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van Heijster, Petrus, Chen, Chao-Nien, Nishiura, Yasumasa, & Teramoto,Takashi(2018)Localized patterns in a three-component Fitzhugh-Nagumo model revis-ited via an action functional.Journal of Dynamics and Differential Equations, 30(2), pp. 521-555.

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https://doi.org/10.1007/s10884-016-9557-z

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https://eprints.qut.edu.au/100412/

https://doi.org/10.1007/s10884-016-9557-z

Localized patterns in a three-componentFitzHugh–Nagumo model revisited via an actionfunctional

Peter van Heijster · Chao-Nien Chen ·Yasumasa Nishiura · Takashi Teramoto

Abstract In this manuscript, we combine geometrical singular perturbationtechniques and an action functional to revisit – and further study – the ex-istence and stability of stationary localized structures in a singularly per-turbed three-component FitzHugh–Nagumo model. In particular, the actionfunctional replaces the Melnikov integral approach used in [A. Doelman etal., J. Dyn. Differ. Equ. (2009)] to explicitly derive existence conditions forstationary localized structures. In addition, the action functional also givescritical information on the stability of these stationary localized structures,thus circumventing a tedious Evans function computation. This highlights thestrength of the action functional approach.

Keywords Reaction-diffusion system · Activator-inhibitor model · Calculusof variations · Singular perturbations · Existence · Stability

Mathematics Subject Classification (2000) 35B25 · 35B35 · 35K57 ·49J40

PvH was supported under the Australian Research Councils Discovery Early Career Re-searcher Award funding scheme DE140100741. He also acknowledges support from Queens-land University of Technology (QUT) under the ECARD/VCRF Travel Fellowships awardscheme. Part of the work was done when CNC was visiting QUT and he is grateful for thesupport by the Ministry of Science and Technology of Taiwan. YN is partially supported byKAKENHI Grant-in-Aid for Scientific Research (A)26247015. TT is partially supported byKAKENHI Grant-in-Aid for Scientific Research 15KT0100.

Peter van HeijsterSchool of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4001,AustraliaTel.: +61-7-3138-1671E-mail: [email protected]

Chao-Nien ChenDepartment of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Yasumasa NishiuraWPI Advanced Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan

Takashi TeramotoSchool of Medicine, Asahikawa Medical University, Asahikawa, 078-8510, Japan

2 Peter van Heijster et al.

1 Introduction

1.1 Three-component FitzHugh–Nagumo model

The singularly perturbed three-component FitzHugh–Nagumo model underconsideration is

Ut = ε2Uxx + U − U3 − ε(αV + βW + γ) ,

τVt = Vxx + U − V ,θWt = D2Wxx + U −W ,

(1)

with 0 < ε 1;D > 1; τ, θ > 0; (x, t) ∈ R×R+;α, β, γ ∈ R, and all parametersare assumed to be O(1) with respect to ε. A dimensional version of this modelwas derived in [34,42] to study gas-discharge dynamics on a phenomenolog-ical level. Here, U represents the current density in the nonlinear S-shapedconductor, V the voltage drop in a high ohmic linear conductor and W thesurface charge. See [37] for a recent review article related to the backgroundof this model.

Localized structures of different versions of this model, i.e. solutions thatare close to the model’s background states except for several localized re-gions in space, have been studied intensively afterwards by scientists [1,18,34,42,44] and mathematicians [9,15,32,38,19,21–24,46]. From a mathemat-ical perspective, model (1) is particular interesting since it is arguably thefirst three-component singularly perturbed reaction-diffusion equation studiedin this much detail. In particular, the fact that the activator U is a bistableNagumo-like equation weakly coupled to two linear slow inhibitors V,W (forα, β > 0) makes this equation amenable to rigorous mathematical analysis.

1.2 Analyses with the Melnikov integral

In one spatial dimension, the most common method to study the existence ofstationary localized structures in singular perturbed coupled reaction-diffusionequations like (1) is Geometric Singular Perturbation Theory (GSPT), see forexample [17,25,26] and references therein. Generally, the final step in thesecomputations is to derive explicit conditions for when the unstable manifoldof the asymptotic end state of the localized structure at −∞ transversallyintersects with the stable manifold of the (potentially different) asymptoticend state of the localized structure at +∞. This verification is based on aMelnikov-type integral computation [41] and in the end results in conditionsdetermining the potential width(s) of the constructed localized structures. Theabove described approach is for example used in [15] to prove the existence oflocalized stationary pulse solutions in (1), see also upcoming Theorem 1.

To subsequently determine the linear (and nonlinear) stability of the sta-tionary localized structure, one generally needs an Evans function approachto locate the point spectrum. (The essential spectrum is generally straightfor-ward to compute as one only needs the asymptotic dynamics of the localized

Localized patterns in a three-component FitzHugh–Nagumo model 3

structure). In short, an Evans function is an analytic function whose zeroescoincide with the point eigenvalues of the linearized stability problem. Werefer to [27] – and references therein – for a detailed explanation of (the con-struction of) Evans functions. In [12–14], the NonLocal Eigenvalue Problem(NLEP) approach was developed to simplify the study of the Evans functionfor singularly perturbed problems like (1) and the approach heavily utilizesthe singular perturbed structure of the problem. In [19], the NLEP approachwas extended and used to derive stability criteria for the localized stationarypulse solutions of (1) as constructed in [15], see also upcoming Theorem 1.

In this manuscript, we combine GSPT and an action functional to de-termine existence and stability conditions of stationary localized structuressupported by (1). So, the action functional replaces – at least at a formal level– both the Melnikov-type integral of [15] and the NLEP computation of [19].We first show (see §2) that this new approach with the action functional re-produces the results of [15] and [19] related to stationary localized structures.Therefore, we restate the main results related to stationary localized structuresobtained in these two articles.

Theorem 1 (Thm. 2.1 and Thm. 5.1 in [15] and Thm. 4.1 and Thm. 6.1in [19])

Stationary 1-pulse solutions: Let (α, β, γ,D) be such that

αe−2x + βe−2x/D = γ , (2)

has a positive solution x = x∗. Then, for ε small enough, (1) supports astationary 1-pulse solution Z1p(x) = (U1p, V1p,W1p)(x) that asymptotes tou(1, 1, 1) as x→ ±∞. Here, u is the most negative root of the cubic equation

u3 − u+ ε((α+ β)u+ γ) = 0. (3)

In particular, u = −1 + 12ε(α + β − γ) + O(ε2). The width of a stationary

1-pulse solution is, to leading order in ε, 2x∗. In addition, a 1-pulse solutionZ1p is nonlinearly stable if, and only if,

f(2x∗) := αe−2x∗

+β

De−2x

∗/D > 0 . (4)


The profile of a stationary 1-pulse solution Z1p = (U1p, V1p,W1p)t is, to leadingorder – and up to translations in x due to translation invariance – given by

Z1p(x) =

−1

−1 + (ex∗ − e−x∗)ex

−1 + (ex∗/D − e−x∗/D)ex/D

χI−s +

tanh(

1√2ε

(x+ x∗))

−e−2x∗−e−2x∗/D

χI−f

+

1

1− e−x∗(ex + e−x)

1− e−x∗/D(ex/D + e−x/D)

χI0s +

− tanh(

1√2ε

(x− x∗))

−e−2x∗−e−2x∗/D

χI+f

+

−1

−1 + (ex∗ − e−x∗)e−x

−1 + (ex∗/D − e−x∗/D)e−x/D

χI+s .

(5)Here, χI±,0

sand χI±f

are the characteristic functions for the three slow intervals

I−s = (−∞,−x∗−√ε), I0s = (−x∗+√ε, x∗−√ε), I+s = (x∗+

√ε,∞) and the

two fast intervals I−f = [−x∗ −√ε,−x∗ +√ε], I+f = [x∗ −√ε, x∗ +

√ε].

Stationary symmetric 2-pulse solutions: Let (α, β, γ,D) be such that

0 =α

(e−x1 − e−x2

)2+ β

(e−x1/D − e−x2/D

)2,

−2γ =α(e−2x2 − e−2x1

)− 2αe−x1+x2 + β

(e−2x2/D − e−2x1/D

)− 2βe−x1/D+x2/D ,

(6)

has a solution pair (x1, x2) = (x∗1, x∗2) with 0 < x∗2 < x∗1. Then, for ε small

enough, (1) supports a stationary symmetric 2-pulse solution Z2p asymptotingto u(1, 1, 1). The width of both symmetric pulses is, to leading order in ε,x∗1 − x∗2, and the distance between the two pulses is, to leading order, 2x∗2. Inaddition, a stationary symmetric 2-pulse solution Z2p is nonlinearly stable if,and only if,

λ1 := f(x∗1 − x∗2)− f(x∗1 + x∗2) > 0 , and

λ2 := f(2x∗1) + f(2x∗2) + λ1

−√

(f(2x∗1)− f(2x∗2))2 + (f(x∗1 + x∗2) + f(x∗1 − x∗2))2 > 0 ,

(7)

with f as defined in (4) and with (x∗1, x∗2) a solution pair of (6). (Note that

λ1,2 are, up to multiplicative negative constants, the leading order eigenvaluesof the symmetric 2-pulse solution as determined in [19].)


The profile of the stationary symmetric 2-pulse solution Z2p = (U2p, V2p,W2p)t

is, to leading order – and up to translations in x – given by

Z2p(x) =

U2p(x)V2p(x)W2p(x)

=

−12 (sinh (x∗1)− sinh (x∗2)) ex − 1

2(

sinh(x∗1D

)− sinh

(x∗2D

))ex/D − 1

χI1s

+

tanh(

12ε

√2 (x+ x∗1)

)−e−2x∗1 − e−x∗1+x∗2 + e−x

∗1−x∗2

−e−2x∗1/D − e(−x∗1+x∗2)/D + e(−x∗1−x∗2)/D

χI2f

+

1

−e−(x+x∗1) − ex−x∗1 − 2 sinh (x∗2)ex + 1

−e−(x+x∗1)/D − e(x−x∗1)/D − 2 sinh (x∗2D )ex/D + 1

χI3s

+

− tanh(

12ε

√2 (x+ x∗2)

)e−2x

∗2 − e−x∗1+x∗2 − e−x∗1−x∗2

e−2x∗2/D − e(−x∗1+x∗2)/D − e(−x∗1−x∗2)/D

χI4f

+

0

−e−(x+x∗1) + e−(x+x∗2) + ex−x

∗2

−e−(x+x∗1)/D + e−(x+x∗2)/D + e(x−x

∗2)/D

+

−1

−ex−x∗1 − 1

−e(x−x∗1)/D − 1

χI5s

+

tanh(

12ε

√2 (x− x∗2)

)e−2x

∗2 − e−x∗1+x∗2 − e−x∗1−x∗2

e−2x∗2/D − e(−x∗1+x∗2)/D − e(−x∗1−x∗2)/D

χI6f

+

1

−e−(x+x∗1) − ex−x∗1 − 2 sinh (x∗2)e−x + 1

−e−(x+x∗1)/D − e(x−x∗1)/D − 2 sinh (x∗2D )e−x/D + 1

χI7s

+

− tanh(

12ε

√2 (x− x∗1)

)−e−2x∗1 − e−x∗1+x∗2 + e−x

∗1−x∗2

−e−2x∗1/D − e(−x∗1+x∗2)/D + e(−x∗1−x∗2)/D

χI8f

+

−12 (sinh (x∗1)− sinh (x∗2)) e−x − 1

2(

sinh (x∗1D )− sinh (

x∗2D ))e−x/D − 1 .

χI9s .

(8)Here, χI1,3,5,7,9s

and χI2,4,6,8fare the characteristic functions for the five slow

intervals I1s = (−∞,−x∗1 −√ε), I3s = (−x∗1 +

√ε,−x∗2 −

√ε), I5s = (−x∗2 +√

ε, x∗2 −√ε), I7s = (x∗2 +

√ε, x∗1 −

√ε), I9s = (x∗1 +

√ε,∞), and the four

fast intervals I2f = [−x∗1 −√ε,−x∗1 +

√ε], I4f = [−x∗2 −

√ε,−x∗2 +

√ε], I6f =

[x∗2 −√ε, x∗2 +

√ε], I8f = [x∗1 −

√ε, x∗1 +

√ε].


0010 10 1010

11

0 0

1 1

x x

UU

VV

WW

2x2

x1 x

2

2x

Fig. 1 Left panel: Stable stationary 1-pulse solution Z1p supported by (1). The sys-tem parameters are (α, β, γ,D, τ, θ, ε) = (3, 1, 2, 5, 1, 1, 0.01). Right panel: Stable sta-tionary symmetric 2-pulse solution Z2p supported by (1). The system parameters are(α, β, γ,D, τ, θ, ε) = (2,−1,−0.25, 5, 1, 1, 0.01). Note that this figure is an adaptation ofFigure 1 in [15] and Figure 1 in [19].

See Figure 1 for a stable stationary 1-pulse solution and a stable stationarysymmetric 2-pulse solution obtained by numerically simulating (1).

Remark 1 Note that the original activator-inhibitor framework of the model(1) required that both α, β > 0. From a mathematical point of view, thisrestriction is not necessary and it is shown in [15,24] that the dynamics ofthe model is much richer when this restriction is dropped. For instance, from(6) it directly follows that stationary symmetric 2-pulse solutions do not existfor α, β > 0. Therefore, we do not explicitly require that α, β > 0 in thismanuscript.

1.3 Action functional via a variational approach

In Theorem 1, the profiles Z1p (5) and Z2p (8) of the stationary structures (asfunction of x and x∗, x∗1,2) have been determined by GSPT. Next, a Melnikov-type integral is used to derive the actual existence conditions (2) (determiningx∗) and (6) (determining x∗1,2). Finally, the NLEP approach is utilized tocompute the leading edge of the point spectrum to determine the stability ofthe localized structures, i.e. to derive (4) and (7).

In this manuscript, we show that this Melnikov-type integral computationto determine the existence conditions can be replaced by finding the criticalpoints (as function of x∗, respectively x∗1,2) of an action functional formulatedby a variational structure associated with the stationary version of (1):

0 = ε2uxx + u− u3 − ε(αv + βw + γ) ,

0 = vxx + u− v ,0 = D2wxx + u− w .

(9)


Moreover, the minimizers of the action functional coincide with the stableprofiles. This way, the action functional also circumvents the whole NLEPmachinery.

In particular, the action functional for (9) is given by

J(u) =

∫ ∞−∞

(1

2ε2u2x + F (u)− F (u) +

1

2εα(uL1u− u2) (10)

+1

2εβ(uL2u− u2) + εγ(u− u)

)dx ,

with antiderivative

F (u) =1

4u4 − 1

2u2 , (11)

and L1 := (− d2

dx2 + 1)−1, L2 := (−D2 d2

dx2 + 1)−1 and u the most negative rootnear −1 of (3). In particular, for given u, L1u exactly solves the second equa-tion of (9): vxx+u− v = 0. Similarly, L2u exactly solves the third equation of(9). Observe that the terms F (u), u and u2 in the action functional compen-sate for the asymptotic behavior of the pulse solutions. That is, to ensure thatthe action functional is well-defined and bounded. In particular, we requirethat u− u ∈ H1(R).

Heuristically, the relationship between the stationary pulse solutions of (9)and the critical points of J (10) is clear from the directional derivative of J .The directional derivative of J(u) for φ ∈ C∞0 is given by

J ′(u)φ = limt→0

J(u+ tφ)− J(u)

t. (12)

We are going to look at the separate terms of J (10) independently. For ex-ample, the u2x-term in (10) gives

limt→0

(u+ tφ)2x − u2xt

= limt→0

(2uxφx + tφ2x) = 2uxφx .

Similarly,

limt→0

F (u+ tφ)− F (u)

t= (u3 − u)φ , lim

t→0

(u+ tφ)− ut

= φ,

and

limt→0

(u+ tφ)L1,2(u+ tφ)− uL1,2u

t= φL1,2u+ uL1,2φ = 2φL1,2u ,

since L1,2 are both self-adjoint.In total, we get

J ′(u)φ =

∫ ∞−∞

(ε2uxφx + (u3 − u)φ+ εφ(αL1u+ βL2u+ γ)

)dx

=

∫ ∞−∞

(−ε2uxx + (u3 − u) + ε(αL1u+ βL2u+ γ)

)φdx ,


where we used integration by parts in the last step. In other words, J ′(u)φ = 0for all φ ∈ C∞0 if u solves the first equation of (9) (recall L1u = v andL2u = w). That is, the critical points of J (10) coincide with the stationarypulse solutions of (9).

Furthermore, if the critical point is a minimizer of the action functionalthan the stationary pulse solution is expected to be stable. As a brief explana-tion for this, we observe that (9) is a second-order Hamiltonian system and theMaslov index of a homoclinic orbit is zero if it is a non-degenerate minimizerof J [5]. The stability now follows from a modified version of Theorem 1.2 of[6] and its proof, using index theory, is similar.

The same type of action functional was used in a series of papers by Chen etal. [2,3,6,7] to study the following two-component FitzHugh-Nagumo model:

Ut = dUxx + U(U − β)(1− U)− V ,τ Vt = Vxx + U − γV , (13)

with all the parameters being positive and 0 < β < 12 , i.e. the non-symmetric

case. The variational approach allows for more flexibility on the ranges of theparameters. Most notably, this method has been used to study various rangesof diffusion coefficient d in (13) to find different phenomena for standing waves,including the situation where the activator u and inhibitor v interact strongly(as can also be seen from the action functional). In particular, by investigat-ing the critical points in the variational structure, the existence of front andpulse solutions of (13) was established through innovative new techniques thatdealed with a nonlocal term in the associated action functional [2,3,7]. Thestability question of standing pulse solutions has been justified in [6] usingthe Maslov index theory [5] and the stability of standing front solutions wasinvestigated in [7].

We will not provide a fully detailed proof in this manuscript why thisaction functional approach as discussed above works. However, such a proof– based on calculus of variations – would essentially follow the argumentsdemonstrated in [2,6].

In [21], a system of 2N first order ordinary differential equations (ODEs)describing the evolution of the interfaces of an N -pulse solution-like initialcondition of (1) is rigorously derived using a renormalization group approach[36]. For N = 1, respectively N = 2, these N -pulse solution-like initial con-ditions used in [21] coincide – to leading order – with the profiles stated in(5), respectively (8). The system of ODEs describes the dynamically evolvinginterfaces of these initial conditions and its fixed points thus determine theunknown(s) x∗, respectively x∗1,2.

More generally, the fixed points of the ODE system for N -pulse solution-like initial conditions correspond to stationary N -pulse solutions, i.e. theycoincide with the critical points of the action functional J for stationary N -pulse solution ZNP, see §3. Moreover, this system of ODEs is a gradient flowwith functional G, see Remark 4.3 in [21]. Unsurprisingly, this functional Gis similar to J(ZNP). In other words, the action functional J yields a direct


approach in computing this gradient flow – up to the scaling of a multiplicativeconstant which can be seen as a mere rescaling of time and the addition of aregular constant that does not influence the gradient flow.

The rigorous analyses in [21] also shows that the leading edge of the pointspectrum determines the motion of the interfaces and that this leading edgeis the only part of the spectrum in the right half plane, see, for instance,Figure 3 in [21]. Therefore, and as long as the interfaces are well-separated,the dynamically evolving N -pulses are characterised by the location of theirinterfaces (i.e. by x∗, x∗1,2, etc.) and instabilities – other than the motion of theinterfaces – are not expected. In that sense, it is also not surprising that thereis a direct connection between leading edge of the point spectrum and the firstand second derivatives of the action functional. Note that it is mathematicallyunclear what happens when two interfaces meet and that the ODE descriptionof [21] is no longer valid in this case. However, numerically it appears that twocolliding interfaces of (1) disappear, see, for instance, Figure 10 in [21].

In this manuscript we assume that τ and θ are O(1) with respect to ε. Since(9) is independent of τ and θ, the action functional approach for determiningthe existence of stationary localised structures also works for the parameterregime where τ and/or θ are O(ε−2). However, the approach cannot be useddirectly to derive stability results since the index argument used in [6] toconclude stability from the fact that the orbit is a non-degenerate minimizerof J is not sufficient anymore. See also [6] for more details. The parameterregime τ and/or θ are O(ε−2) is of particular interest since in this regimethe essential spectrum is asymptotically close to the origin and additionaleigenvalues pop out of the essential spectrum [19]. Furthermore, stationarypulse solutions potentially bifurcate to travelling pulse solutions and breathersolutions [19]. Observe that while existence results for travelling pulse solutionsin a variational setting of (13) were established in [3], the action functionalapproach has not been established for travelling pulse solutions of (1) yet. Seealso §5.2.1.

1.4 Results & Outline

In this manuscript, we are interested in stationary localized structures of (1),i.e. localized solutions of (9), in the case of weak interaction between the acti-vator U and the two inhibitors V and W . Our analysis of the action functional(10) shows that there is a connection between the action functional, the Mel-nikov integral and the NLEP approach. In particular, critical points of theaction functional determine the existence conditions of these localized struc-tures (such as stated in Theorem 1). In addition, the minimizers of the actionfunctional – which can often be obtained without significant additional com-putations – coincide with the stable structures, while the other critical pointsyield unstable localized structures. These stability results are obtained withoutusing an Evans function. This highlights the usefulness of an action functional


as an extremely powerful tool: it significantly reduces the computational effortto derive existence – and especially stability – results.

In §2, we show that the action functional (10) can reproduce the resultsrelated to existence and stability of stationary 1-pulse solutions and stationarysymmetric 2-pulse solutions supported by (1) as stated in Theorem 1. Duringthis derivation, we also obtain the new result that α > 0 and β < 0 is anecessary condition for stable stationary symmetric 2-pulse solutions – seeLemma 4. The results of this section further demonstrate – and benchmark –the action functional approach.

In §3, we generalize the results and use the action functional to derive ex-istence and stability conditions for general stationary N -pulse solutions ZNP

supported by (1). We apply the new results to study stationary asymmetric2-pulse solutions and stationary 3-pulse solutions. In both cases, we find that– unlike the existence results for stationary 1-pulse solutions and stationarysymmetric 2-pulse solutions – these localized structures only exist in very spe-cific parameter regimes. In both cases, a necessary, but not sufficient, conditionfor existence is that α and β have opposite sign, see Remark 1. Note that theresults of this section have not appeared in the literature before.

In §4, we restrict the unbounded domain R to [−L,L] and we use the ac-tion functional approach to study stationary spatially-periodic localized struc-tures on this bounded domain. In particular, we derive existence and stabilityconditions for stationary spatially-periodic 1-pulse solutions supported by (1)on [−L,L] with periodic boundary conditions using the action functional ap-proach. In principle, this approach could also be easily generalized to deriveconditions for general stationary spatially-periodic N -pulse solutions. How-ever, we decided not to pursue this direction. Observe that periodic solutionsto (1) have – to the best of our knowledge – not been studied in detail in theliterature before.

We end this manuscript with a summary and short outlook on futureprojects, see §5.

2 Stationary 1-pulse and 2-pulse solutions

In this section, we show that the variational approach as described in theintroduction can be used to derive the existence conditions and stability resultsfor stationary 1-pulse solution Z1p and stationary symmetric 2-pulse solutionZ2p as stated in Theorem 1.

2.1 Stationary 1-pulse solutions

The profile of a stationary 1-pulse solution Z1p (5) is derived by GSPT. Weuse the action functional J (10) to reproduce the results of Theorem 1 forstationary 1-pulse solutions Z1p. In particular, the critical points of J – forZ1p given by (5) – are given by the same expression as the existence condition


for stationary 1-pulse solutions (as stated in (2)), and only the condition forthe minimizers of J(Z1p) coincide with the stability criterion for stationary1-pulse solutions (as stated in (4)).

Lemma 1 The action functional J(Z1p) of a stationary 1-pulse solution Z1p

(5) is given by

J(Z1p) = 2εα(e−2x∗−1)+2εβD(e−2x

∗/D−1)+4εγx∗+4√

2

3ε+O(ε

√ε) . (14)

Proof We split the full indefinite integral of J(Z1p) over the three slow regionsI±,0s and the two fast regions I±f as stated in Theorem 1

J =

∫ ∞−∞

dx =

∫I−s

dx︸︷︷︸I1

+

∫I−f

dx︸︷︷︸I2

+

∫I0s

dx︸︷︷︸I3

+

∫I+f

dx︸︷︷︸I4

+

∫I+s

dx︸︷︷︸I5

. (15)

Next, we use a regular expansion for the three components of the stationary1-pulse solution

U1p = u0 + εu1 +O(ε2) , V1p = v0 + εv1 +O(ε2) , W1p = w0 + εw1 +O(ε2)

with u0, v0 and w0 given by the leading order formulas in (5). As it turnsout, only the leading order terms u0, v0 and w0 are needed to compute theleading order O(ε)-term of the action functional J(Z1p). For example, a directcomputation show that the antiderivative term F (U1p)−F (u) in the slow fieldsI±,0s is O(ε2), and thus independent of u1 up to O(ε2).

We compute the five integrals as stated in (15) separately. In the three slowregions I±,0s , we use the slow x-scaling, while we use the fast ξ := x/ε-scalingin the two fast regions I±f , see for example [15] for more details with regardto the slow-fast scaling. The first integral I1 gives:

I1 =

∫ −x∗−√ε−∞

(1

2ε2(U1p)2x + F (U1p)− F (u) +

1

2εα(U1pL1U1p − u2

)+

1

2εβ(U1pL2U1p − u2

)+ εγ(U1p − u)

)dx

=

∫ −x∗−√ε−∞

(1

2ε2(U1p)2x + F (U1p)− F (u) +

1

2εα(U1pV1p − u2

)+

1

2εβ(U1pW1p − u2

)+ εγ(U1p − u)

)dx

=

∫ −x∗−√ε−∞

((−1

2εα(ex∗ − e−x∗

)ex)

+

(−1

2εβ(ex∗/D − e−x∗/D

)ex/D

)+O(ε2)

)dx

=1

2εα(e−2x

∗ − 1) +1

2εβD(e−2x

∗/D − 1) +O(ε√ε) .


The second integral I2 gives

I2 =

∫ −x∗+√ε−x∗−√ε

(1

2ε2(U1p)2x + F (U1p)− F (u) +

1


)+

1


)+ εγ(U1p − u)

)dx

=ε

∫ −ξ∗+ 1√ε

−ξ∗− 1√ε

(1

2(U1p)2ξ + F (U1p)− F (u) +

1

2εα(U1pV1p − u2

)+

1

2εβ(U1pW1p − u2

)+ εγ(U1p − u)

)dξ

(16)

=ε

∫ −ξ∗+ 1√ε

−ξ∗− 1√ε

(1

4sech4

(1

2

√2 (ξ + ξ∗)

)+

1

4tanh4

(1

2

√2 (ξ + ξ∗)

)−1

2tanh2

(1

2

√2 (ξ + ξ∗)

)+

1

4+O(ε)

)dξ

=ε

∫ 1√ε

− 1√ε

(1

2sech4

(1

2

√2ξ

)+O(ε)

)dξ

=2√

2

3ε+O(ε

√ε) ,

where we used that V1p and W1p are constant and O(1) in the fast regionsand the identity

tanh4 (x)− 2 tanh2 (x) + 1 = (tanh2 (x)− 1)2 = sech4(x) .

The third integral I3 gives

I3 =

∫ x∗−√ε

−x∗+√ε

(1

2ε2(U1p)2x + F (U1p)− F (u) +

1


)+

1


)+ εγ(U1p − u)

)dx

=

∫ x∗−√ε

−x∗+√ε

(−1

2εαe−x

∗(ex + e−x)− 1

2εβe−x

∗/D(ex/D + e−x/D) + 2εγ

+O(ε2)

)dx

= εα(e−2x∗ − 1) + εβD(e−2x

∗/D − 1) + 4εγx∗ +O(ε√ε) .

By the symmetry x 7→ −x, the fourth integral I4, respectively fifth integralI5, follow directly from I2, respectively I1. In particular,

I4 =2√

2

3ε+O(ε

√ε) , I5 =

1

2εα(e−2x

∗ − 1) +1

2εβD(e−2x

∗/D − 1) +O(ε√ε) .

Adding the above five parts gives (14).


Taking the derivative of J(Z1p) (14) with respect to x∗ gives

d

dx∗J(Z1p) = −4εαe−2x

∗ − 4εβe−2x∗/D + 4εγ +O(ε

√ε) . (17)

A priori, the derivatives of the higher order correction terms of J(Z1p) (14)could have a leading order influences on (17). However, it is straightforward –but technical – to show that this is actually not the case here. So, the criticalpoints of J(Z1p) are given by the zeroes of the above expression. To leadingorder this yields the existence condition for stationary 1-pulse solutions ofTheorem 1, see (2). Moreover, a critical point is a minimizer if, in addition to(2),

d2

d(x∗)2J(Z1p) = 8εαe−2x

∗+ 8ε

β

De−2x

∗/D +O(ε√ε) > 0 .

This condition is identical to the stability criteria for stationary 1-pulse solu-tions of Theorem 1, see (4). In other words, both the existence and stabilitycriteria for stationary 1-pulse solutions Z1p are encoded in the expression ofthe action functional J(Z1p) (14).

Observe that for α, β, γ positive with α+β > γ the stability criterium im-plies that all 1-pulse solutions are stable (i.e. critical points of the the actionfunctional J(Z1p) are always minimizers), while all 1-pulse solutions are unsta-ble for α, β, γ negative with |α|+ |β| > |γ| (i.e. critical points of the the actionfunctional J(Z1p) are always maximizers). If α and β have opposite signs –see Remark 1 – and |αD| > |β| it is possible to have a saddle-node bifurcationof 1-pulse solutions [19]. For example, taking (α, β, γ,D) = (4,−1,−0.2, 5),we find two stationary 1-pulse solutions from the existence condition (2). Inother words, the action functional J(Z1p) has two critical points. One pulsewith half-width x∗1,1 ≈ 1.1 and the other pulse with half-width x∗1,2 ≈ 4.0.The graph of the action functional J(Z1p) as function of x∗ shows that thefirst pulse solution is stable, while the second pulse solution is unstable, seeFigure 2. Or, similarly,

1

ε

d2

d(x∗)2J(Z1p,2) ≈ −0.312 < 0 <

1

ε

d2

d(x∗)2J(Z1p,1) ≈ 2.52 .

Upon using γ as a decreasing bifurcation parameter (and keeping the otherparameters fixed), these two 1-pulses theoretically coincide and disappear for

γ = − 2√2

55/4≈ −0.378 at half-width x∗ = 5

8 log 20 ≈ 1.872. See also [19].

2.1.1 Numerics

To confirm our obtained analytic results, we numerically simulate (9). We firstpresent a brief description of the numerical method used for path following thesolution branches. This method is based on the predictor-corrector method ofpseudo-arclength continuation [11,28,43].


1 2 3 4 5

-3

-2

-1

1

2

x

J(Z1p)/"

Fig. 2 Plot of the leading order component of J(Z1p)/ε for (α, β, γ,D) = (4,−1,−0.2, 5).The minimizer around 1.1 yields a stable stationary 1-pulse solution, while the maximizeraround 4 yields an unstable stationary 1-pulse solution.

We begin with the general form of the stationary problem of a scalarreaction-diffusion equation (so for one of the components) in a truncated onedimensional domain Ω := [−L,L] with L large enough

0 = DUxx + F (U ; γ) ,

where D is the diffusion coefficient and F : R× R→ R, and γ ∈ R representsa continuation parameter.

We spatially discretize U by setting ∆x = 2L/n. In other words, U becomesU := (U0, U1, · · · , Un−1)T where Ui = U(−L + i∆x), i = 0, 1, . . . , n − 1. So,we get

G(U ; γ) := DUxx + F (U ; γ) = 0,∂U

∂x

∣∣∣∣∂Ω

= 0, (18)

where D = DI with I the n × n identity matrix, F and G : Rn × R → Rn,and 0 is the n-dimensional zero-vector.

Next, we show a single continuation step with respect to the continuationparameter γ. Namely, the transition from a j-th calculated solution set to thenext solution set of the branch. If the Jacobian matrix ∂G(U j ; γj)/∂U is non-singular, the implicit function theorem assures the existence of the solutionbranch in the neighborhood of a j-th solution set. Since system (18) consistsof n equations for (n + 1) unknowns, a quadratic scalar equation is attachedfor the small distance ∆s := s− sj between two consecutive solution sets,

n−1∑i=0

(Ui − Ui(sj))2∆x+ (γ − γ(sj))2 − (∆s)2 = 0 (19)

where the j-th solution set Zj := (U(sj), γ(sj))T are implicitly parameterizedas function of the arclength parameter s along the branch.


Using the Taylor expansion Z −Zj =dZj

ds∆s+O((∆s)2), we replace (19)

by the following linear form with respect to the increments

N(U ; γ) =

n−1∑i=0

dU jids

(Ui − Ui(sj))∆x+dγj

ds(γ − γ(sj))−∆s = 0, (20)

where dZj/ds is supposed to be an unit vector tangent to the branch curveat the current position Zj .

By solving the (n+1) equations of (18) and (20) for each continuation step,a solution branch is obtained as a chain of solutions Zj . The initial guess forthe next solution set is obtained in the direction of dZj/ds. We then iterativelysolve the equations using Newton’s method,

∂(G, N)(Zj)

∂U∂γ∆Z = −

(G(Zj)

N(Zj)

), (21)

where ∆Z := Zj+1 − Zj . If the step size ∆s is given small enough, theNewton’s iteration converges to the next solution set Zj+1 on the branch inthe direction perpendicular to dZj/ds.

After converging, we compute the new tangent vector dZj+1/ds by solving(n+ 1) equations using the Jacobian matrix evaluated at Zj

∂(G, N)(Zj)

∂U∂γ

dZj+1

ds=

(01

). (22)

The new tangent vector is rescaled to satisfy |dZj+1/ds|2 = 1, thus preservingto the right direction along the branch.

For the problem at hand, we can actually restrict our numerical simulationsto the half-line [0, L] (with no-flux boundary conditions at zero) since theproblem is translation invariant and the pulse solutions are symmetric in x 7→−x (where we assumed for convenience that a solution is centered aroundzero). That is, if we for example want to find stationary 1-pulse solutions on[−L,L], we can instead look for half 1-pulse solutions on [0, L]. See also theright panels of Figure 3.

The numerical action functional can also easily be computed by appro-priately truncating the indefinite integral of the action functional (10). Thatis,

Jnum(u) = 2

∫ L

0

(1

2ε2u2x + F (u)− F (u) +

1

2εα(uL1u− u2) (23)

+1


)dx ,

with antiderivative F as given in (11), u the asymptotic background state near−1 and L1,2 as before. In principle, there can be a large difference between thenumerical action functional Jnum (23) and the asymptotic action functional


x

1.0

10

0.1

0.1 0 1.0 2.0 3.0

I

I

IIII

III

III

8.0 16 24x

1.0

1.0

0

1.0

1.0

0

1.0

1.0

0

stable

unstable

unstable

0

8.0 16 240

8.0 16 240

U

U

U

W

W

W

V

V

V

x

JInum 1.32

JIInum 4.08

Fig. 3 Left panel: Numerically obtained bifurcation diagram with bifurcation parameter γversus the half-width x∗ of 1-pulse solutions. The red curve corresponds to (numerically)stable solutions, while the green curves represent (numerically) unstable branches. The bluedashed curve represents the asymptotically predicted half-width (2). The other system pa-rameters are kept fixed at (α, β,D, τ, θ, ε) = (4,−1, 5, 1, 1, 0.02), while the numerical simula-tions were done on the half-line [0, 24] with no-flux boundary conditions. Right panels: Theright half of the profiles of the three stationary 1-pulse profiles obtained for γ = −0.2. Thesecond pulse labelled II is stable, while the first one is unstable. This is confirmed by thenumerically computed action functionals Jnum(Z) (23). That is, JII

num < JInum. Compare

to Figure 2. The third stationary 1-pulse solution is a pulse solution in the strong interac-tion regime (very small width) and has not been studied by the asymptotic methods in thismanuscript. Note that to obtain numerically stability results, we always simulate (1) with(τ, θ) = (1, 1) (so also in the upcoming Figures 5 – 7 and Figure 10).

J (10) due to the change from an indefinite integral to a definite integral andthe fact that the pulse solutions do not asymptote to zero.

Using the above described continuation method, the asymptotically pre-dicted fold scenario for 1-pulse solutions with γ as bifurcation – and con-tinuation – parameter and (α, β,D) = (4,−1, 5) is confirmed numerically,see Figure 3. Moreover, the 1-pulse solution related to the minimizer of theasymptotic action functional (10) is numerically stable (all eigenvalues arenegative), while the maximizer of asymptotic action functional (10) is numer-ically unstable (positive eigenvalue). This is also confirmed by the numericalaction functional (23): the asymptotic minimizer has a smaller numerical ac-tion functional than the asymptotic maximizer. That is, JIInum < JInum, seeFigure 3.

The numerical simulation also reveals the existence of a third unstablestationary 1-pulse solution. This 1-pulse solution has very small width and is nolonger in the semi-strong interaction regime. For example, the fast componentis not close to the background state u+γ (the largest positive root of u− u3 −ε((α + β)u + γ) = 0) near x = 0 as can be seen from the solution displayedin panel III of Figure 3. In other words, this solution can not be capturedwith the GSPT analysis and is therefore not part of the asymptotic resultspresented in this manuscript.


2.2 Stationary symmetric 2-pulse solutions

The existence and stability criteria for stationary symmetric 2-pulse solutionssupported by the original three-component FitzHugh-Nagumo model (1) asdescribed in the second part of Theorem 1 can also be derived by the actionfunctional J (10). The function J(Z2p) will be a function of the two unknownwidths x∗1,2. In particular, 2x∗2 represents the distance between the two sym-metric pulses, while x1 − x2 represents the width of the two pulses, see theright panel of Figure 1.

Lemma 2 The action functional J(Z2p) of a stationary symmetric 2-pulsesolution Z2p (8) is given by

J(Z2p) =8√

2

3ε+ 2εα

(−2 + e−2x

∗1 + e−2x

∗2 + 2e−x

∗1+x

∗2 − 2e−x

∗1−x∗2

)+ 2εβD

(−2 + e−2x

∗1/D + e−2x

∗2/D + 2e−x

∗1/D+x∗2/D − 2e−x

∗1/D−x∗2/D

)+ 4εγ(x∗1 − x∗2) +O(ε

√ε) .

(24)

Proof The proof of Lemma 2 is conceptual the same as the proof of Lemma 1– though algebraically more involved as we have to split the integral of theaction functional over nine different regions

J =

∫ ∞−∞

dx =

9∑i=1

Ii =

5∑i=1

∫I2i−1s

dx+

4∑i=1

∫I2if

dx,

with I2i−1s , I2if given in Theorem 1.The computation of the four fast integrals are identical to the computation

of the fast integrals of the 1-pulse solution Z1p as given in (16). Therefore, weomit the details of these computations

4∑i=1

I2i =

4∑i=1

∫I2if

· dx =8√

2

3ε+O(ε

√ε) .

In other words, the integrals over the fast regions only contribute to the con-stant term of the action functional J(Z2p) as stated in Lemma 2. So, it doesnot contribute (to leading order) to the existence and stability properties ofsymmetric 2-pulse solutions.

The computations of the five slow integrals are more involved than thecomputations of the slow integrals for the 1-pulse solution – though againsimilar in spirit. We get

I1 =

∫I1s

(1

2ε2(U2p)2x + F (U2p)− F (u) +

1


)+

1


)+ εγ(U2p − u)

)dx


=

∫ −x∗1−√ε−∞

(−1

2εα(ex(ex∗1 − e−x∗1 − ex∗2 + e−x

∗2

))−1

2εβ(ex/D

(ex∗1/D − e−x∗1/D − ex∗2/D + e−x

∗2/D)))

dx

=− 1

2εα(

1− e−2x∗1 − e−x∗1+x∗2 + e−x∗1−x∗2

)− 1

2εβD

(1− e−2x∗1/D − e−x∗1/D+x∗2/D + e−x

∗1/D−x∗2/D

)+O(ε

√ε) .

Similarly, we find that to leading order

I3 = 2εγ(x∗1 − x∗2) +1

2εα(−2 + e−2x

∗1 + e−2x

∗2 + 2e−x

∗1+x

∗2 − 2e−x

∗1−x∗2

)+

1

2εβD

(−2 + e−2x

∗1/D + e−2x

∗2/D + 2e−x

∗1/D+x∗2/D − 2e−x

∗1/D−x∗2/D

)I5 = − εα

(1− e−2x∗2 − e−x∗1+x∗2 + e−x

∗1−x∗2

)− εβD

(1− e−2x∗2/D − e−x∗1/D+x∗2/D + e−x

∗1/D−x∗2/D

)I7 = 2εγ(x∗1 − x∗2) +

1

2εα(−2 + e−2x

∗1 + e−2x

∗2 + 2e−x

∗1+x

∗2 − 2e−x

∗1−x∗2

)+

1

2εβD

(−2 + e−2x

∗1/D + e−2x

∗2/D + 2e−x

∗1/D+x∗2/D − 2e−x

∗1/D−x∗2/D

)I9 = − 1

2εα(

1− e−2x∗1 − e−x∗1+x∗2 + e−x∗1−x∗2

)− 1

2εβD

(1− e−2x∗1/D − e−x∗1/D+x∗2/D + e−x

∗1/D−x∗2/D

)Summing up I1 to I9 gives (24) and completes the proof.

Now, the widths x∗1,2 of the stationary symmetric 2-pulse solutions aredetermined by the critical points of J(Z2p) as function of both x∗1 and x∗2. Tothis end, we compute both partial derivatives:

∂

∂x∗1J(Z2p)) =4εγ + 4εα

(−e−2x∗1 − e−x∗1+x∗2 + e−x

∗1−x∗2

)+ 4εβ

(−e−2x∗1/D − e−x∗1/D+x∗2/D + e−x

∗1/D−x∗2/D

)+O(ε

√ε) ,

(25)and

∂

∂x∗2J(Z2p) =− 4εγ + 4εα

(−e−2x∗2 + e−x

∗1+x

∗2 + e−x

∗1−x∗2

)+ 4εβ

(−e−2x∗2/D + e−x

∗1/D+x∗2/D + e−x

∗1/D−x∗2/D

)+O(ε

√ε) .

(26)The critical points of (24) are given by x∗1,2 such that both right-hand sides of(25) and (26) are zero. Upon adding the right-hand sides, we obtain (a multipleof) the first condition of (6) as stated in Theorem 1. While subtracting the


right-hand sides, gives (a multiple of) the first condition of (6) as stated inTheorem 1. In other words, the critical points of the action functional J(Z2p)(given that 0 < x∗2 < x∗1) are given by expressions that coincide with theexistence condition for the widths of the stationary symmetric 2-pulse solutionsas given in (6).

The second derivative test for functions of two variables (see for exampleTheorem 2.1 and Theorem 3.1 in Chapter 8 in [10]) determines the character ofthe critical points of the action functional. It is determined by the determinantof the Hessian of J(Z2p). This determinant D is given by

D =∂2J

∂(x∗1)2∂2J

∂(x∗2)2−(

∂2J

∂x∗1∂x∗2

)2

and we have a local maximum if D > 0 and ∂2J∂(x∗1)

2 < 0, a local minimum if

D > 0 and ∂2J∂(x∗1)

2 > 0 and a saddle point if D < 0.

First, we compute the leading order contributions of the different compo-nents of D

∂2

∂(x∗1)2J(Z2p) = 4εα

(2e−2x

∗1 + e−x

∗1+x

∗2 − e−x∗1−x∗2

)+ 4ε

β

D

(2e−2x

∗1/D + e−x

∗1/D+x∗2/D − e−x∗1/D−x∗2/D

)= 4ε(λ1 + 2f(2x∗1)) ,

(27)

with f defined in (4) and λ1 in (7). Similarly,

∂2

∂(x∗2)2J(Z2p) = 4εα

(2e−2x

∗2 + e−x

∗1+x

∗2 − e−x∗1−x∗2

)+ 4ε

β

D

(2e−2x

∗2/D + e−x

∗1/D+x∗2/D − e−x∗1/D−x∗2/D

)= 4ε(λ1 + 2f(2x∗2)) ,

(28)

and∂2

∂x∗1∂x∗2

J(Z2p) = −4εα(e−x

∗1+x

∗2 + e−x

∗1−x∗2

)− 4ε

β

D

(e−x

∗1/D+x∗2/D + e−x

∗1/D−x∗2/D

)= −4ε(λ1 + 2f(x∗1 + x∗2)) .

(29)

So, we get that D, the determinant of the Hessian of J(Z2p), is to leadingorder given by

D = 16ε2((λ1 + 2f(2x∗1))(λ1 + 2f(2x∗2))− (λ1 + 2f(x∗1 + x∗2))2

). (30)

The connection between the conditions on the Hessian (30) of the action func-tional J(Z2p) and the conditions on the eigenvalues of the symmetric 2-pulsesolution as given in (7) is not direct obvious. However, a closer inspection ofboth expressions reveals that there is actually a one-to-one correspondence.


Lemma 3 A solution pair (x∗1, x∗2) solving (6) is a local minimizer of J(Z2p)

if, and only if, both λ1,2 (7) are positive.

Proof If (x∗1, x∗2) is a local minimizer of J(Z2p), then the Hessian D (30) is

positive and ∂2J∂(x∗1)

2 > 0 (27). As a consequence of (30), this means that also

∂2J∂(x∗2)

2 > 0 (28). So, 2f(2x∗1) + λ1 > 0 and 2f(2x∗2) + λ1 > 0 and therefore

f(2x∗1) + f(2x∗2) + λ1 > 0. This gives

D > 0

=⇒ (λ1 + 2f(2x∗1))(λ1 + 2f(2x∗2)) > (λ1 + 2f(x∗1 + x∗2))2

=⇒ (f(2x∗1) + f(2x∗2) + λ1)2 > (f(2x∗1)− f(2x∗2))2 + (λ1 + 2f(x∗1 + x∗2))2

=⇒ f(2x∗1) + f(2x∗2) + λ1 >√

(f(2x∗1)− f(2x∗2))2 + (λ1 + 2f(x∗1 + x∗2))2

=⇒ λ2 > 0.(31)

Moreover, the above inequality (f(2x∗1)+f(2x∗2)+λ1)2 > (f(2x∗1)−f(2x∗2))2+(λ1 + 2f(x∗1 + x∗2))2 can be rewritten as

λ1 (f(2x∗1) + f(2x∗2)− 2f(x∗1 + x∗2)) > 2(f(x∗1 + x∗2)2 − f(2x∗1)f(2x∗2)

).(32)

By the definition of f , we have that the right-hand side of the above expressionis always positive:

f(x∗1 + x∗2)2 − f(2x∗1)f(2x∗2) =

∣∣∣∣αβD∣∣∣∣ (e−x∗1−x∗2/D − e−x∗1/D−x∗2)2 > 0 .

So, the left-hand side of (32) is also positive. To prove that λ1 > 0 it is thussufficient to show that f(2x∗1)+f(2x∗2)−2f(x∗1+x∗2) > 0. We use the definitionof λ1 to write this term as

f(2x∗1)+f(2x∗2)−2f(x∗1+x∗2) = (f(2x∗1)+f(2x∗2)+λ1)−(f(x∗1−x∗2)+f(x∗1+x∗2)) .

Since f(2x∗1) + f(2x∗2) + λ1 > 0 and λ2 > 0, we have that f(2x∗1) + f(2x∗2) +λ1 > f(x∗1 − x∗2) + f(x∗1 + x∗2), see the definition of λ2 in (7). Consequently,f(2x∗1) + f(2x∗2) − 2f(x∗1 + x∗2) > 0 and we thus also have that λ1 > 0. Thisconcludes the first part of the proof.

If both λ1,2 > 0, then f(2x∗1) + f(2x∗2) +λ1 > 0 and we can use (31) in theother direction to show that the Hessian D (30) is positive. In particular,

λ2 > 0

=⇒ (f(2x∗1) + f(2x∗2) + λ1)2 > (f(2x∗1)− f(2x∗2))2 + (λ1 + 2f(x∗1 + x∗2))2

=⇒ (λ1 + 2f(2x∗1))(λ1 + 2f(2x∗2))− (λ1 + 2f(x∗1 + x∗2))2 > 0

=⇒ D > 0 .

What remains to show is that ∂2J∂(x∗1)

2 > 0 (27) or ∂2J∂(x∗2)

2 > 0 (28), since D > 0.

Since f(2x∗1)+f(2x∗2)+λ1 > 0, we also have that 2 max f(2x∗1), f(2x∗2)+λ1 >


0. In other words, ∂2J∂(x∗1)

2 > 0 or ∂2J∂(x∗2)

2 > 0. This concludes the second part of

the proof.

This again showcases the fact that the action functional as described in thismanuscript yields the essential stability information for the stationary localizedstructures of (1). This information was previously rigorously determined via anEvans function computation by the NLEP approach [19]. While the obtainedstability results via the action functional are not new, they are much easierto compute – when compared to the NLEP approach of [19]. This shows thestrength of the method.

The proof of the above lemma also reveals to following remarkable fact– which has not been proven, to the best of our knowledge, in the literaturebefore.

Lemma 4 The three-component FitzHugh–Nagumo model (1) supports nostable stationary symmetric 2-pulse solutions for α < 0.

Proof Assume that for α < 0 there is a solution pair (x∗1, x∗2) solving (6) such

that both λ1 and λ2 of (7) are positive. Then, the first existence condition of(6) implies that β > 0 and, since D > 1 by assumption,

f(2x∗1) + f(2x∗2)− 2f(x∗1 + x∗2) = α(e−x∗1 − e−x∗2 )2 +

β

D(e−x

∗1/D − e−x∗2/D)2

< α(e−x∗1 − e−x∗2 )2 + β(e−x

∗1/D − e−x∗2/D)2

= 0 ,

see again the first existence condition of (6). However, the condition λ2 > 0implies that

f(2x∗1) + f(2x∗2)− 2f(x∗1 + x∗2)

= (f(2x∗1) + f(2x∗2) + λ1)− (f(x∗1 − x∗2) + f(x∗1 + x∗2)) > 0 .

So, f(2x∗1)+f(2x∗2)−2f(x∗1 +x∗2) is both smaller and larger than zero. Conse-quently, there is no solution pair (x∗1, x

∗2) solving (6) such that both λ1 and λ2

of (7) are positive, i.e. the stationary symmetric 2-pulse solution is unstable.This concludes the proof.

Note the condition that α and β need to have opposite sign for the ex-istence of stationary symmetric 2-pulse solutions was already proven in [15].However, the fact that they are always unstable for α < 0 was not shown inthat manuscript.

2.2.1 Examples

We conclude this section with two examples. First, for (α, β, γ,D) = (2,−1,−0.25, 5), the existence condition (6) (or (25) and (26)) for stationary sym-metric 2-pulse solutions is solved by (x∗1,1, x

∗2,1) ≈ (3.4, 1.7) and (x∗1,2, x

∗2,2) ≈

(5.3, 1.1). The first solution (x∗1,1, x∗2,1) is a local minimizer of J(Z2p) since

D(x∗1,1, x∗2,1)/ε2 ≈ 0.19 > 0 , and λ1 + 2f(2x∗1,1) ≈ 0.19 > 0 .


x1

x2

3.0 3.81.5

2.0

x1

1.0

1.2

5.1 5.5x1

x2

x2

00

2

2

4

4

6

6

2.5

2.5

2.0

2.5

2.5

2.0

2.0

2.8

2.7

2.7

2.3652.365

2.37

2.37

2.365

2.3652.36

2.36

@

@x2

J(Z2p) = 0

@

@x1

J(Z2p) = 0

(J(Z2p) C)/"

Fig. 4 Left panel: Contour plot of the leading order component of (J(Z2p) − C)/ε with

C = 8√2

3(dotted lines), ∂

∂x∗1J(Z2p) = 0 (solid blue line) and ∂

∂x∗2J(Z2p) = 0 (solid red

line) for (α, β, γ,D) = (2,−1,−0.25, 5). Top right panel: Zoom in around the intersection of∂

∂x∗1J(Z2p) = 0 and ∂

∂x∗2J(Z2p) = 0 near (x∗1, x

∗2) = (3.4, 1.7). Bottom right panel: Zoom in

around the intersection of ∂∂x∗1

J(Z2p) = 0 and ∂∂x∗2

J(Z2p) = 0 near (x∗1, x∗2) = (5.3, 1.1). We

observe that the former intersection is a minimizer of the action functional and thus createsa stable symmetric 2-pulse solution, while the latter intersection is a saddle point and thuscreates an unstable symmetric 2-pulse solution.

So, Lemma 3 implies that both λ1,2 of Theorem 1 are positive and that thisstationary symmetric 2-pulse solution is thus stable. In particular, we have

λ1(x∗1,1, x∗2,1) ≈ 0.29 > 0 , λ2(x∗1,1, x

∗2,1) ≈ 0.032 > 0 .

The second solution (x∗1,2, x∗2,2) is a saddle point of J(Z2p) since

D(x∗1,2, x∗2,2)/ε2 ≈ −0.32 < 0 .

Lemma 3 thus implies that this symmetric 2-pulse solution is unstable. Inparticular, λ1,2 (7) are both negative

λ1(x∗1,2, x∗2,2) ≈ −0.0038 < 0 , λ2(x∗1,2, x

∗2,2) ≈ −0.097 < 0 .

See also Figure 4.As a second example, we set (α, β, γ,D) = (−2, 2.4, 1, 5). The existence

condition (6) (or (25) and (26)) are solved by (x∗1,1, x∗2,1) ≈ (3.2, 1.1) and

(x∗1,2, x∗2,2) ≈ (2.3, 1.5). Both of these roots are not minimizers of (24) and thus

yield unstable symmetric 2-pulse solutions by Lemma 3. This is also confirmedby the signs of λ1,2 (7). In particular, the former solution is a saddle-point as

D(x∗1,1, x∗2,1)/ε2 ≈ −0.056 < 0 ,


andλ1(x∗1,1, x

∗2,1) ≈ −0.10 < 0 , λ2(x∗1,1, x

∗2,1) ≈ −0.16 < 0 .

The latter solution is a local maximizer as

D(x∗1,2, x∗2,2)/ε2 ≈ 0.014 > 0 , λ1(x∗1,1, x

∗2,1) + 2A(x∗1,1) ≈ −0.34 < 0 ,

andλ1(x∗1,2, x

∗2,2) ≈ −0.68 < 0 , λ2(x∗1,2, x

∗2,2) ≈ −0.67 < 0 .

Since α < 0, the result that both stationary symmetric 2-pulse solutions areunstable is as expected by Lemma 4.

Using the continuation method discussed in §2.1.1 on both parameter setsconfirms the above mentioned results. Moreover, for both examples we againobserve a fold scenario for stationary symmetric 2-pulse solutions with γ asbifurcation parameter, see Figure 5 and Figure 6.

I

IIstable

unstable

unstable

stable

xx

1

x2

I

II

III

III

1.0

1.0

10

0.4 0.4 0.80 16 24x

1.0

0

0

III

8

U

WV

x1x

2

I

16 241.0

1.0

0

0 8

U

WV

1.0

II

16 24

1.0

0

0 8

U

WV

JInum 1.37

JIInum 0.976

Fig. 5 Left panel: Numerically obtained bifurcation diagram with bifurcation parameterγ versus x∗1 and versus x∗2 of stationary symmetric 2-pulse solutions. The other systemparameters are kept fixed at (α, β,D, τ, θ, ε) = (2,−1, 5, 1, 1, 0.02), while the numerical sim-ulations were done on the half-line [0, 24] with no-flux boundary conditions. The red curvecorresponds to (numerically) stable solutions, while the green curves represent (numerically)unstable branches. The blue dashed curve represents the asymptotically widths for (x∗1, x

∗2)

(6). Right panels: The right half of the profiles of the three stationary symmetric 2-pulseprofiles obtained for γ = −0.25. The second one is stable, while the first one is unstable,which is confirmed by the numerically computed action functionals. That is, JII

num < JInum.

Compare to Figure 4. The third stationary 2-pulse solution is a pulse solution in the stronginteraction regime (very small width) and has not been studied in this manuscript.

3 Stationary N-pulse solutions

The action functional J (10) can also be used to compute the existence andstability criteria for stationary N -pulse solutions ZNP supported by (1) and


x

1.0

10

x

x

I

II

I

I

II

1.0

1.0

0

1.0

1.0

0

16 240

16 24

0

0

8

8

unstable

unstable

unstable

unstable

0.4 0.8 1.20.1

V

U

W

V

U

W

x1x

2

x2

x1

Fig. 6 Left panel: Numerically obtained bifurcation diagram with bifurcation parameterγ versus x∗1 and versus x∗2 of symmetric 2-pulse solutions. The other system parametersare kept fixed at (α, β,D, τ, θ, ε) = (−2, 2.4, 5, 1, 1, 0.02), while the numerical simulationswere done on the half-line [0, 24] with no-flux boundary conditions. All obtained solutionsare (numerically) unstable, as asymptotically predicted in Lemma 4 since α < 0. The bluedashed curve represents the asymptotically widths for (x∗1, x

∗2) (6). Right panels: The right

half of the profiles of the two unstable stationary symmetric 2-pulse profiles obtained forγ = 1.

that asymptotes to u(1, 1, 1) (see Theorem 1). These criteria have – to the bestof our knowledge – not been derived in the literature before.

3.1 Stationary N -pulse solutions

Following the approach of the previous sections, we divide our domain in 2N+1slow regions – which we label I2i−1, for i = 1, . . . , 2N+1 – and 2N fast regions– which we label I2i, for i = 1, . . . , 2N . The fast regions are centered around x∗i ,the position of the front or back of a pulse (note that this notation is slightlydifferent as the notation used for the stationary symmetric 2-pulse solution).Similar to the stationary 1-pulse solution and the stationary symmetric 2-pulsesolution, the action functional over the fast fields are constant and do thereforenot contribute to the existence and stability criteria of an N -pulse solution.

To compute the leading order contributions over the slow fields, we firstobserve that – to leading order – the fast component in the slow regions is givenby u2i−1(x) = (−1)i, for i = 1, . . . , 2N+1. Next, we compute the leading orderof the slow components in these slow fields using GSPT. We follow §2 of [21]to see that these slow components (v2i−1, w2i−1) in the i-th slow region I2i−1are given by

v2i−1(x) = (−1)i +Avi ex +Bvi e

−x ,

w2i−1(x) = (−1)i +Awi ex/D +Bwi e

−x/D ,

for i = 1, . . . 2N + 1 and Av,wi , Bv,wi integration constants. To determine theseconstants, we observe that (v2i−1, w2i−1)(x) and (v2i+1, w2i+1)(x), as well astheir derivatives, should match to leading order when evaluated at x∗i (in the


fast field I2i). That is, for i = 1, . . . , 2N we have

Avi ex∗i +Bvi e

−x∗i = Avi+1ex∗i +Bvi+1e

−x∗i − 2(−1)i ,

Avi ex∗i −Bvi e−x

∗i = Avi+1e

x∗i −Bvi+1e−x∗i ,

Awi ex∗i /D +Bwi e

−x∗i /D = Awi+1ex∗i /D +Bwi+1e

−x∗i /D − 2(−1)i ,

Awi ex∗i /D −Bwi e−x

∗i /D = Awi+1e

x∗i /D −Bwi+1e−x∗i /D ,

Rearranging these terms gives

Avi = Avi+1 − (−1)ie−x∗i , Bvi+1 = Bvi + (−1)iex

∗i ,

Awi = Awi+1 − (−1)ie−x∗i /D , Bwi+1 = Awi + (−1)iex

∗i /D .

Moreover, the slow components in the outer two regions (v1, w1), respectively(v4N+1, w4N+1), should stay bounded when evaluated at respectively x→ −∞and x→∞. This gives, Bv,w1 = 0 = Av,w2N+1. Thus, to leading order the profilesof the slow components in the (2i− 1)-th slow field are given by

v2i−1(x) = (−1)i + ex2N∑j=i

(−1)j+1e−x∗j + e−x

i−1∑j=1

(−1)jex∗j ,

w2i−1(x) = (−1)i + ex/D2N∑j=i

(−1)j+1e−x∗j /D + e−x/D

i−1∑j=1

(−1)jex∗j /D ,

for i = 1, . . . , 2N + 1 and with the convention that the sum over a negativerange (like

∑0j=1) is identical to zero.

Consequently, the action functional contribution over the slow fields can becomputed explicitly. After some tedious – but straightforward – computations,this gives the following result (which we present without proof).

Lemma 5 The action functional J(ZNP) of a N -pulse solution ZNP is givenby

J(ZNP) =εC + 2εγ

(2N∑i=1

(−1)ix∗i

)− 2εα

N +

2N∑i,j=1,i<j

(−1)i+jex∗i−x∗j

− 2εβD

N +

2N∑i,j=1,i<j

(−1)i+jex∗i /D−x∗j /D

,

(33)where C ∈ R is a particular O(1)-constant coming from the fast fields that canbe determined explicitly.

As a result, the location of the interfaces, i.e. the fronts and backs, and thusthe complete profiles of stationary N -pulse solutions are determined by thecritical points of J(ZNP) (33) and these stationary N -pulse solutions are stableif and only if the corresponding critical point is a local minimum (since the


1

1

0816 1624 248x

0

W

V U

y1

y3

y2

Fig. 7 A numerically obtained unstable stationary asymmetric 2-pulse solution of (1). Thesystem parameters are (α, β, γ,D, ε) = (−4, 5, 0.66658, 2, 0.02) and the numerical simula-tions was done on the domain [−24, 24] with no-flux boundary conditions. The obtainedwidths between the interfaces are (y∗1 , y

∗2 , y∗3) ≈ (2.7629, 0.8142, 1.3872) and these are in

perfect agreement with the asymptotically predicted widths, see the example after Lemma 6.

essential spectrum for stationary N -pulse solutions asymptoting to u(1, 1, 1)is the same as the essential spectrum for stationary 1-pulse solutions and thuslies well in the left half plane [19]).

Formal Result 1 The location of the interfaces of stationary N -pulse solu-tions are determined by the critical points of J(ZNP) (33). That is, by thesolutions x∗i , i = 1, . . . , 2N with x∗i < x∗i+1 of

J(ZNP)

∂x∗i= 0

2N

i=1

. (34)

A solution of (34) is stable if, and only if, it is a local minimum of J(ZNP).That is, we have a local minimum if, and only if, the Hessian of J(ZNP) ispositive definite [10].

Formal Result 1 can, in principle, be made rigorous with the GSPT techniquesand NLEP techniques of respectively [15] and [19]. These computations willbe straightforward – but extremely tedious – and we decided not to pursuethis direction.

3.2 Example: stationary asymmetric 2-pulse solutions

Theorem 1 discusses the existence and stability properties of stationary sym-metric 2-pulse solutions supported by (1) and we reproduced these results withthe action functional in §2.2. However, (1) potentially also posses stationaryasymmetric 2-pulse solutions Za2p. That is, stationary 2-pulse solutions wherethe width of the two pulses are different, see Figure 7 for an example of anumerically obtained stationary asymmetric 2-pulse solution.


Lemma 5 yields that the action functional of such a stationary asymmetric2-pulse solutions is given by

J(Za2p) =εC + 2εγ (x∗4 − x∗3 + x∗2 − x∗1)− 2εα(

2− e−(x∗2−x∗1) + e−(x∗3−x∗1)

−e−(x∗4−x∗1) − e−(x∗3−x∗2) + e−(x∗4−x∗2) − e−(x∗4−x∗3)

)− 2εβD

(2− e−(x∗2−x∗1)/D + e−(x

∗3−x∗1)/D − e−(x∗4−x∗1)/D

−e−(x∗3−x∗2)/D + e−(x∗4−x∗2)/D − e−(x∗4−x∗3)/D

),

(35)with x∗i the location of the interfaces of the two pulses with the assumptionx∗1 < x∗2 < x∗3 < x∗4. In other words, x∗i is the location of i-th fast field. Becauseof translation invariance, we can rewrite the action functional as function ofonly three variables: the distances between the fast fields. In particular, weintroduce y∗i := x∗i+1 − x∗i for i = 1, 2, 3 and observe that y∗i > 0 by definitionand y∗1 6= y∗3 – since we are looking for asymmetric 2-pulse solutions. Now, wecan write (35) as

J(Za2p) =εC + 2εγ (y∗3 + y∗1)− 2εα(

2− e−y∗1 + e−(y∗1+y

∗2 )

−e−(y∗1+y∗2+y∗3 ) − e−y∗2 + e−(y∗2+y

∗3 ) − e−y∗3

)− 2εβD

(2− e−y∗1/D + e−(y

∗1+y

∗2 )/D − e−(y∗1+y∗2+y∗3 )/D

−e−y∗2/D + e−(y∗2+y

∗3 )/D − e−y∗3/D

).

(36)

Taking the partial derivatives of the above expression with respect to y∗i andequating them to zero gives the existence condition for stationary asymmetric2-pulse solutions

−γ = αe−y∗1

(−1 + e−y

∗2 − e−(y∗2+y∗3 )

)+ βe−y

∗1/D

(−1 + e−y

∗2/D − e−(y∗2+y∗3 )/D

),

0 = αe−y∗2

(−1 + e−y

∗1 + e−y

∗3 − e−(y∗1+y∗3 )

)+ βe−y

∗2/D

(−1 + e−y

∗1/D + e−y

∗3/D − e−(y∗1+y∗3 )/D

),

−γ = αe−y∗3

(−1 + e−y

∗2 − e−(y∗1+y∗2 )

)+ βe−y

∗3/D

(−1 + e−y

∗2/D − e−(y∗1+y∗2 )/D

).

(37)

We write the second equation of (37) as

α =− β e−y∗2/D

(−1 + e−y

∗1/D + e−y

∗3/D − e−(y∗1+y∗3 )/D

)e−y

∗2

(−1 + e−y

∗1 + e−y

∗3 − e−(y∗1+y∗3 )

)=− β e

−y∗2/D(1− e−y∗1/D

) (1− e−y∗3/D

)e−y

∗2 (1− e−y∗1 ) (1− e−y∗3 )

,

(38)


to observe that α and β necessarily have opposite signs as the above fraction isalways positive (remember that y∗i > 0), see Remark 1. Subtracting the thirdequation from the first equation yields

α(e−y∗1 − e−y∗3 )

(1− e−y∗2

)= −β(e−y

∗1/D − e−y∗3/D)

(1− e−y∗2/D

).

Observe that both sides of the above expression are unequal to zero sincey∗1 6= y∗3 (unless of course α = 0 or β = 0). So, we get

α = −β (e−y∗1/D − e−y∗3/D)

(1− e−y∗2/D

)(e−y

∗1 − e−y∗3 ) (1− e−y∗2 )

. (39)

Comparing (38) with (39) we get that β = 0 or

(e−y∗1/D − e−y∗3/D)

(1− e−y∗2/D

)(e−y

∗1 − e−y∗3 ) (1− e−y∗2 )

=e−y

∗2/D

(1− e−y∗1/D

) (1− e−y∗3/D

)e−y

∗2 (1− e−y∗1 ) (1− e−y∗3 )

.

(40)The first case implies that α = β = γ = 0, which would imply that stationaryasymmetric 2-pulse solutions do not exist.

So, for stationary asymmetric 2-pulse solutions to exist, we need (40) tohold. While it is too complicated to make general statements about (40), it iseasy to verify that

y∗3 = 2 log

(1− e−(y∗1+y∗2 )/2e−y

∗2/2 − e−y∗1/2

)> 0 , (41)

solves (40) for D = 2 as long as y∗1 > y∗2 (since otherwise the denominator ofthe above fraction is non-positive). For the 2-pulse solution to be asymmetricy3 should differ from y1. So, we necessarily need

y∗1 6= 2 log

(e−y

∗2/2

1−√

1− e−y∗2

). (42)

Solving the first equation of (37) with D = 2 and with α given in terms of βby (38) (or equivalently by (39)), yields

γ = βe−y∗1/2

(e−y

∗1/2(−1 + e−y

∗2 − e−(y∗2+y∗3 )

)(e−y

∗1/2 + e−y

∗3/2)

(1 + e−y

∗2/2) + 1− e−y∗2/2 + e−(y

∗2+y

∗3 )/2

).

(43)with y∗3 given by (41). In summary, for D = 2 and for given β, y∗1 and y∗2(with y∗1 > y∗2 and such that (42) holds), we get that a stationary asymmetric2-pulse solution exists if and only if (38) (with D = 2), (41) and (43) hold.

We use the second derivative test to determine the stability of this sta-tionary asymmetric 2-pulse solution. In particular, we have a local minimumif, and only if, the Hessian of J(Za2P) evaluated at a critical point of J ispositive definite, e.g. [10]. We use the following lemma to determine whetherthe Hessian is positive definite or not.


Lemma 6 (Theorem 2.2 in Chapter 8 of [10]) The Hessian H(f), f :Rn → R at a critical point of f is positive definite if, and only if,

det (Hj(f)) > 0 , for 1 ≤ j ≤ n . (44)

Here, Hj(f) is the j × j matrix formed by the first j rows and columns of theHessian H(f).

Moreover, the Hessian is negative definite if, and only if,

(−1)j det (Hj(f)) > 0 , for 1 ≤ j ≤ n , (45)

while the Hessian is indefinite if (44) and (45) do not hold and det (Hj(f)) 6=0, for 1 ≤ j ≤ n.

For example, if we set β = 5, y∗1 = 2 log 2 ≈ 1.3863, and y∗2 = 2 log (3/2) ≈0.81093 (note that (42) holds), we get from (38), (41) and (43) that y∗3 =4 log 2 ≈ 2.7759, α = −4 and γ = 2

3 . Thus, we have an asymmetric 2-pulsesolution for (α, β, γ,D) = (−4, 5, 23 , 2) with widths between the interfaces ofrespectively, 2 log 2, 2 log (3/2) and 4 log (2). See Figure 7 for a numericallyobtained stationary asymmetric 2-pulse solution for very similar parametervalues and with – after interchanging the role of y∗1 and y∗3 (which is alloweddue to the symmetry x 7→ −x of the problem) – very similar widths. Thisstationary asymmetric 2-pulse solution is however unstable since the Hessian ofthe action functional JZa2p (36) at the above stated locations y∗1,2,3 is indefinite– and thus not positive definite. In particular, the Hessian (divided by ε) isgiven by

H(JZa2p) =

112 − 5

121336

− 512 − 5

4 − 14

1336 − 1

412

,

and det (H2(JZa2p)) < − 5

18 < 0, i.e. the minor M33(H(JZa2p)) has a negative

determinant. So, by Lemma 6 the Hessian is not positive definite and there-fore this critical point of the action functional is not a local minimizer. Conse-quently, Formal Result 1 implies that the stationary asymmetric 2-pulse solu-tion is unstable. Moreover, since det (H1,3(JZa2p)) 6= 0 and det (H2(JZa2p)) >0, we actually have that the Hessian is indefinite, so the stationary asymmetric2-pulse solution is also not a local maximizer of the action functional.

A more thorough investigation of the second derivatives of the action func-tional J(Za2p) (36) for D = 2 and arbitrary given β, y∗1 , and y∗2 (with β 6= 0,y∗1 > y∗2 and such that (42) holds) and with α, y∗3 and γ given by respectively(38), (41) and (43), reveals that all entries of the Hessian matrix are multiplesof β. Consequently, the stability results for general β can be easily derived fromthe results for β = 1. In particular, if a critical point of the action functionalassociated to a stationary asymmetric 2-pulse solution is a local minimizer ofthe action functional for β = 1 and given y∗1,2 (and with α, γ and y∗3 given byrespectively (38), (41) and (43)), then the corresponding stationary asymmet-ric 2-pulse solution for 0 < β 6= 1 is also a local minimizer. In contrast, the


corresponding stationary asymmetric 2-pulse solution for 0 > β is a local max-imizer. If the Hessian is indefinite for β = 1, then the Hessian will be indefinitefor all β. Setting β = 1 (and D = 2), reduces the variables in the Hessian totwo: y∗1 and y∗2 . Next, we the introduce Y1,2 := e−y1,2/2 and write the Hes-sian in terms of the new variables. Using a Findroot procedure in for exampleMathematica with variables Y1,2, reveals that the determinant of H3 = H isalways positive, while the determinant of H1, respectively H2, changes signat the curve f1(Y1, Y2) = 0, respectively f2(Y1, Y2) = 0, see Figure 8. As itturns out, both determinants of H1 and H2 are never simultaneously positive– again see Figure 8. So, for a given arbitrary β, we never expect to have alocal minimizer since, for β > 0, we never have that detHi > 0 for j = 1, 2, 3,while for β negative, we have det (H3) < 0.

Y1 = ey1/2

Y2 = ey2/2

Y1 = Y3

0 0.5 1

0.5

1

f2(Y1, Y2) = 0

f1(Y1, Y2) = 0

f1 > 0

f2 < 0

Y1 = Y2

Fig. 8 The Hessian associated to a stationary asymmetric 2-pulse solution is never positivedefinite for D = 2, β = 1 (without loss of generality) and with α, y∗3 and γ given byrespectively (38), (41) and (43). The determinant of H3(JZa2p ) > 0 – as long as Y1 6= Y3(42) (which is indicated by the dotted line). The determinant of H1(JZa2p

), respectivelyH2(JZa2p ), changes sign at f1 = 0 (blue curve), respectively f2 = 0 (red curve). In particular,det (H1(JZa2p

)) > 0 above the blue curve, while det (H2(JZa2p)) > 0 below the red curve.

Consequently, there is no region in (Y1, Y2)-space where all three determinants are positive.So, Formal Result 1 combined with Lemma 6 implies that the stationary asymmetric 2-pulse

solutions are unstable. Note that we used the rescaled variables Y1,2 = e−y∗1,2/2 in the plotsand the gray area indicates the area where 0 < Y1 < Y2 < 1, i.e y∗2 < y∗1 .

This leads to the following conjecture:

Conjecture 1 For D = 2, all stationary asymmetric 2-pulse solutions are un-stable.

Note that for D = 2 and y∗1 > y∗2 such that y∗1 = 2 log

(e−y∗2/2

1−√

1−e−y∗2

), i.e. y∗3 =

y∗1 (see (42)), the determinant of the Hessian matrix of the action functionalJ(Za2p) (36) actually becomes zero (since the first and third row are identical)


and consequently the Hessian has at least one zero eigenvalue and Lemma 6is inconclusive.

3.3 Example: stationary symmetric 3-pulse solutions

For stationary symmetric 3-pulse solutions, Lemma 5 gives the following actionfunctional

J(Z3p) =εC − 2εα(

3− e−2x∗1 − e−2x∗2 − e−2x∗3 − 2e−(x∗1−x∗2)

+2e−(x∗1+x

∗2) + 2e−(x

∗1−x∗3) − 2e−(x

∗1+x

∗3) − 2e−(x

∗2−x∗3) + 2e−(x

∗2+x

∗3))

− 2εβD(

3− e−2x∗1/D − e−2x∗2/D − e−2x∗3/D − 2e−(x∗1−x∗2)/D

+2e−(x∗1+x

∗2)/D + 2e−(x

∗1−x∗3)/D − 2e−(x

∗1+x

∗3)/D − 2e−(x

∗2−x∗3)/D

+2e−(x∗2+x

∗3)/D

)+ 4εγ (x∗1 − x∗2 + x∗3) ,

with x∗i the location of the fast fields such that −x∗1 < −x∗2 < −x∗3 < 0 < x∗3 <x∗2 < x∗1. Taking the partial derivatives of the above expression with respectto x∗i and equating them to zero gives the existence condition for stationarysymmetric 3-pulse solutions

−γ =α(−e−2x∗1 − e−(x∗1−x∗2) + e−(x

∗1+x

∗2) + e−(x

∗1−x∗3) − e−(x∗1+x∗3)

)+ β

(−e−2x∗1/D − e−(x∗1−x∗2)/D + e−(x

∗1+x

∗2)/D + e−(x

∗1−x∗3)/D

−e−(x∗1+x∗3)/D),

γ =α(−e−2x∗2 + e−(x

∗1−x∗2) + e−(x

∗1+x

∗2) − e−(x∗2−x∗3) + e−(x

∗2+x

∗3))

+ β(−e−2x∗2/D + e−(x

∗1−x∗2)/D + e−(x

∗1+x

∗2)/D − e−(x∗2−x∗3)/D

+e−(x∗2+x

∗3)/D

),

−γ =α(−e−2x∗3 − e−(x∗1−x∗3) − e−(x∗1+x∗3) + e−(x

∗2−x∗3) + e−(x

∗2+x

∗3))

+ β(−e−2x∗3/D − e−(x∗1−x∗3)/D − e−(x∗1+x∗3)/D + e−(x

∗2−x∗3)/D

+e−(x∗2+x

∗3)/D

).

(46)

Proceeding in a similar fashion as in the previous example, we add the firsttwo equations to get

α =− β g(x∗1/D, x∗2/D, x

∗3/D)

g(x∗1, x∗2, x∗3)

, (47)

with g(x, y, z) := −(e−x − e−y)2 − (e−y − e−x)(ez − e−z) and g < 0 sincex > y > z > 0. So, it follows directly that for symmetric 3-pulse solutions to


exist we – again – necessarily need to have that α and β have opposite sign,see Remark 1.

Equating the second and third equation in (46) gives

α =− β h(x∗1/D, x∗2/D, x

∗3/D)

h(x∗1, x∗2, x∗3)

, (48)

with h(x, y, z) := −(e−y−e−z)2 +e−x((ey+e−y)− (ez +e−z)) . So, comparing(47) with (48) gives β = 0 (and hence stationary symmetric 3-pulse solutionsdo not exist) or

g(x∗1, x∗2, x∗3)h(x∗1/D, x

∗2/D, x

∗3/D) = h(x∗1, x

∗2, x∗3)g(x∗1/D, x

∗2/D, x

∗3/D) .

(49)As it turns out, this equation can have solutions. However, we are not able(as in case of the asymmetric 2-pulse solution) to write it down explicitly andwe are restricted to a numerical solver to determine the roots. For example,for D = 2 and x∗3 = 2 and x∗2 = 4, we get that x∗1 ≈ 6.323 solves (49) (notethat there are more solutions to (49)). Using (47) this gives α = −1.891β –so α and β indeed have opposite sign – and the first equation of (46) thengives γ ≈ 0.0489β. A numerical investigation of the Hessian related to thissymmetric 3-pulses solutions shows that all terms in the Hessian are multiplesof β (as expected). So, again, all the stability information can be obtainedfrom the β = 1-case. For β = 1, we have det (H1(J(Z3p))) ≈ −0.110 <0 ,det (H2(J(Z3p))) ≈ 0.0208 > 0 , and det (H3(J(Z3p))) ≈ 0.00126 > 0 . So,by Formal Result 1 and Lemma 6 the symmetric 3-pulses solution is unstablefor both β < 0 and β > 0.

4 Periodic solutions on [−L,L]

In this section, we determine the existence and stability properties of station-ary spatially-periodic 1-pulse solutions to (1) on the bounded interval [−L,L]with periodic boundary conditions using an action functional restricted to theperiodic domain [−L,L]. That is, we assume that the spatial period of thesolution is 2L and – without loss of generality – we furthermore assume thatthe solution is symmetric in x = 0. Using the standard GSPT machinery asoutlined in the introduction, it is straightforward to determine that the profileof a stationary spatially-periodic 1-pulse solution on [−L,L] and half-width


x∗ (with 0 < x∗ < L) is given by

Zper(x) =

Uper(x)Vper(x)Wper(x)

=

−1

2 sinh (x∗)sinh (L) cosh (x+ L)− 1

2 sinh (x∗/D)sinh (L/D) cosh ((x+ L)/D)− 1

χI−s

+

tanh(

12ε

√2 (x+ x∗)

)sinh (2x∗−L)

sinh (L)sinh ((2x∗−L)/D)

sinh (L/D)

χI−f

+

1

2 sinh (x∗−L)sinh (L) cosh (x) + 1

2 sinh ((x∗−L)/D)sinh (L/D) cosh (x/D) + 1

χI0s

+

− tanh(

12ε

√2 (x− x∗)

)sinh (2x∗−L)

sinh (L)sinh ((2x∗−L)/D)

sinh (L/D)

χI+f

+

−1

2 sinh (x∗)sinh (L) cosh (x− L)− 1

2 sinh (x∗/D)sinh (L/D) cosh ((x− L)/D)− 1

χI+s .

(50)

Here, with slight abuse of notation, χI±,0s

and χI±fare the characteristic func-

tions for the three slow intervals I−s = [−L,−x∗ −√ε), I0s = (−x∗ +√ε, x∗ −√

ε), I+s = (x∗+√ε, L], and the two fast intervals I−f = [−x∗−√ε,−x∗+

√ε],

I+f = [x∗ −√ε, x∗ +√ε].

To determine the width of a stationary spatially-periodic 1-pulse solutionon [−L,L], we restrict the indefinite action functional J (10) to the periodicdomain [−L,L]. That is,

Jper(u) =

∫ L

−L

(1

2ε2u2x + F (u)− F (u) +

1

2εα(uL1u− u2) (51)

+1


)dx ,

with F (u) still the antiderivative given in (11), L1,2 as before and u :=u(±L) = −1 +O(ε). A direct computation of this integral – which we presentwithout proof – yields the following result.

Lemma 7 The action functional Jper(Zper) of a stationary spatially-periodic1-pulse solution Zper (50) on [−L,L] and width 2x∗ is to leading order, givenby

J(Zper) =4εαsinh (x∗ − L) sinh (x∗)

sinh (L)+ 4εβD

sinh ((x∗ − L)/D) sinh (x∗/D)

sinh (L/D)

+ 4εγx∗ +4√

2

3ε .

(52)


stable

x

L

(↵,, , D) = (3, 2, 1, 5)

5 10 15 20

0.5

1

1.5

2

Fig. 9 The half-width x∗ of stationary spatially-periodic 1-pulse solutions on [−L,L] asfunction of L for given (α, β, γ,D) = (3, 2, 1, 5). For every fixed L there exists a unique sta-tionary spatially-periodic 1-pulse solution. Moreover, for L→∞, the half-width approachesthe half-width as determined for the stationary 1-pulse solutions as discussed in Theorem 1(dotted line). The gray area indicates the region where x∗ > L.

To determine the existence condition, we determine the critical points of (52).They are given by the solution of

dJ(Zper)

dx∗> 0 =⇒ α

sinh (L− 2x∗)sinh (L)

+ βsinh ((L− 2x∗)/D)

sinh (L/D)= γ . (53)

Observe that this is the same condition that you would get if you computedthe Melnikov-type integral as in [15]. These solutions are stable if

d2J(Zper)

(dx∗)2= 0 =⇒ α

cosh (L− 2x∗)sinh (L)

+β

D

cosh ((L− 2x∗)/D)

sinh (L/D)> 0 ,

and unstable otherwise. For L→∞, the existence, respectively stability, con-dition converge to the existence, respectively stability, condition for the local-ized stationary 1-pulse solution as discussed in Theorem 1. Note that pointspectrum of these periodic structures on bounded domains can – in general –be computed rigorously with the Singular Limit Eigenvalue Problem (SLEP)approach for singularly perturbed reaction-diffusion equations [30,29,31] andnote that this SLEP approach is closely related to the Evans function [31].

From the existence and stability conditions, we immediately get that – forgiven L – stationary spatially-periodic 1-pulse solutions exist, are unique andare stable if α, β > 0 and 0 < |γ| < α + β. See Figure 9 for an example.So, even for α, β > 0 and γ < 0 there are stable stationary spatially-periodic1-pulse solutions (note that this is not possible for localized stationary 1-pulsesolution). Similarly, stationary spatially-periodic 1-pulse solutions exist, areunique and are unstable if α, β < 0 and 0 < |γ| < |α+ β|. For α, β of oppositesign – see Remark 1 – the situation is more complicated. For example, (53) canhave three solutions as is shown in the left panels of Figure 10. In particular,we observe a fold bifurcation at L = L∗ for increasing domain size L, changingthe number of solutions of (53) from one (for L < L∗) to three (for L > L∗).Numerical simulations of the periodic ODE problem with ε = 0.01 confirm theasymptotic findings, see the right panels of Figure 10.


x

I

II

1.0

1.0

0

0 84

1.0

1.0

0

0 84

1.0

1.0

0

0 84

x

x

III

(↵,, , D) = (3,1, 0.25, 5)

1

0.1

x

L2 4 6 8 10

10

unstable

stable

stable

2 4 6 8 10 12 L

x

1

0.1

10

unstable

unstable

stable

(↵,, , D) = (3, 1, 0.2, 5)

III

III

WV

U

W

V

U

WV

U

x

L

L

JIper,num 4.67

JIIper,num 3.96

JIIIper,num 9.97

Fig. 10 Left panels: Half-widths x∗ of stationary spatially-periodic 1-pulse solutions on[−L,L] as function of L for α and β with opposite sign. In particular, in the top left panel weset (α, β, γ,D) = (−3, 1, 0.2, 5), while the bottom left panel has (α, β, γ,D) = (3,−1, 0.25, 5).In both cases, we observe a fold bifurcation at a particular L∗-value: for L < L∗ only onesolution exists, while for L > L∗, the existence condition (53) has three solutions. ForL → ∞, some of the half-widths approach the half-widths for stationary 1-pulse solutionsasymptoting to u(1, 1, 1) (dotted lines). The other solutions actually approach the half-widths for stationary 1-pulse solutions asymptoting to u(1, 1, 1), with u the solution of (3)near +1. The existence and stability results of these type of localized structures followimmediately from the symmetry of (1) and we refer to [15] for more details. Note thata log-scale is used on the vertical axes (for the sake of presentation) and that the grayarea indicates the region where x∗ > L. Right panels: The right half of the profiles of thethree – numerically obtained – stationary spatially-periodic 1-pulse solutions for L = 8 and(α, β, γ,D, ε) = (−3, 1, 0.2, 5, 0.01). So, corresponding to the parameter set in the top leftpanel. The numerically computed widths, x∗I = 0.9466, x∗II = 2.810, and x∗III = 7.442, arein excellent correspondence with the widths obtained from the asymptotic analysis, thatis, the solutions of (53) for this parameter set are given by x∗I ≈ 0.9465, x∗II ≈ 2.793,and x∗III ≈ 7.441. Furthermore, we observe that the action functional for the (numerically)stable solution in panel II indeed has the lowest value – as predicted by the asymptoticanalysis.

Of course, we could also determine the existence and stability properties ofstationary spatially-periodic symmetric 2-pulse solutions, asymmetric 2-pulsesolutions, and N -pulse solutions on [−L,L]. However, these derivations willbe very similar in spirit as in the rest of this manuscript and only the algebrawill differ (slightly). Therefore, we decided not to pursue this direction.

Remark 2 The existence of stationary spatially-periodic 1-pulse solutions sup-ported by (1) on [−L,L] immediately implies the existence of stationaryspatially-periodic pulse solutions with period 2L supported by (1) on R. Thegeneralisation of stability results is, however, more subtle since the action


functional approach does not provide information on the location of the fullspectrum. For localized structures of (1) – like the solutions studied in thefirst part of the manuscript – this is not a problem since the action functionalapproach finds the crucial part of the spectrum determining the fate of thestructures. The spectrum of a stationary spatially-periodic pulse solution on Ris more complicated. For example, it necessarily connects to the origin, see forinstance [35,40], and the stability of such a spatially-periodic pulse solution isthus also determined by the exact location of this “small spectrum” [35]. Theaction functional approach does not provide information regarding the smallspectrum – nor to other parts of the spectrum that can be of influence on thestability for these type of solutions. So, the action functional approach can notbe used to determine stability of stationary spatially-periodic pulse solutionson R.

5 Summary and outlook

5.1 Summary

In this manuscript, we introduced a new approach to study stationary localizedstructures in the singularly perturbed three-component FitzHugh–Nagumomodel (1). This new approach is a combination of the geometrical singularperturbation approach used in [15] for the same model (1) with a variationalapproach of Chen et al. [2,3,6,7] for a non-singularly perturbed two-componentFitzHugh–Nagumo model (13). In particular, we computed the values of anaction functional J (10) for several different stationary pulse solutions – ob-tained by standard geometrical singular perturbation techniques – with unde-termined widths. The critical points of this action functional J for a particularstationary pulse solution – say a stationary 1-pulse solution for convenience –then coincide with the potential widths of this stationary 1-pulse solution. Inother words, the critical points of J yield the existence condition for station-ary 1-pulse solution. In addition, if a critical point is a local minimizer of J– which, in general, can be determined by a second derivative test – then thestationary 1-pulse solution with the corresponding width is stable. In otherwords, the local minimizers of J yield the stability condition for stationary1-pulse solution.

In the first part of the manuscript, we validated the new approach with theaction functional and showed that the new approach reproduces the results of[15,19] with respect to stationary 1-pulse solutions and stationary symmetric2-pulse solutions. In particular, we derived the existence and stability condi-tions of these stationary localized structures as previously determined in [15,19] and as stated in Theorem 1. In this process, we also proved a previouslyunknown result relating to the stability of symmetric 2-pulse solutions, seeLemma 4.

In the second part of the manuscript, we used the new approach to studythe existence and stability of general N -pulse solutions, see Formal Result 1,


and we showed that the more exotic stationary localized structures such asstationary asymmetric 2-pulse solutions and stationary 3-pulse solutions onlyexists in very specific parameter regimes. We also used the new approach tostudy stationary spatially-periodic 1-pulse solutions on the bounded interval[−L,L].

In a sense, the action functional (formally) replaces the Melnikov-type in-tegral of [15] – which is a standard computation used for geometrical singularperturbation problems like (1) – to determine the existence condition of sta-tionary localized pulse solutions supported by (1), and it (formally) replacesthe NLEP approach of [19] – which is a standard technique to compute pointspectrum for geometrical singular perturbation problems like (1) – to deter-mine the stability condition of stationary localized pulse solutions supportedby (1). This new approach – while only formal – has the advantage that itis computational more efficient, especially with respect to the derivation ofthe stability criteria. Furthermore, in [21] it was shown that pulse-like initialconditions of (1) behave like a gradient flow and the new approach can also beseen as a direct approach to compute (a scaled version of) this gradient flow.

5.2 Outlook

5.2.1 Travelling localized structures and planar localized structures

As eluded to in the introduction, localized structures supported by the sin-gularly perturbed three-component FitzHugh–Nagumo model (1) has beenstudied extensively. Besides stationary pulse solutions, existence and stabilityconditions for travelling pulse solutions [15,19], travelling front solutions [21],and planar stationary – and travelling – spots and front solutions [22–24] havebeen derived in a rigorous and/or formal fashion. The action functional usedin this manuscript is not suitable to study these travelling objects. However,an existence result for travelling pulse solutions in a variational setting of (13)was established in [3]. In that manuscript, a mono-stable system is treated,while – due to the asymptotic scaling – (1) is a bi-stable system. Similarly,an action functional has been used in [4,7,8,39] to obtain existence results forstanding waves of (13) in higher dimensions – so with ·xx replaced by ∆·. It is achallenge to see whether the travelling pulse solutions and the planar localizedstructures of (1) can also be studied with an adjusted action functional.

5.2.2 Heterogeneous Media

Recently, a heterogeneous version of (1) modelling an imperfection in thebackground of the medium received severe attention [16,33,20,45,46]. In theformulation of this manuscript – and similar to [20] – the heterogeneous version


of (1) is given byUt = ε2Uxx + U − U3 − ε(αV + βW + γ(x)) ,

τVt = Vxx + U − V ,θWt = D2Wxx + U −W ,

(54)

with 0 < ε 1;D > 1; τ, θ > 0; (x, t) ∈ R× R+;α, β ∈ R, and

γ(x) =

γ1 for x < 0 ,γ2 for x ≥ 0 ,

(55)

with γ1,2 ∈ R. Here, all parameters are again assumed to be O(1) with respectto ε. Due to the heterogeneity the problem is no longer translation invariant.Therefore, a stationary pulse solution to the heterogeneous model (54) is typi-cally isolated and does not come as a family of solutions as in the homogeneouscase. In other words, the heterogeneity is able to pin the stationary localizedstructures [20]. This pinning can actually happen at the defect (near x = 0)or away from the defect (O(1) distance away from x = 0), see Figure 11 and[20]. The pinned solutions near the defect have been studied in [20] and thereexistence and stability criteria have been formally derived. However, currentlythere are virtual no asymptotic results with respect to pinned solutions awayfrom the defect – only that they exist under certain parameter conditions [16].For example, it is unknown what the distance between the pulse solution andthe defect is for a given defect (55). It is a challenge to see whether this dis-tance can be determined by a higher order computation of the adjusted actionfunctional

Jd(Z) =

∫ ∞−∞

(1

2ε2u2x + F (u)− F (uγ(x)) +

1

2εα(uL1u− (uγ(x))

2) (56)

+1

2εβ(uL2u− (uγ(x))

2) + εγ(x)(u− uγ(x)))dx ,

with antiderivative F and L1,2 as before and uγ(x) the solution of (3) nearu = −1 with γ(x) = γ1 for x < 0 and with γ(x) = γ2 for x ≥ 0, see againFigure 11.

Acknowledgements PvH thanks the National Changhua University of Education in Tai-wan, the National Center for Theoretical Sciences in Taiwan and Tohoku University in Japanfor their hospitality. CNC is grateful for the warm hospitality of Queensland University ofTechnology in Australia.

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