R, then from (6.10) we obtain a positivewavespeed c = dR/dt > 0. This means that the solution grows in


size as the front moves outward. Conversely, if we start with a solutionwith a core radius that is smaller than the equilibrium value, R < R,then we get a negative wavespeed, c = dR/dt < 0. This situationdescribes a front moving inward toward the origin and a solution that is“collapsing.” Consequently, we conclude that the equilibrium solutionis unstable to perturbations that either diminish or increase its size. Infact, we can obtain the general solution of (6.10) in implicit form,

(6.11) t(R) = t0 +1c

(R +

n− 1c

ln |cR− (n− 1)|).

Solution (6.11) is specified up to a time-shift constant t0, which isdetermined by the initial data. In Figure 9 we compare direct numericalsimulations of the solution the radially symmetric version of (6.1)with the results predicted by our analysis for axisymmetric travelingwaves. Indeed, instability of the equilibrium solution is verified andthe growing/decaying threshold behavior is apparent. Our result for thefront position R(t), (6.11), is shown to match the numerical simulationsvery well except when the solution collapses inward (and the largeradius approximation cannot be expected to hold) and the structureof the front changes significantly. For the unstable growing solution,the front position is accurately given by (6.11) and in fact, since R(t)is increasing, we expect the accuracy of our approximation to steadilyincrease in time and the numerical solution should show convergenceto the axisymmetric traveling wave.

The instability of the solutions smaller or larger than the critical nu-cleation front solution has been part of the focus of extensive studiesin geometric models for the motion of diffusive interfaces (Fife [1988],Rubinstein et al. [1989]). In describing curved fronts in pattern formingreaction diffusion systems in mathematical biology, equation (6.10) isa prototype for unstable threshold behavior (Aronson and Weinberger[1978], Grindrod [1996]). Our solution of the axisymmetric problemfor (6.1) is in fact just one special case of a more general theory forthe motion of curved fronts in reaction-diffusion problems (Aronsonand Weinberger [1978], Fife [1988], Jones [1983a], and [1983b], Kuzinand Pohozaev [1997], Nefedov [1993], Rubinstein and Sternberg [1992],Rubinstein et al. [1989], Terman [1987]). The general theory, as pre-sented in these references and other works not cited here, characterizesall nonincreasing monotone radial solutions, shows that asymptotically

u(r, t)

r

3020r1R0

U0

0

u(r, t)

r

10Rr150

U0

0

R(t)

t

150100500

30

20

r1

10

R(t)

t

806040200

10

r1

5

0

FIGURE 9. Numerical solutions of the radially symmetry reaction-diffusion equation (6.1) exhibitingnucleation threshold behavior. In both simulations the initial data is a step function u(r, 0) = u0(r) =H(r − r1). Solution profiles at equally spaced times are shown (left) and the position of the front as

a function of time (right). Solutions larger than the nucleation radius, r1 > R, will grow indefinitely(top). Solutions smaller than the nucleation radius, r1 < R, collapse inward (bottom).


along each ray expanding solutions approach planar wave fronts, andincorporates the effects of local curvature. Other general structuresthat can be described within this general theory include rotating spiralwaves (Bernoff [1991], Murray [1990]) and more general helical waves(Grindrod [1996], Tyson and Keener [1988]). The results of these stud-ies show that the speed of the motion of curved interfaces is given bythe speed corresponding to one-dimensional planar interfaces, modifiedby corrections due to the mean curvature of the fronts. Recalling thatthe curvature is inversely proportional to the radius of curvature of ashape, κ = 1/R, we see that equation (6.9) is an example of this result,c = c(ω) − (n − 1)κ. Numerous more advanced studies have derivedthis general result using locally orthogonal coordinate systems thatare aligned with the dynamically evolving fronts. By contrast, in ourpresentation, we capture the leading order behavior using elementaryapplications of regular perturbation series and phase plane arguments.

7. Transversal intersections of invariant manifolds and theheteroclinic orbits. In this section we review some results fromdynamical systems theory showing that the heteroclinic orbits foundfor c = c(ω) in system (2.2) lie in the transverse intersections of certaininvariant manifolds in the three-dimensional (U, V, c) phase space. Thisresult, stated in subsection 7.2, demonstrates that the heteroclinicorbits are locally unique, and it is especially useful in establishing thestability of the solutions in the full partial differential equation. Asa preliminary step in subsection 7.1, we study an earlier result thatjustifies the statements made in subsection 2.1 about the phase planesand the existence of a heteroclinic orbit at c = c(ω).

7.1. Phase planes for ω ∈ (ωmin, ωmax) and c > 0. In thissubsection we will report on results for the equation (1.1) with c > 0.In particular, we show that, for each ω ∈ (ωmin, ωmax), there existsa unique value of c = c(ω) for which (1.1) has a ‘lower’ heteroclinicsolution connecting (UR, 0) to (UL = 0, 0). This demonstration thenjustifies the statements made in subsection 2.1 about the stable andunstable manifolds in this case.


The equation (1.1) may be written in system form as:

(7.1)U ′ = V

V ′ = −cV − F (U ;ω).

In order to follow the presentation of Fife [1979], we make the followingcoordinate change: V = −V , c = −c, and z = −z. This coordinatechange makes c < 0 and transforms the ‘lower,’ V < 0, heteroclinicsolution connecting (UR, 0) to (UL = 0, 0) for (7.1) into an ‘upper,’V > 0, heteroclinic solution connecting (UL, 0) to (UR, 0) of the system

(7.2)U ′ = V

V ′ = −cV − F (U ;ω).

It is the existence of this upper connection that is established inChapter 4.4 of Fife [1979], see the treatment there of ‘saddle-saddleconnections,’ and we choose to follow this presentation for historicalreasons, dropping hats from now on.

Let us examine the forward evolution under (7.2) of the trajectory onthe upper right branch of the manifold WU

loc(0, 0) with the parametriza-tion chosen so that U(z = 0) = UM . Since U is strictly increasing whenV > 0, this trajectory must exit the semi-infinite strip 0 ≤ U ≤ UR

and V ≥ 0 either through the U -axis with U ∈ (0, UR) or through theline U = UR with V > 0. Denote the exit point by (Uc, Vc). Also, asin previous sections, it is useful to study the reformulation of (1.1) asa first-order equation for V as a function of U :

(7.31)dV

dU+F (U ;ω)

V= −c.

The desired result is shown in three short steps. In step 1 it is shownthat there exists a large negative value of c such that the solution of(7.3) with V (U = 0) = 0 is a strictly increasing function on [0, UR).Hence, Uc = UR and Vc > 0. Next, in step 2, we consider what happensas c increases from the large negative value found in step 1, showingthat there exists a c0 such that Vc0 = 0, with either Uc0 = UR orUc0 < UR. We remark that Step 2 is taken directly from part of alemma due to Kanel, see Lemma 4.14 in Fife [1979]. Finally, in step 3,the possibility of Uc0 < UR is ruled out for the nonlinear functions


F (U ;ω) of the form studied in this article. Therefore, for the valueof c0 found in step 2, it will be the case that Uc0 = UR, so that thetrajectory is precisely the heteroclinic connection to the saddle (UR, 0)that was sought.

Step 1. Let us explicitly assume c < 0 and that |c| is large. The proofwill be by contradiction. Let us start by assuming that the conclusionis false, i.e., assume there exists a first maximum point (Umax, Vmax)with 0 < Umax < UR. This maximum is given explicitly as

(7.4) Vmax = −F (Umax;ω)c

.

On the one hand, Vmax > 0. Hence, (7.4) implies that F (Umax;ω) > 0,and from the properties of F , one therefore sees that Umax > UM .On the other hand, since F (U ;ω) < 0 for U < UM , one sees thatdV/dU ≥ −c. Hence, by integrating from UL = 0 to UM , one finds

(7.5) V (UM ) ≥ |c|UM ,

since c < 0. Therefore, combining (7.4) and (7.5), it is seen that

|c|UM ≤ Vmax =F (Umax;ω)|c| ,

which cannot be true for |c| large. This demonstrates, for c < 0 and |c|large, that the solution along the local unstable manifold WU

loc(UL, 0) is

a strictly increasing function for U ∈ (0, UR), and it completes step 1.

Step 2. Here we consider what happens for values of c less negative thanthose just obtained in step 1. The main work here is to compare twosolutions V1(U) and V2(U) of (7.3), both with V = 0 at U = UL = 0,but such that c1 > c2. From (7.3), one directly gets

d

dU(V1 − V2)− F (U ;ω)

V1V2(V1 − V2) = −(c1 − c2).

Introducing an integrating factor, this equation may be rewritten as:

(7.6)

d

dU[(V1 − V2)Z(U)] = −(c1 − c2)Z(U)

Z(U) = exp−

∫ U

UM /2

(F (U ′;ω)

V1(U ′)V2(U ′)

)dU ′

.


The lower limit in the integrating factor was taken to be UM/2 here fordefiniteness. Now, as U → 0+, the difference (V1 − V2) vanishes, whilethe exponential factor stays bounded since F < 0 and the integrationis from right to left. Hence, the term in square brackets also vanishesin the limit. Finally, we observe that this term in square brackets is astrictly decreasing function by (7.6), since c1 − c2 > 0; and, therefore,V1(U)− V2(U) must approach zero from below.

The above estimate establishes the desired comparison result: c1 > c2implies V1(U) < V2(U) for U > 0 along the trajectories on WU

loc(0, 0). Ifone increases the value of c from that obtained in step 1 (i.e., one takesa sequence of less negative values), then the graphs of the functionsV (U) giving these manifolds form a monotonically decreasing sequenceof curves, and there exists a value of c, call it c0, such that Vc0 = 0.

Step 3. Here we rule out the possibility that Uc0 < UR. This will implythat Uc0 = UR, so that in the system with the value c0 just obtained instep 2, the manifold WU

loc(0, 0) coincides with the manifold WSloc(U

R, 0),i.e., there is an upper heteroclinic connection from (0, 0) to the othersaddle at (UR, 0), as desired.

If Uc0 < UR, then the trajectory on WUloc(0, 0) crosses the U -axis with

a vertical tangent at (Uc0 , 0). Hence, the manifold WUloc(0, 0) obtained

for the system with c slightly less than c0 will have zero slope at morethan one point to the left of (UR, 0), i.e., dV/dU will have two rootsU1, U2 < UR. One can then readily show that this implies that F (U ;ω)has an additional root less than UR and not equal to UM , whichcontradicts our assumptions about F .

This completes the proof of the existence of a value of c0 = c(ω) forwhich (1.1) has an upper heteroclinic orbit for each ω ∈ (ωmin, ωmax),as stated in subsection 2.1.

7.2. Transversality for (2.2) in (U, V, c) space. In this subsectionwe review well-known results, see Jones [1995], Jones et al. [1991],showing that the heteroclinic orbit of (1.1) studied above for c > 0 liesin the transverse intersection of two invariant manifolds in the three-dimensional (U, V, c) phase space. As a consequence, the heteroclinicorbit will be locally unique. Also, this transversality is important fordemonstrating stability of traveling waves in the full reaction diffusionpartial differential equations, see Jones [1984].


For each ω ∈ (ωmin, ωmax), we will analyze the sequence of (U, V ) phaseplanes obtained in sections 2.1 and 7.1 for c ∈ (c(ω) − δ, c(ω) + δ),where δ > 0 is fixed, in the three-dimensional (U, V, c) space. LetL = (UL(c), 0, c)|c ∈ (c(ω)− δ, c(ω) + δ), and R = (UR(c), 0, c)|c ∈(c(ω)− δ, c(ω) + δ), for a fixed δ > 0. Now, one directly obtains two-dimensional surfaces (manifolds) by taking the unions over c of the localstable and unstable manifolds of the saddle equilibria obtained for eachindividual c. For example, WU

loc(R) ≡ ∪c∈(c(ω)−δ,c(ω)+δ)WUloc(U

R(c), 0).At each point on both of these manifolds, the tangent spaces are two-dimensional, and we recall that two surfaces (manifolds) in R3 aresaid to intersect each other transversely at a point if the sum of thedimensions of their tangent spaces at that point is three. We show inthis subsection that, for each fixed value of ω ∈ (ωmin, ωmax), the twosurfaces WU

loc(R) and WSloc(L) intersect each other transversely in the

three-dimensional (U, V, c) space precisely along the one-dimensionalheteroclinic orbit in the c = c(ω) plane. The results stated hereare taken directly from Section 4.5 of Jones [1995], with only minormodifications to adapt the argument to the lower heteroclinic orbitfrom (UR, 0) to (0, 0) studied here, and to use vectors as opposed todifferential forms.

We will use the notation η±1 ≡(∆U±1 ,∆V ±

1 ,∆c±1 ) and η±2 ≡(∆U±2 ,∆V ±

2 ,∆c±2 ), to denote two independent vectors on the tangent plane to a sta-ble manifold (+) or an unstable manifold (−). A normal vector, n, tosuch a tangent plane is obtained by taking cross-products of η1 and η2:

n = (n1, n2, n3)= (∆V1∆c2 −∆c1∆V2,−∆U1∆c2 + ∆U2∆c1,∆U1∆V2 −∆V1∆U2).

In addition, we will need to have a convenient way in which to measurethe distance between the manifolds WU (UR, 0) and WS(0, 0), as well asthe rate at which that distance changes as c changes. Hence, we followJones [1995] and, for fixed ω ∈ (ωmin, ωmax), define h∓(c) to be thepoints at which the manifolds WU (UR, 0) and WS(0, 0), respectively,intersect the vertical line U = UM in the (U, V ) plane. On the c = c(ω)plane, this distance will be zero, since the manifolds coincide there.We will be interested in the rates at which h∓(c) change as c changesthrough c(ω), and in particular to show that

(7.7)(∂h−

∂c− ∂h+

∂c

)∣∣∣c=c(ω)

= 0.


U

Vc

c(ω)

U R(ω)

FIGURE 10. Schematic picture of transversal intersection of the unstable andstable manifolds in (U, V, c) phase space. The thick solid curves are L and R,and they depend on ω.

When this inequality holds, the manifolds transversely intersect in thethree-dimensional (U, V, c) space.

The evolution of the components of the tangent and normal vectors isdetermined directly from the equation of variations, see, e.g., Codding-ton and Levinson [1987]. For (2.2) with the equation c′ = 0 appended,we find:

(7.8)∆U ′ = ∆V

∆V ′ = −c∆V − V (z)∆c− f ′(U(z))∆U + ω∆U

∆c′ = 0.

Here, the terms U(z) and V (z) are evaluated along the heteroclinicorbit for c = c(ω), and the quantities ∆U,∆V,∆c denote the variationsin the U, V, and c variables from their values along that heteroclinicorbit. Starting from the definition of n and using the chain rulederivative, we find:

(7.9)n′

1 = −cn1 + (f ′(U)− ω)n2

n′2 = −n1

n′3 = V (z)n2 − cn3.

We shall only need the n3 equation.


Having set up all of the general theory, we now make a specific choice,following the judicious selection in Jones [1995], Jones et al. [1991], ofthe two vectors in the tangent planes:

η±1 ≡ (∆U±1 ,∆V ±

1 ,∆c±1 ) =(

0,∂h±

∂c, 1

)η2 ≡ (∆U±

2 ,∆V ±2 ,∆c±2 ) = (V,−cV − F (U ;ω), 0).

Of course, by selecting the vector field of the original system (2.2) asthe second vector, the vector η2 is the same on the tangent planes toboth manifolds. Therefore, we see by explicit calculation that, for thischoice, n1(z) = cV (z) + F (U(z);ω), n2(z) = V (z), and

(7.10) n3(0) = −V (z = 0)(∂h±/∂c).

The first two components are known for all z, since the vector field andheteroclinic solution (U(z), V (z)) are functions of z, whereas the terms∂h±/∂c are only defined on the line U = UM (i.e., at z = 0), so thatn3 is known so far only in terms of the unknown derivatives of h andonly at z = 0.

From (7.9), we now have the explicit differential equation for n3(z):

(7.11) n′3 = −cn3 + V (z)2.

This equation may be solved along each of the manifolds in termsof V (z) with the aid of an integrating factor, and by integratingalong WU (UR, 0) we get: n3(z) = e−cz

∫ z

−∞ V (z′)2ecz′dz′. Hence, we

immediately obtain:

(7.12) n3(0) =∫ 0

−∞V (z′)2ecz′

dz′ > 0.

By combining (7.10) and (7.12) and because V < 0 along the hetero-clinic orbit, we get

(7.13)∂h−

∂c

∣∣∣c=c(ω)

> 0.

Now, in precisely the same fashion, one can show that

(7.14)∂h+

∂c

∣∣∣c=c(ω)

< 0.


Hence, combining (7.13) and (7.14), one obtains the desired result:

(7.15)(∂h−

∂c− ∂h+

∂c

)∣∣∣c=c(ω)

> 0,

and the proof of transversality is complete.

8. Discussion. In this article we have reviewed how a combinationof phase plane analysis and perturbation methods can be used toobtain radially symmetric solutions of semilinear elliptic equation andreaction-diffusion systems. By treating the nonautonomous coefficientsin the radial differential equation as slowly varying parameters, we wereable to obtain an asymptotically accurate autonomous solution fromthe phase plane solution of the damped equation (1.1). Use of a familyof exact solutions for the autonomous problem proved very helpful inthe analysis of the problem and in the examination of general propertiesof the radial solutions.

We also reviewed the demonstration due to Jones [1995], Jones etal. [1991] that the heteroclinic orbit of (2.2) lies in the transverseintersection of the stable and unstable manifolds of equilibria in thefull three-dimensional (U, V, c) phase space, precisely in the planec = c(ω). This result directly implied the local uniqueness of theheteroclinic orbit. It is hoped that our review of this result for travelingwaves in one-space dimension, or planar interfaces in higher spacedimensions, will motivate the reader to study the more advanced resultsderived for curved fronts in axisymmetric higher-dimensional problems,n > 1. In the papers by Aronson and Weinberger [1978], Atkinson andPeletier [1986], Berestycki and Lions [1980], Brezis and Lieb [1984], andJones [1983b], the existence and local uniqueness of radially decreasingsolutions are demonstrated for the nonautonomous problems (1.10).These results are more general than that obtained here via perturbationmethods, since no large interface assumptions are made. Moreover,the existence of a threshold effect (recall Section 6) is established ina much more general context than that shown here, and it is proventhat asymptotically, along every ray emanating from the origin in then-dimensional space, the shapes of the solutions approach that of theone-dimensional traveling wave, see Aronson and Weinberger [1978],Jones [1983b].

A careful review of our approach suggests methods for use of theinterface solutions to construct more general geometric structures, or


r

U

R(ωb) R(ωa)00

u(x, y)

(a)

(b)

FIGURE 11. Construction of the pulse from increasing and decreasing frontsolutions with R = R(ωa) and R = R(ωb) (a), and a three-dimensionalrepresentation of the corresponding annular ring solution.

more general nonautonomous equations of the form

(8.1)d2U

dr2+ G(r)

dU

dr+ F (U ;ω(r)) = 0.

A classical perturbation expansion of the interface structure beginswith the leading order problem of a stationary planar front. This isan interface with zero curvature, or equivalently, an infinite radius ofcurvature, R→∞. The interface structure is given by the lower hete-roclinic orbit in the Hamiltonian phase plane shown in Figure 4b withω = ω0. While zero curvature is maintained, the governing differentialequation (1.1) is autonomous and the solution has translation invari-ance. When the front has any finite amount of curvature, translation


invariance is lost, as in the nonautonomous equation (4.1), and theHamiltonian structure is lost due to the introduction of a slowly vary-ing damping term. We have constructed monotone decreasing solutionsfor this case, in the range ωmin < ω < ω0. However, we noted fromthe phase plane analysis of the damped equation (2.1) that heteroclinicorbits, corresponding to interface solutions, exist in a larger range, forall ωmin < ω < ωmax. Earlier, the solutions for ω0 < ω < ωmax weredismissed as representing unphysical solutions with a negative dampingcoefficient, c < 0, describing a negative radial position, R < 0. How-ever, using the invariance of equation (2.1) under the transformationz → −z and c → −c, these solutions can be shown to describe mono-tone increasing fronts with a positive radial position. For the familyof problems with closed-form solutions described in Section 2.1, theseresults are given by

(8.2)U(z) =

UR

[1 + (UR)k exp −k(UR)k(z − z0)]1/k,

c(ω) =ω

UR(ω)k− UR(ω),

for ω ∈ (ω0, ωmax), and the radial position is still given by (4.2),R = (n − 1)/c(ω) > 0. The radial position for these increasing frontsis plotted in Figure 8 for ω > ω0. For ω ω0 this branch of solutionsis the finite damping continuation of the upper heteroclinic orbit fromthe Hamiltonian phase plane system.

Using these observations, it becomes clear how to construct annular ringsolutions, the radially symmetric analogue of one-dimensional pulsesolutions. If the parameter ω is allowed to slowly vary from above ω0

to below ω0 as r increases then the increasing and decreasing fronts canbe used to form a localized pulse. For example if ω(r) is given by

(8.3) ω(r) = 12 (ωa + ωb) + 1

2 (ωb − ωa) tanh(r − r0δ

),

with δ r0 and ωb > ω0 > ωa then ω ∼ ωa for r < r0 and ω ∼ ωb forr > r0. This form of ω(r) was used to construct the annular solutionshown in Figure 11. While equation (8.3) gives an arbitrary formfor ω(r), we can imagine this front-like solution coming from anothersemilinear elliptic equation that couples ω(r) to U(r), and similarly


for a time-dependent system of reaction-diffusion equations like theFitzHugh-Nagumo model. Other forms of ω(r) coupled to our equationfor u, (6.1) could give periodic trains of axisymmetric pulses; theseare often called “target patterns” (Murray [1990]). Similarly, a weakdependence on a variation in the angular direction, ω = ω(r, θ) wouldperturb such solutions to produce spiral wave patterns (Bernoff [1991])from these rings.

Acknowledgments. TW thanks Thomas Beale, Andrew Bernoff,Robert Almgren and Radica Sipcic for helpful discussions. TK grate-fully acknowledges support from an Alfred P. Sloan Fellowship and anNSF CAREER award.

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