Pinned Solutions in a Heterogeneous ThreeComponent FitzHugh–Nagumo Model
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Abstract
We analyse pinned front and pulse solutions in a singularly perturbed threecomponent FitzHugh–Nagumo model with a small jumptype heterogeneity. We derive explicit conditions for the existence and stability of these type of pinned solutions by combining geometric singular perturbation techniques and an action functional approach. Most notably, in certain parameter regimes we can explicitly compute the pinning distance of a localised solution to the defect.
Keywords
Reaction–diffusion equations Defects Calculus of variations Singular perturbations Existence Stability Localised defect solutionsMathematics Subject Classification
34A34 34A36 34C37 35B25 35B35 35K57 49J401 Introduction
For completeness, we first recall the definitions of the big\(\mathcal {O}\)notation and the big\(\varTheta \)notation.
Definition 1

\(h_1 = \mathcal {O}(\phi _1)\) as \(\varepsilon \downarrow \varepsilon _0\) if there are constants \(k_0>0\) and \(\varepsilon _1\) such that \(h_1(\varepsilon ) \le k_0\phi _1(\varepsilon )\) for \(\varepsilon _0< \varepsilon < \varepsilon _1\).

\(h_2 = \varTheta (\phi _2)\) as \(\varepsilon \downarrow \varepsilon _0\) if there are constants \(k_0, k_1>0\) and \(\varepsilon _1\) such that \(k_1\phi _2(\varepsilon ) \le h_2(\varepsilon ) \le k_0\phi _2(\varepsilon )\) for \(\varepsilon _0< \varepsilon < \varepsilon _1\).
By combining Geometrical Singular Perturbation Theory (GSPT) [25, 31, 32] with an action functional approach [52], we derive the following result related to the the existence and stability of local defect front and pulse solutions.
Main Result 1
Let \(\varepsilon \) be small enough and let \(\tau \) and \(\theta \) be bounded by some \(\mathcal {O}(1)\)constant.^{1}
To understand the implications of Main Result 1, we take a closer look at the existence conditions (3) and (5), and stability conditions (4) and (6). If \(\alpha \) and \(\beta \) have the same sign, then f is monotonic. Consequently, the existence condition (3) for local defect front solutions pinned to the left of the defect has at most one solution, while the existence conditions (5) for local defect pulse solutions pinned to the right of the defect has no solutions. In other words, if \(\alpha \) and \(\beta \) have the same sign—and if the other system parameters are chosen appropriately—then Main Result 1 gives the existence of a local defect front solutions pinned to the left of the defect. After noting that \(g(x_d) = f'(x_d)\), we get from (4) that this solution is stable only if \(\alpha \) and \(\gamma _2\) have the same sign.
Remark 1
The results presented in Main Result 1 are not rigorous since not all the functional analytic details of the methodology of combining geometric singular perturbation techniques and an action functional approach have been fully worked out. Many of the functional analytic details of the approach are given in all detail in series of papers by Chen and collaborators [5, 6, 7, 8, 9, 10, 11, 12] for slightly different problems and these methods can be generalised to the setting of the current manuscript, see also [52]. However, these generalisations are a nontrivial exercise and we decided to not proceed this direction (for the readability of the manuscript). Instead, we explain the essentials of the approach in some detail in Sect. 2, see, in particular, Sect. 2.2, and we test the results of Main Result 1 against numerical simulations and we get excellent agreement, see, in particular, Figs. 5, 6, 10, and 11. Further, in [52] we introduced this methodology for the homogeneous version of (1) and used it to explicitly replicate known rigorous results regarding the existence and stability of localised solutions from [23, 53, 55].
1.1 Background of the Model
A dimensional homogeneous version of (1)—so with \(\gamma _1=\gamma _2=\gamma (x)\)—was introduced in the nineties to study gasdischarge systems [42, 48]. Versions of the dimensional—and nondimisionalised—homogeneous model have been studied intensively afterwards, see [2, 15, 23, 28, 39, 46, 51, 52, 53, 55, 56, 57, 58, e.g.] and references therein. From a mathematical point of view, the homogeneous version of (1) is arguably the most mathematically rigorously studied singularly perturbed threecomponent reaction–diffusion equation. It is not a surprise that the homogeneous version of (1) supports stable slowly travelling front solutions with speed \(c = \frac{3}{2} \sqrt{2} \varepsilon ^2 \gamma \) [55], since the homogeneous model can be seen as a weakly perturbed AllenCahn type equation [4, 15, 26]. Consequently, stationary front solutions exist only for \(\gamma \equiv 0\). In [23], it was shown there exist a (family of) stationary pulse solution(s) with leading order width \(2x^*\) if there is an \(x^*>0\) solving \(f(2x^*) =\gamma \), where f is defined in (3). By using the NonLocal Eigenvalue Problem approach for Evans functions [20, 21, 22], it was shown in [53] that the critical part of the spectrum associated with a stationary pulse solution consists of a translation invariance eigenvalue at the origin and a critical eigenvalue \(\lambda = 3\sqrt{2}\varepsilon ^2 g(2x^*)\), where g is defined in (4). The remaining subset of the spectrum is contained in the left halfplane bounded away from the imaginary axis with an \(\mathcal {O}(1)\)bound and there are no complexvalued eigenvalues for \(\tau \) and \(\theta \) of \(\mathcal {O}(1)\). Thus, the stationary pulse solution is stable if \(g(2x^*)>0\). In other words, the existence and stability conditions for homogeneous stationary pulse solutions coincide with the first conditions for the existence and stability of local defect pulse solutions of Main Result 1.
(Versions of) the heterogeneous model (1), and the effect of the defect, have also been studied [24, 41, 54, 61, 62, e.g.]. We shortly discuss the results of [24, 54] as they are most relevant for this manuscript. Since the heterogeneous model (1) is, in contrast to the homogeneous model, not translation invariant, a stationary solution is typically isolated and does not come as a family of solutions. Moreover, since the defect is small it does not alter the spectrum in a leading order fashion and the perturbed translation invariant eigenvalue (at the origin for the homogeneous case) will determine the fate of the stability of a defect solution (there are no leading complexvalued eigenvalues since \(\tau \) and \(\theta \) are \(\mathcal {O}(1)\) [53]). In [54] it was shown that, under certain parameter conditions, (1) supports socalled stable pinned global defect solutions [24]. That is, stationary front and pulse solutions with one of the interfaces pinned at the defect (this in contrast to local defect solutions were the interfaces are pinned away from the defect). In particular, it was shown that global defect front solutions exist if \(\gamma _1 \gamma _2 < 0\) and that they are stable if, in addition, \(\gamma _1>0\), see also panel “e” of Fig. 1. For the global defect pulse solutions it was shown that the widths of the pinned pulse solutions are, to leading order, given by \(2x^*\), where \(2x^*\) solves \(f(2x^*)=\min \{\gamma _1,\gamma _2\}\), see also panel “a” of Fig. 1. That is, they correspond to the widths of the pulse solutions in the homogeneous case with \(\gamma =\min \{\gamma _1,\gamma _2\}\).
Local defect solutions were investigated numerically in [54] since the analytic methods employed in [54] cannot be used directly to study local defect solutions. This is due to the fact that the interaction of the defect with the localised interfaces is weak—due to the \(\varTheta (1)\)distance between them—and higher order computations are needed. For instance, a leading order GSPT analysis appended with a Melnikov integral [47] gives that the leading order width \(2x^*\) of a local defect pulse solution is—again—given by the roots of \(f(2x^*) = \gamma _i\), see also panels “b” and “c” of Fig. 1. In other words, the widths of the pinned pulse solutions are in essence not affected by the introduction of the small defect. However, the pinning distance \(x_d\) cannot be determined from this leading order analysis.
Since the system parameters used for the simulations of the pinned pulse (front) solutions in Fig. 1 are all the same, we have the coexistence of local and global defect pulse (front) solutions. Moreover, the numerically stable local defect pulse solution shown in panel “c” of Fig. 1 is pinned in the region to the right of the defect (where \(\gamma (x)= \gamma _2\)), while the numerically stable global defect pulse solution shown in panel “a” of Fig. 1 is pinned in the opposite region to the left of the defect (where \(\gamma (x)= \gamma _1\)). The two stable pinned pulse solutions are separated by a numerically unstable pinned pulse solution shown in panel “b” of Fig. 1. This unstable pulse is pinned at the defect in the \(\gamma _2\)region and has a width similar as the stable local defect pulse solution. This unstable pinned pulse solution acts as the separatrix between the two stable pinned pulse solutions and is called the scatter solution [39, 40, 41, 50, 62].
In [24], the authors used geometric methods to study the persistence of heteroclinic and homoclinic orbits for a general system of ordinary differential equations (ODEs) with a weak defect and under generic conditions on the nonlinearities. The ODE associated to pinned defect solutions of (1) (see (10)) fits into an extended version of this general system, see also Remark 1.13 of [24]. Consequently, some of the results of [24] are directly applicable here.
Theorem 2

If \(\gamma _1=0\), then (1) supports a local defect front solution \(Z_{f,ld}^{\ell }=(U_{f,ld}^{\ell }, V_{f,ld}^{\ell },\) \(W_{f,ld}^{\ell })\) that asymptotes to \(\pm 1 +\mathcal {O}(\varepsilon )\) as \(x \rightarrow \pm \infty \) and with its front pinned to the left of the defect.

If \(0<\gamma _1< \alpha +\beta \), then (1) supports a local defect pulse solution \(Z_{p,ld}^{\ell }=(U_{p,ld}^{\ell }, V_{p,ld}^{\ell }, W_{p,ld}^{\ell })\) that asymptotes to \(1 +\mathcal {O}(\varepsilon )\) as \(x \rightarrow \infty \) and with its pulse pinned to the left of the defect.
Whilst Theorem 2 partly settles the question related to the existence of local defect front and pulse solutions supported by (1), it has several limitations. Firstly, it requires that both \(\alpha \) and \(\beta \) are positive and does not provide any insights for \(\alpha \) and/or \(\beta \) negative. See, however, Remark 2. Secondly, Theorem 2 does not provide any information regarding the profiles—and thus also not regarding the pinning distances—of the local defect solutions. Thirdly, for the existence of local defect front solutions it is required that \(\gamma _1 = 0\)—this to ensure the existence of a stationary front solution in the homogeneous case [55]—while one would also expect local defect front solutions for \(\gamma _1\) small, but not zero. Finally, Theorem 2 does not provide any information regarding the stability of the local defect solutions. The results of this manuscript as stated in Main Result 1 (partly) address the above issues and thus significantly extend the results of [24]. In particular, we put a priori no additional restrictions on the parameters and determine leading order expressions for the pinning distances \(x_d\). In addition, we also determine the stability of the local defect solutions.
Remark 2
From Main Result 1 it follows that the most interesting results of this manuscript relate to the case where \(\alpha \beta <0\). For instance, the second existence condition of (5) implies that, in this case only, the pinning distance \(x_d = \varTheta (1)\) (since f is monotonic for \(\alpha \beta >0\)). The original activator–inhibitor framework of the homogeneous version of (1) [42, 48, e.g.] actually required that both \(\alpha \) and \(\beta \) were positive. However, this restriction is mathematically not necessary and the dynamics of the homogeneous and heterogeneous model is much richer without it. See, for instance, [23, 24, 58].
1.2 Outlook
We derive Main Result 1 by combining GSPT techniques with the action functional approach of [52]. In short, GSPT techniques will provide the profile of a local defect solution—with unknown width (for a pulse) and unknown pinning distance (for both a front and a pulse). The critical points of the action functional landscape of the derived profile determine the potential widths and pinning distances of the local defect solution under consideration and only the minimisers of the action functional yield stable local defect solutions. In Sect. 2, we discuss this action functional approach in more detail and show how to append the action functional (9) of the homogeneous case as to deal with the heterogeneity of (1).
Besides local and global defect solutions, another type of defect solutions—the trivial defect solution – was introduced in [24]. This type of defect solution is characterised by the fact that it stays \(\mathcal {O}(\varepsilon )\)close to both asymptotic end states over the whole spatial domain. That is, a trivial defect solution is—in some sense—a small perturbation of a steady state solution and can thus be seen as the heterogeneous equivalent of a homogeneous steady state solution. It was shown in [24] that, under generic conditions, trivial defect solutions exist and are unique—in the sense that there is exactly one trivial defect solution near each of the steady states. The homogenous version of (1) has two steady state solutions (near \((U,V,W)=\pm (1,1,1)\)) that fulfil these generic conditions, and, consequently, (1) has two trivial defect solutions \(Z_{td}^{\pm }=(U_{td}^{\pm }, V_{td}^{\pm }, W_{td}^{\pm })\). Whilst pinned local defect solutions are the main subject of interest of this manuscript, we also explicitly determine the profiles of these trivial defect solutions \(Z_{td}^{\pm }\) in “Appendix A”. We add the derivation of these profiles for completeness, but also to illustrate how the region around the defect should be handled from an asymptotic perspective.^{3} Specifically, we show that the defect introduces two new fast regions where the dynamics of the Ucomponent dominates, one just to the left of the defect and one just to the right of the defect. To leading order, these additional fast regions do not contribute to the profile of the trivial defect solutions, i.e. the profiles are to leading order \(\pm 1\) in both fast regions, but they do contribute at an \(\mathcal {O}(\varepsilon )\)level. Heuristically, local defect solutions can be seen as a concatenation of the equivalent stationary solution to the homogenous model with one of the trivial defect solutions.^{4} Therefore, obtaining insights in these trivial defect solutions is also a first crucial step towards understanding local defect solutions.
In Sect. 3, we derive the part of Main Result 1 related to local defect front solutions \(Z_{f,ld}^\ell \) pinned to the left of the defect and determine the profile—up to and including \(\mathcal {O}(\varepsilon )\)terms—of these solutions. We also use the action functional approach to study global defect front solutions \(Z_{f,gd}\) and reproduce the key results of [54] related to the existence (\(\gamma _1\gamma _2<0\)) and stability (\(\gamma _1>0>\gamma _2\)) of these global defect front solutions (note that only leading order computations are needed to obtain these results). Finally, we combine the results for local and global defect front solutions and we numerically investigate the stationary version of (1) to confirm the asymptotic findings.
In Sect. 4, we follow the same procedure as in Sect. 3 but now for defect pulse solutions. That is, we derive the part of Main Result 1 related to local defect pulse solutions \(Z_{p,ld}^r\) pinned to the right of the defect and determine the profile—up to and including \(\mathcal {O}(\varepsilon )\)terms—of these solutions. Most of this derivation has been placed in “Appendix B” since it is very similar to the derivation in Sect. 3 and it is only algebraically more involved (the GSPT procedure now determines the profiles up to two unknowns, the pinning distance \(x_d\) and the pulse width \(2x^*\)). We also discuss the connection of the action functional approach and the results for global defect pulse solutions \(Z_{p,gd}\) from [54] and combine these results to obtain a broader picture for pinned pulse solutions. Finally, we numerically investigate the stationary version of (1) to confirm the asymptotic findings.
We end this manuscript with a summary and a short outlook on future projects.
2 GSPT and the Action Functional Approach
2.1 GSPT
Remark 3
The defining difference between a local defect solution \(Z_{ld}\) and a global defect solution \(Z_{gd}\) is the location of the interfaces of the solution with respect to the defect at \(x=0\). If this pinning distance is \(\varTheta (1)\) in the slow scaling x, then we have a local defect solution \(Z_{ld}\), while we have a global defect solution \(Z_{gd}\) if the pinning distance is \(\mathcal {O}(1)\) in the fast scaling \(\xi :=x/\varepsilon \). In other words, for global defect solutions \(Z_{gd}\) the defect and one of the interfaces lie in the same fast field, while they lie in different fast fields—and are separated by a slow field—for local defect solutions \(Z_{ld}\). So, the spatial domain has to be divided in \(N+1\) fast and \(N+1\) slow regions to study global defect solutions.
2.2 The Action Functional Approach
3 Pinned Front Solutions
In this section, we focus on pinned front solutions supported by (1) and we first derive the part of Main Result 1 related to the existence and stability of local defect front solutions \(Z_{f, ld}^\ell \) pinned to the left of the defect. In particular, we explicitly derive the relationship (3) determining the pinning distance of the interface of the local defect front solution to the defect. In Sect. 3.2, we combine these results with the results of [54] related to global defect front solutions to obtain a broader picture for pinned front solutions. Finally, we confirm our asymptotic findings by numerically investigating the stationary version of (1), that is, by investigating (10).
3.1 Local Defect Front Solutions \(Z_{f, ld}^\ell \) Pinned to the Left of the Defect
Remark 4
3.1.1 The Next Order
3.1.2 The Derivation of the First Part of Main Result 1
3.2 Pinned Front Solutions
Properties of \(f(x) := \alpha e^{x} + \beta e^{x/D}\) and \(g(x) := f'(x) = \alpha e^{x} + \frac{\beta }{D} e^{x/D}\) for \(x \in [0,\infty )\)
f(x)  f(0)  \(\mathcal {R}(f)\)  \(f_{ex}\)  g(x)  

\(\alpha>0, \beta >0\)  \(\searrow \)  \(>0\)  \((0,\alpha +\beta ]\)  –  \(\searrow , >0\) 
\(\alpha<0,\beta <0\)  \(\nearrow \)  \(<0\)  \([\alpha +\beta , 0)\)  –  \(\nearrow , <0\) 
\(\alpha >0,\beta<0, \alpha D<\beta \)  \(\nearrow \)  \(<0\)  \([\alpha +\beta , 0)\)  –  \(<0\) 
\(\alpha >0,\beta<0, \alpha<\beta <\alpha D\)  \(\searrow \nearrow \)  \(<0\)  \([f_{ex}, 0)\)  \(<0\)  \(g(x_{ex})=0\) 
\(\alpha>0,\beta <0, \alpha >\beta \)  \(\searrow \nearrow \)  \(>0\)  \([f_{ex},\alpha +\beta ]\)  \(<0\)  \(g(x_{ex})=0\) 
\(\alpha<0,\beta >0, \alpha D<\beta \)  \(\searrow \)  \(>0\)  \((0,\alpha +\beta ]\)  –  \(>0\) 
\(\alpha<0,\beta >0, \alpha <\beta <\alpha D\)  \(\nearrow \searrow \)  \(>0\)  \((0,f_{ex}]\)  \(>0\)  \(g(x_{ex})=0\) 
\(\alpha <0,\beta>0, \alpha  >\beta \)  \(\nearrow \searrow \)  \(<0\)  \([\alpha +\beta ,f_{ex}]\)  \(>0\)  \(g(x_{ex})=0\) 
Number of local defect front solutions \(Z_{f,ld}^\ell \) pinned to the left of the defect with pinning distances \(x_d=\varTheta (1)\), and their stability properties, as derived from Main Result 1
\(\# Z_{f,ld}^\ell =0\)  \(\# Z_{f,ld}^\ell =1\)  \(\# Z_{f,ld}^\ell =2\)  

\(\alpha ,\beta >0\) or  \(4 \tilde{\gamma }_1/\gamma _2 \le 0\) or  \(0< 4 \tilde{\gamma }_1/\gamma _2 < f(0)\)  – 
(\(\alpha <0,\beta >0\) and \(\alpha D <\beta \))  \(4\tilde{\gamma }_1/\gamma _2 \ge f(0)\)  \(\gamma _2>0\): stable  
\(\alpha ,\beta <0\) or  \(4 \tilde{\gamma }_1/\gamma _2 \ge 0\) or  \(f(0)< 4 \tilde{\gamma }_1/\gamma _2 <0^*\)  – 
(\(\alpha >0, \beta <0\) and \(\alpha D < \beta \))  \(4\tilde{\gamma }_1/\gamma _2 \le f(0)\)  \(\gamma _2<0\): stable  
\(\alpha >0, \beta <0\) and  \(4 \tilde{\gamma }_1/\gamma _2 \ge 0\) or  \( f(0) \le 4 \tilde{\gamma }_1/\gamma _2 <0^* \)  \(f_{ex}< 4 \tilde{\gamma }_1/\gamma _2< f(0)\) 
\(\alpha<\beta <\alpha D\)  \(4 \tilde{\gamma }_1/\gamma _2 < f_{ex}\)  \(\gamma _2<0\): stable  \(\gamma _2>0\): \(x_d^1\) stable 
\(\gamma _2<0\): \(x_d^2\) stable  
\(\alpha >0, \beta <0\) and  \(4 \tilde{\gamma }_1/\gamma _2 \ge f(0)\) or  \(0 \le 4 \tilde{\gamma }_1/\gamma _2 <f(0)\)  \(f_{ex}< 4 \tilde{\gamma }_1/\gamma _2 <0^*\) 
\(\alpha > \beta \)  \(4 \tilde{\gamma }_1/\gamma _2 < f_{ex}\)  \(\gamma _2>0\): stable  \(\gamma _2>0\): \(x_d^1\) stable 
\(\gamma _2<0\): \(x_d^2\) stable  
\(\alpha <0, \beta >0\) and  \(4 \tilde{\gamma }_1/\gamma _2 > f_{ex}\) or  \( 0 < 4 \tilde{\gamma }_1/\gamma _2 \le f(0) \)  \(f(0)<4 \tilde{\gamma }_1/\gamma _2< f_{ex}\) 
\(\alpha < \beta < \alpha D\)  \(4 \tilde{\gamma }_1/\gamma _2 \le 0\)  \(\gamma _2>0:\) stable  \(\gamma _2>0\): \(x_d^2\) stable 
\(\gamma _2<0\): \(x_d^1\) stable  
\(\alpha <0, \beta >0\) and  \(4 \tilde{\gamma }_1/\gamma _2 > f_{ex}\) or  \(f(0) < 4 \tilde{\gamma }_1/\gamma _2 \le 0^* \)  \(0< 4 \tilde{\gamma }_1/\gamma _2 <f_{ex}\) 
\(\alpha  > \beta \)  \(4 \tilde{\gamma }_1/\gamma _2 \le f(0)\)  \(\gamma _2<0\): stable  \(\gamma _2>0\): \(x_d^2\) stable 
\(\gamma _2<0\): \(x_d^1\) stable 
3.2.1 Numerical Results
4 Pinned Pulse Solutions
In this section, we focus on pinned pulse solutions supported by (1) and we derive the part of Main Result 1 related to the existence and stability of local defect pulse solutions \(Z_{p, ld}^r\) pinned to the right of the defect. We follow the same procedure as for pinned front solutions and we combine GSPT techniques with the action functional approach. However, for the clarity of the presentation, and since the computations are similar in spirit (though algebraically more involved) as the computations for pinned front solutions, the derivation of the profile of a local defect pulse solution \(Z_{p,ld}^r\)—with unknown pulse halfwidth \(x^*\) and unknown pinning distance \(x_d\)—and the computation of its action functional \(J_p\) (15) is placed in “Appendix B”. In addition, we show that the action functional approach also reproduces some of the previously obtained results of [54] related to the existence of global defect pulse solutions \(Z_{p,gd}\) and the nonexistence of local defect pulse solutions \(Z_{p,ld}^m\) with the defect pinned in between the two interfaces [54]. Finally, in Sect. 4.2, we confirm our asymptotic findings by numerically investigating (10).
4.1 The Derivation of the Second Part of Main Result 1
We need to compute the action functional \(J_p(U_{p,ld}^r)\) (15) associated to a local defect pulse solution \(Z_{p,ld}^r\) pinned to the right of the defect to derive the parts of Main Result 1 related to local defect pulse solutions. We state the action functional below and refer to “Appendix B” for its proof.
Lemma 1
Proof of Lemma 1
See “Appendix B”. \(\square \)
4.2 Pinned Pulse Solutions
We further investigate the results of Main Result 1 related to local defect pulse solutions, and we combine these results with the results of [54] related to global defect pulse solutions. So, we aim to get a broad picture of pinned pulse solutions supported by (1).
4.2.1 Numerical Results: Pulse Widths
4.2.2 Local Defect Pulse Solutions \(Z_{p,ld}^m\) Pinned Around the Defect
In addition, the part of the leading order term of the action functional \(J_p(U_{p,ld}^m)\) that depends on the pulse halfwidth \(x^*\) is identical to the leading order term of the action functionals \(J_p(U_{p,ld}^{r,\ell })\) (38). Consequently, they have the same critical points \(\{x^*  f(2x^*) = \gamma _i\}\) and thus also the same pulse halfwidths and stability properties. For instance, for \( \gamma _2>\gamma _1\), \(J_p(U_{p,ld}^m)\) is minimal for \(y=0, \{x^*  f(2x^*) = \gamma _1\}\) and \(g(2x^*)>0\) and maximal for \(y=2x^*\), \(\{x^*  f(2x^*) = \gamma _2\}\) and \(g(2x^*)<0\). So, again, smaller \(\gamma _i\)’s are favourable.
4.2.3 Numerical Results: Pinning Distances
Pulse width \(2x^*\), pinning distance \(x_d\), pinning region, and stability properties (S \(=\) stable, U \(=\) unstable) of pinned pulse solutions obtained from Main Result 1, symmetry (8) and [54] for \((\alpha , \beta , D, \gamma _1, \gamma _2) = (4,1,5,0.2,0.3)\), as well as their relation to the profiles of Fig. 8 obtained from simulating (10) on a domain of length \(2L=48\)
\(2x^*\)  \(x_d\)  Region  Stab.  Figure 8  Num. stab.  \(J_p^{NUM}\)  

\(Z_{p,gd}^{\ell ,1}\)  2.20  –  \(x<0\)  U  f  U  \(\,2.166\varepsilon \) 
\(Z_{p,gd}^{\ell ,2}\)  8.01  –  \(x<0\)  U  a  U  \(\,1.301\varepsilon \) 
\(Z_{p,gd}^{r,1}\)  2.61  –  \(x>0\)  S  e  S  \(\,2.623\varepsilon \) 
\(Z_{p,gd}^{r,2}\)  5.83  –  \(x>0\)  U  b  U  \(\,2.469\varepsilon \) 
\(Z_{p,ld}^{\ell ,1}\)  2.20  2.88  \(x<0\)  S  g  S  \(\,2.172\varepsilon \) 
\(Z_{p,ld}^{\ell ,2}\)  8.01  2.01  \(x<0\)  U  –  –  – 
\(Z_{p,ld}^{r,1}\)  2.61  2.76  \(x>0\)  U  d  U  \(\,2.617\varepsilon \) 
\(Z_{p,ld}^{r,2}\)  5.83  2.20  \(x>0\)  U  c  U  \(\,2.465\varepsilon \) 
As for the front case, the results of Main Result 1 related to local defect pulse solutions are more interesting for \(\alpha \beta <0\), see also Remark 2. In this case, stable and unstable local defect pulse solutions \(Z_{p,ld}\) with different widths can be pinned in the same \(\gamma \)region, see, for instance, Figs. 8, 10 and 11. Combining the result of this manuscript with the results for global defect pulse solutions \(Z_{p,gd}\) of [54], and since the first existence condition \(f(2x^*)= \gamma _i\) of Main Result 1 can have up to two solutions, shows that there can actually be a myriad of different pinned defect pulse solutions, especially for \(0< \gamma _{1,2} < \alpha + \beta \) and \(\alpha D > \beta \), see also Table 1. For instance, for \((\alpha , \beta , D, \gamma _1, \gamma _2) = (4,1,5,0.2,0.3)\)—the parameter values used in Figs. 6, 8, 10 and 11—the asymptotic results of this manuscript and [54] predict the existence of (at least) eight different pinned pulse solutions. Two of these eight pinned solutions are stable, while the other six are unstable. The asymptotic results for this particular parameter set are summarised in Table 3. Numerical simulations of (10) for the same parameter set on a domain of length \(2L=48\) yield the same stable pinned pulse solutions, as well as six unstable pinned pulse solutions (and some small pulse profiles similar to the ones shown in panels “d” and “e” of Fig. 7). However, one of the unstable pinned pulse solutions from the asymptotic results of Main Result 1 is not found numerically on this domain of integration. This is due to the relative small size of the domain and we remark that we did find this solution by simulating on a larger domain, see Fig. 10 and also Remark 5. In addition, one of the numerically computed unstable pinned pulse solutions—shown in panel “h” of Fig. 8—is not found by the asymptotic results of Main Result 1. We postulate that this stems from the fact that the pinning distance for this pinned pulse solution is—asymptotically—much larger than one and a higher order analysis is needed to also find the pinning distance for this solution, see again Remark 5. Actually, we believe that there are potentially three more of these pinned pulse solutions that are pinned far away from the defect and that are not captured by the asymptotic results of this manuscript.
Remark 5
Main Result 1 predicts that, under the conditions of Theorem 2, the pinning distance of a local defect front and pulse solution is much larger than one. However, the numerically observed pinning distances of local defect front and pulse solutions under these conditions do not appear to be much greater than one (see, for instance, the panel “c” of Fig. 1, Figures 4 and 5 in [54] and Figure 1 in [24]). We postulate that this stems from the fact that the numerical simulations are necessarily done on a bounded domain and this obviously influences the results on the pinning distances. Moreover, the asymptotic results from Main Result 1 hold only for small enough \(\varepsilon \). See also Fig. 12, where we show numerically simulated profiles of local defect pulse solutions for \(\alpha \) and \(\beta \) positive—one of the conditions of Theorem 2—but for two different \(\varepsilon \)values and on two different domains of integration.
5 Discussion
5.1 Summary
In this manuscript, we studied a heterogeneous threecomponent FitzHugh–Nagumo model (1) and we derived existence and stability conditions for the simplest, but fundamental, localised pinned solutions—namely pinned front and pulse solutions—that are pinned away from the heterogeneity, see Main Result 1. In certain parameter regimes, we explicitly computed the relationship between the jumptype heterogeneity (2) and the pinning distance \(x_d\) of a pinned localised solution, see, in particular, (3) and (5). In other parameter regimes, e.g. for \(\alpha \beta >0\) for pinned pulse solutions, we showed that the pinning distances \(x_d\) are much larger than one, see also Remarks 2 and 5, and higher order computations are needed to explicitly determine these pinning distances. These results were derived by combining GSPT techniques with an action functional approach. This combined approach was pioneered by us in [52] to study stationary localised solutions for the homogeneous threecomponent FitzHugh–Nagumo model. By appending the homogeneous action functional to deal with the defect, see (15) and (17), we showed that—due to the asymptotic scaling of the defect—the leading order width of the localised solution is not affected by the defect (see, however, Remark 1). In contrast, the pinning distance is determined at the next order (in \(\varepsilon \)) of the action functional. In essence, the defect destroys the translation invariance of the homogeneous problem and pinpoints—from the family of stationary localised solutions in the homogeneous case—a set of isolated locations for the localised pinned solutions. In addition, the defect also determines the fate of the translation invariance eigenvalue at zero—and thus the stability—of the localised pinned solutions, see (4) and (6).
5.2 Concluding Remarks and Future Work
5.2.1 Designing a Defect
Localised pinned solutions to (1) have been studied in detail before, see, in particular, [24, 54]. However, this is the first time that (1) has been studied in this generality and that the pinning distance \(x_d\) has been computed explicitly. Actually, to the best of our knowledge this is the first time that this pinning distance has been computed analytically for local defect solutions for such systems of heterogeneous nonlinear reaction–diffusion equations. A direct consequence of explicitly characterising the pinning distance \(x_d\) in terms of the system parameters and the strength of the defect is that it opens the path to solving the inverse problem of controlling the pinned solutions. That is, for a given width and location, can we find suitable system parameters and a heterogeneity such that (1) supports a stable localised pinned solution satisfying these prescribed conditions. For instance, say \((\alpha ,\beta ) = (3,1)\) are given and we want to design a stable local defect pulse solution pinned to the right of the defect with width \(2x^*=3\) and pinning distance \(x_d=2.3\). Then, (5) of Main Result 1 holds for \(D \approx 4.94\) and consequently \(\gamma _2 = 3 e^{3}  e^{3/D} \approx 0.395\) (5). To ensure this pinned solution is also stable, i.e. such that (6) holds, it is required that \(\gamma _1 < \gamma _2\) and \(\gamma _1 \not \approx \gamma _2\).
5.2.2 Extensions Within the Action Functional Framework
The homogeneous version of (1) was originally developed for both \(\alpha \) and \(\beta \) positive, see also Remark 2. In contrast, the results of this manuscript give explicit information regarding the pinning distances \(x_d\) of pinned localised solutions for \(\alpha \) and \(\beta \) of opposite sign only. For \(\alpha \) and \(\beta \) positive, the results only imply that the pinning distances \(x_d\) are much larger than one. It is expected that an even higher order computation of the action functional will also provide the crucial information regarding the explicit pinning distances \(x_d\) of pinned localised solutions for \(\alpha \) and \(\beta \) positive. The exploration of the next order term of the action functional is part of future work.
The homogeneous version of (1) also supports stable stationary symmetric 2pulse solutions [23, 53]—and, under the right parameter conditions, also stationary asymmetric 2pulse solutions or stationary symmetric 3pulse solutions [52]—so it can be expected that (1) also supports these more exotic types of localised solutions. It is interesting to see if the explicit pinning distances of these more exotic solutions—as well as their stability—can also be explicitly determined by the action functional approach of this manuscript.
The heterogeneous model (1) has one small jumptype heterogeneity (2). However, there is a priori no reason to restrict to one jump and different types of heterogeneities have been investigated numerically before [41, 50, 54, e.g.]. For instance, in [54] (1) with a bumptype heterogeneity, i.e. \(\gamma (x)=\gamma _1\) for \(x \notin [A,A]\) and \(\gamma (x)=\gamma _2\) for \(x \in [A,A]\), was numerically simulated and it was observed that—under the right conditions on the parameters—local defect pulse solutions with the expected widths still form, see, in particular, the left panel of Figure 8 in [54]. The approach of the action functional used in this manuscript can be easily extended to handle several of these jumptype heterogeneities and is interesting to see if, for instance, the numerical observations of [54] can be shown analytically with the current approach. This bumptype heterogeneity is particularly interesting since a pinned solution in the region \([A, A]\) (with \(A=\varTheta (1)\)) cannot have an asymptotically large pinning distance (as is the case for the jumptype heterogeneity for \(\alpha \) and \(\beta \) both positive).
5.2.3 Interaction Dynamics
5.2.4 Collision Dynamics
The importance of various types of defect solutions, such as scatter solutions, originated from numerical explorations of the collision dynamics between travelling pulse solutions and defects [41, 50, 62, e.g.], since understanding the structure of defect solutions would help understanding the dynamics of travelling pulse solutions when colliding with the heterogeneity. In the current setting with \(\tau \) and \(\theta \) \(\mathcal {O}(1)\), the homogeneous version of (1) does not support travelling pulse solutions [23]. However, for \(\tau \) and/or \(\theta \) large (in particular \(\mathcal {O}(\varepsilon ^{2})\)) the homogeneous version does support stable and unstable travelling pulse solutions and these solutions travel with speed \(c = \mathcal {O}(\varepsilon ^2)\) [23, 53, 55]. In addition, the homogeneous version also supports socalled breathing pulse solutions [23]. In [54], several numerical simulations were performed in this parameter regime and it appeared that, under the right parameter conditions, an initially breathing pulse solution could evolve to a travelling pulse solution after colliding with the defect, see, in particular, Figure 10 in [54]. It would be interesting to see whether this collision dynamics can be studied analytically and whether we can develop an understanding on how a defect influences travelling pulse solutions and breathing pulse solutions. One of the complications arising in this parameter regime stems from the fact that the essential spectrum of the linearised operator needed to determine stability lies asymptotically close to the imaginary axis and additional point eigenvalues (compared to the \(\tau , \theta \) \(\mathcal {O}(1)\)case) potentially pop out of this essential spectrum, see [53] for more detail. Unfortunately, the action functional approach used in the current setting is not directly suitable to study these travelling pulse solutions, not even in the homogeneous case, see [52] for more details. However, an existence result for travelling pulse solutions supported by a monostable twocomponent FitzHugh–Nagumo model was established in [6] using a variational approach. Finally, observe that the general results of [24] are not applicable in this situation, since the underlying hypotheses in [24] are not met.
Footnotes
 1.
 2.
 3.
Observe that the trivial defect solutions \(Z_{td}^{\pm }\) can also be studied with variational methods.
 4.
 5.
The third root relates to an unstable homogeneous steady state solution (in PDE sense) and is not of interest for localised structures, see [23] for more details.
 6.
This is due to the fact that stationary front solutions exist only for \(\gamma =0\) and they are trivially stable in the homogeneous case [55].
 7.
This action functional can be directly obtained from (38) by interchanging the role of \(\gamma _1\) and \(\gamma _2\).
 8.
We do not show all the details of the computations since these are very similar to the computations for the front case.
Notes
Acknowledgements
PvH thanks the National Changhua University of Education in Taiwan, the National Center for Theoretical Sciences in Taiwan, and Tohoku University in Japan for their hospitality. CNC is grateful for the warm hospitality of Queensland University of Technology in Australia. YN and TT also thank Queensland University of Technology in Australia and the National TsingHua University in Taiwan for their hospitality. The authors also acknowledge support from the Mathematics Research Promotion Center in Taiwan and they note that part of this research was finalised during the first joint AustraliaJapan workshop on dynamical systems with applications in life sciences at Queensland University of Technology in Australia.
Open Access
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