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Bichromatic travelling waves for lattice Nagumo equations

We discuss bichromatic (two-color) front solutions to the bistable Nagumo lattice differential equation. Such fronts connect the stable spatially homogeneous equilibria with spatially heterogeneous 2-periodic equilibria and hence are not monotonic like the standard monochromatic fronts. We provide explicit criteria that can determine whether or not these fronts are stationary and show that the bichromatic fronts can travel in parameter regimes where the monochromatic fronts are pinned. The presence of these bichromatic waves allows the two stable homogeneous equlibria to both spread out through the spatial domain towards each other, buffered by a shrinking intermediate zone in which the periodic pattern is visible.


Introduction
In this paper we consider the Nagumo lattice differential equation (LDE) u j (t) = d u j−1 (t) − 2u j (t) + u j+1 (t) + g u j (t); a , (1.1) posed on the spatial lattice j ∈ Z, with t ∈ R. We assume d > 0 and use the standard cubic bistable nonlinearity g(u; a) = u(1−u)(u−a) with a ∈ (0, 1). This LDE is well-known as a prototypical model that describes the competition between two stable states u = 0 and u = 1 in a discrete spatial environment. A crucial role is reserved for so-called travelling front solutions, which have the form Such solutions are often referred to as invasion waves, as they provide a mechanism by which the energetically preferred state can invade the spatial domain.
Our work focuses on the case where c = 0 holds for these primary invasion waves, indicating a delicate balance between the two competing states. In this case (1.1) can admit stable spatially periodic reststates. Numerical results indicate that these states can act as a buffer between regions of space where u = 0 and u = 1 dominate the dynamics. This buffer shrinks as these two stable states appear to move towards each other. This latter process is governed by secondary two-component invasion waves that we analyze in detail in this paper.
Nagumo PDE The LDE (1.1) can be seen as the nearest-neighbour discretization of the Nagumo reaction-diffusion PDE [35] u t = u xx + g(u; a), on a spatial grid with size h = d −1/2 . This PDE has been used as a highly simplified model for the spread of genetic traits [1] and the propagation of electrical signals through nerve fibres [3]. In higher space dimensions it also serves as a desingularization of the standard mean-curvature flow that is often used to describe the evolution of interfaces [18]. Fife and McLeod [21] used phase plane analysis to show that (1. for some smooth waveprofile Φ and wavespeed c that has the same sign as a − 1 2 . These fronts hence connect the two stable spatially homogeneous equilibria u(x, t) ≡ 0 and u(x, t) ≡ 1.
Exploiting the comparison principle, Fife and McLeod were able to show that these front solutions have a surprisingly large basin of attraction. Indeed, any solution to (1.3) with an initial condition u(x, 0) = u 0 (x) that has u 0 (x) ≈ 0 for x −1 and u 0 (x) ≈ 1 for x +1 will converge to a shifted version of this front as t → ∞.
These front solutions can be used as building blocks to capture the behaviour of a more general class of solutions to (1.3). Consider for example the two-parameter family of functions u plt;α0,α1 (x, t) = Φ(x − ct + α 0 ) + Φ(−x − ct + α 1 ) − 1, (1.5) with α 1 ≥ α 0 . Each of these functions can be interpreted as a shifted version of the front solution (1.4) that is reflected in a vertical line to form a plateau. If c < 0, then any initial configuration that has u 0 (x) ≈ 0 for |x| L and u 0 (x) ≈ 1 for |x| ≤ L will converge to a member of the family (1.5) as t → ∞. This provides a mechanism by which compact regions where u ∼ 1 can spread out to fill the entire domain.
On the other hand, when c > 0 one can construct entire solutions that converge to an element of (1.5) as t → −∞ and tend to zero as t → +∞. These solutions are stable under small perturbations [44]. In particular, they can be viewed as a robust elimination process whereby compact regions that have u ∼ 1 are annihilated by two incoming travelling fronts that collide as t → ∞.
Nagumo LDE For many physical phenomena such as crystal growth in materials [6], the formation of fractures in elastic bodies [39] and the motion of dislocations [10] and domain walls [14] through crystals, the discreteness and topology of the underlying spatial domain have a major impact on the dynamical behaviour. It is hence important to develop mathematical modelling tools that can incorporate such structures effectively. Indeed, by now it is well known that discrete models can capture dynamical behaviour that their continuous counterparts can not.
The LDE (1.1) has served as a prototype system in which such effects can be explored. It arises as a highly simplified model for the propagation of action potentials through nerve fibers that have regularly spaced gaps in their myeline coating [3]. Two-dimensional versions have been used to describe phase transitions in Ising models [2], to analyze predator-prey interactions [38] and to develop pattern  recognition algorithms in image processing [12,13]. Recently, an interest has also arisen in Nagumo equations posed on graphs [40], motivated by the network structure present in many biological systems [36]. Many authors have studied the LDE (1.1), focusing primarily on the richness of the set of equilibria [31] and the existence of travelling and standing front solutions [33,45]. Such solutions have the form (1.2), which leads naturally to the waveprofile equation (1.6) Since the behaviour of every lattice point is governed by the same profile Φ, we refer to these front solutions as monochromatic waves in this paper (in order to distinguish them from the bichromatic waves we discuss in the sequel). The seminal results by Mallet-Paret [33] show that for each a ∈ [0, 1] and d > 0 there exists a unique c = c mc (a, d) for which such monochromatic (mc) solutions exist.
Pinning Upon fixing a ∈ (0, 1) \ { 1 2 }, Zinner [45] established that c mc (a, d) = 0 for d 1, while Keener [30] showed that c mc (a, d) = 0 for 0 < d 1. Upon fixing d > 0, Mallet-Paret established [33] that c mc (a, d) = 0 for a ≈ 1 and a ≈ 0. In addition, again for fixed d > 0, the results in [24,34] strongly suggest that there exists δ > 0 so that c mc (a, d) = 0 whenever a − 1 2 ≤ δ; see Figure 1. This last phenomenon is called pinning and distinguishes the LDE (1.1) from the PDE (1.3). It is a direct consequence of the fact that we have broken the translational invariance of space. Indeed, (1.6) becomes singular in the limit c → 0 and the corresponding waveprofiles indeed typically lose their smoothness. Many results suggest that this phenomenon is generic for discrete systems [2,7,16,17,29]. However, by carefully tuning the nonlinearity g it is possible to design systems for which this pinning is absent [15,26]. Understanding the pinning phenomenon is an important and challenging mathematical problem that also has practical ramifications.
Periodicity In this paper we study waves that connect spatially homogeneous stationary solutions of (1.1) with spatially heterogeneous 2-periodic stationary solutions. It is well known that many physical systems exhibit spatially periodic features [22,23,37]. Examples that also feature spatial discreteness include the presence of twinning microstructures in shape memory alloys [4] and the formation of domainwall microstructures in dielectric crystals [41].
In many cases the underlying periodicity comes from the spatial system itself. For example, in [19,20,25] the authors consider chains of alternating masses connected by identical springs (and vice versa). The Figure 2: A monochromatic travelling wave of (1.1) (left panel) connects two spatially homogeneous stationary solutions. A bichromatic travelling wave of (1.1) (right panel) connects a spatially homogeneous stationary solution with a spatially heterogeneous one.
dynamical behaviour of such systems can be easily modelled by LDEs with periodic coefficients. In certain limiting cases the authors were able to construct so-called nanopterons, which are multi-component wave solutions that have low-amplitude oscillations in their tails.
However, periodic patterns also arise naturally as solutions to spatially homogeneous discrete systems. Indeed, we shall see in §3 that the LDE (1.1) with d > 0 admits many periodic equilibria. In addition, the results in [31] explore the periodicity and chaos present in the set of equilibria to homogeneous LDEs with simplified nonlinearities.
It is also possible to introduce a natural periodicity into the structure of (1.1) by taking d < 0. This can be seen by introducing new variables v j = (−1) j u j , which restores the applicability of the comparison principle. This choice essentially decomposes the lattice sites Z into two groups Z odd and Z even that each have their own characteristic behaviour.
Such anti-diffusion models have been used to describe phase transitions for grids of particles that have visco-elastic interactions [8,9,42]. In [5] this problem has been analyzed in considerable detail. The authors show that the resulting two component system admits co-existing patterns that can be both monostable and bistable in nature. Similar results with piecewise linear nonlinearities but more general couplings between neighbours can be found in [43].
Bichromatic waves In this paper we are interested in the parameter region where c mc (a, d) = 0. In a subset of this region it is possible to show that (1.1) has spatially heterogenous stable equilibria. We focus on the simplest case and consider so-called bichromatic (two-color) equilibria, which are spatially periodic with period two. As such, they are closely connected to solutions of the Nagumo equation posed on a graph with two vertices. We set out to construct bichromatic front-solutions to (1.1), which can be seen as waves that connect the spatially homogeneous equilibrium u ≡ 0 with such a 2-periodic state. We emphasize that these differ from the traditional front-solutions (1.2) in the sense that the odd and even lattice sites each have their own waveprofile, as illustrated in Figure 2. Consequently, the bichromatic front-solutions are not monotone.
Our first main contribution is contained in §3, where we give a detailed description of the set of parameters (a, d) where such 2-periodic equilibria exist and where they are stable. In contrast to the setting encountered in [5], the relevant bifurcation curves cannot all be described explicitly. Besides a global result stating that the number of such equilibria decreases as d is increased, we also obtain precise asymptotics that describe the boundaries near the three corners (a, d) ∈ {(0, 0), (1/2, 1/24), (1, 0)} in Figure 1.
As in [5], these preparations allow the existence of bichromatic fronts to be established in a straightforward fashion. Indeed, one can apply the general theory developed by Chen, Guo and Wu in [11] for discrete periodic systems that admit a comparison-principle. These results imply that there exists a unique wavespeed c bc (a, d) for which such bichromatic (bc) fronts exist. If c bc (a, d) = 0, the Fredholm theory developed in [28] together with the techniques from [27, §3] can be used to show that these travelling fronts depend smoothly on (a, d) and are nonlinearly stable.
However, these general results cannot distinguish between the cases c bc (a, d) = 0 and c bc (a, d) = 0 where we have standing respectively travelling fronts. This should be contrasted to the situation for the PDE (1.3), where the sign of the wavespeed is given by the sign of a simple integral [21]. Indeed, there is a large set of parameters (a, d) for which the discrete bichromatic fronts fail to travel, even though the analogous integral does not vanish.
Our second main contribution is that we provide explicit criteria in §4 that can guarantee c bc = 0 or c bc > 0. Together these results cover most of the parameter region where bichromatic fronts exist. In any case, they provide a two-component generalization of the coercivity conditions introduced in [33], which ensure c mc (a, d) = 0 for the boundary regions a ≈ 1 and a ≈ 0.
Our arguments to guarantee c bc = 0 are closely related to the setup used by Keener [30] to establish that monochromatic waves are pinned for 0 < d 1. In particular, for small values of d one can neglect the diffusion term in §1.1 and use properties of the cubic to show that the derivative of the waveprofile must change signs. This contradicts the fact that waveprofiles must be strictly monotonic if they travel.
On the other hand, in §4 we develop an intuitive geometric construction involving reflections to describe a planar recurrence relation that standing bichromatic fronts must satisfy. This allows us to rule out the presence of such fronts when a scalar inequality is violated. This consequently implies the presence of travelling bichromatic fronts.
The parameter regimes where these two arguments apply both converge towards the corner points (0, 0) and (1,0). Near these corners we need the delicate asymptotics described above to distinguish between the two cases.
Colliding fronts One of the main reasons for our interest in these bichromatic fronts is that they present mechanisms via which the stable homogeneous states u = 0 and u = 1 can spread throughout the domain, even though the primary invasion waves are blocked from propagation. By using similar techniques as in [44], we believe it should be possible to construct entire solutions consisting of a righttravelling bichromatic front connection between the homogeneous equilibrium u ≡ 0 and a 2-periodic intermediate state that collides with a left-travelling bichromatic connection between 2-periodic intermediate state and the homogeneous equilibrium u ≡ 1; see Figure 3. The resulting state after the collision is then a pinned monochromatic front that connects 0 with 1. We have been able to numerically verify the existence of these solutions in the parameter regions predicted by the theory developed in this paper.
Acknowledgements HJH acknowledges support from the Netherlands Organization for Scientific Research (NWO) (grant 639.032.612). LM acknowledges support from the Netherlands Organization for Scientific Research (NWO) (grant 613.001.304).

Main Results
Our interest here is in the lattice differential equatioṅ posed on the one-dimensional lattice, i.e., j ∈ Z. The bistable nonlinearity is explicitly given by with a ∈ (0, 1). Our results concern so-called bichromatic (two-colour) travelling wave solutions to the LDE (2.1). Such solutions can be written in the form for some wavespeed c ∈ R and R 2 -valued waveprofile Substituting this Ansatz into (2.1) we obtain the travelling wave system (2.5) Upon introducing the functions we see that any stationary solution to (2.5) must satisfy the nonlinear algebraic equation The full bifurcation diagram for this equation is described in §3. For our purposes here however it suffices to summarize a subset of the conclusions from this analysis, which we do in our first result below. In particular, there exists a region Ω bc in the (a, d)-plane for which the spatially homogeneous system (u,v) = G(u, v; a, d) has a stable equilibrium u bc (a, d), v bc (a, d) that can be interpreted as a bichromatic equilibrium state for the LDE (2.1). We note that the statements (2.11)-(2.12) are also valid upon replacing the bichromatic rest-state (u bc , v bc ) by the monochromatic equilibria (0, 0) and (1, 1). We will be interested in waves that connect these mono-and bichromatic equilibria together. More precisely, we set out to find solutions to (2.5) that satisfy either the 'lower' boundary conditions (2.13) or the 'upper' boundary conditions 14) The result below summarizes several key facts concerning the existence and uniqueness of such waves. It introduces subregions of Ω bc denoted by T low and T up where the bichromatic travelling waves (2.3) exist with nonzero speeds c low > 0 and c up < 0; see Figure 4. With the exception of the inequalities c low ≥ 0 and c up ≤ 0, these properties follow directly from the theory developed in [11,28]. (ii) There exist C ∞ -smooth functions  (iii) For any # ∈ {low, up} and any (a, d) ∈ Ω bc \ T # , there exists a non-decreasing function Φ : R → R 2 that satisfies (2.5) with c = 0 together with the boundary condition (2.13) if # = low or (2.14) if # = up.
We numerically determined the locations of the sets T low and T up in Figure 4. In particular, we simulated (2.1) with an initial condition that consists of the stable periodic pattern multiplied by a hyperbolic tangent. By checking if this solution converges to a travelling or stationary pattern one can decide whether (a, d) ∈ T low . We now introduce the notation Writing α a for the inverse of the strictly increasing function we formally introduce the quantity for any 0 < a < 1. Here the notation d − bc (a) refers to the limit d ↑ d bc (a). The geometric interpretation of this definition will be clarified in §4.2. However, we wish to emphasize here that one only needs information concerning the quantities (d bc , u bc , v bc ) associated to the two-dimensional algebraic problem G(u, v; a, d) = 0 in order to compute Γ(a). In particular, there is an essential difference 1 between computing Γ(a) and using the numerical procedure above to check whether c = 0.
The main contribution of the present paper is contained in our final result, which provides analytical bounds for the parameter regions T low and T up where the bichromatic waves actually travel (i.e., where c low > 0 respectively c up < 0). Both regions contain a neighbourhood of the cusp (a, d) = ( 1 2 , 1 24 ). In addition, the corners (0, 0) and (1, 0) are accumulation points for the sets T up respectively T low . (iii) There exists > 0 so that we have the inclusions The expression (2.21) is well-defined for all 0 < a < 1. If Γ(a * ) > 0 for some 0 < a * < 1, then there exists > 0 so that (a, d) ∈ T low for all (a, d) ∈ Ω bc that have The inequality Γ(a) > 0 holds whenever 1 − a > 0 is sufficiently small. In particular, we have (0, 0) ∈ T up and (1, 0) ∈ T low .
Using numerics we have verified that Γ(a) > 0 holds for a ∈ .498, .999]; see §4.3. Together with (ii) and (v) above, this strongly suggests that T low is a connected set that extends towards the right boundary of Ω bc . By symmetry, this is equivalent to the statement that T up is a connected set that extends towards the left boundary of Ω bc .

Bichromatic stationary solutions
In this section we uncover the structure of the solution set to G(u, v; a, d) = 0 as a function of the parameters (a, d). Our first result shows that for d 1 this equation only has the monochromatic roots (0, 0), (a, a) and (1, 1). The threshold d + (a) between this region and the region with five distinct roots can be explicitly computed. However, we only have qualitative and asymptotic results for the boundary d − (a) where the root-count increases to the maximal value of nine. In Figure 5 we compare these asymptotics to numerically computed values for d − (a). We remark here that the monotonicity of the root count with respect to d does not hold for general bistable nonlinearities g.
that satisfy the following properties. 1 The first problem is three dimensional, while the second problem is infinite dimensional and hence involves truncations.
(iv) Pick any a ∈ (0, 1). The equation G(u, v; a, d) = 0 has seven distinct roots for as d ↑ 1 24 . In order to break the symmetry caused by the swap u ↔ v, we set out to describe the roots of G(u, v; a, d) = 0 that have v > u. To this end, we introduce two regions that are studied separately in the two results below. In Ω − there are three such bichromatic equilibria with v > u. These equilibria can be ordered and the middle one is the only stable one. Two (or three) of these equilibria collide at d = d − (a) in a saddle node (or pitchfork) bifurcation, leaving a single unstable bichromatic equilibrium in Ω + . This equilibrium in turns collides with its swapped counterpart and the monochromatic equilibrium (a, a) on the boundary d + (a). These processes are illustrated in Figure 6. In particular, we see that Ω − coincides with the set Ω bc introduced in §2; cf.
that satisfy the following properties.
(i) Pick any (a, d) ∈ Ω − . Then we have for all # ∈ {A, B, C}. If also (a, d) ∈ Ω − , then the matrix has two strictly negative eigenvalues if # = B or one strictly positive and one strictly negative eigenvalue if # ∈ {A, C}.
(ii) For any 0 ≤ a ≤ 1 we have the identities that satisfy the following properties.
(i) Pick any (a, d) ∈ Ω + . Then we have If also (a, d) ∈ Ω + , then the matrix has one strictly positive and one strictly negative eigenvalue.
(ii) For any 0 ≤ a ≤ 1 we have the identity while for any a ∈ [ 1 2 , 1] we have (3.21) In addition, for any (a, d) ∈ Ω + we have the identity In addition, we have G(u, v; a, d) = 0 if and only if G(v, u; a, d) = 0. The statements hence follow from the ordering (3.11).
Our final result concerns the special case a = 1 2 , in which case it is possible to be more explicit. In particular, the bichromatic roots (u B , v B ) and (u D , v D ) lie on the line u + v = 1 and collide precisely when g (u; while for any 1 24 In addition, we have the identities (3.26)

Saddle-nodes around a = 0
In this section we construct the branches (u B , v B ) and (u C , v C ) of solutions to G(u, v; a, d) = 0 in the regime where (a, d) ≈ (0, 0). In particular, we define and determine the zeroes of H for which (u, v, a, d) are small.
(i) For every 0 < a < δ a and 0 < d < d c (a) the equation H(u, v; a, d) = 0 has precisely two solutions on the set {|u| + |v| < }.
(ii) For every 0 < a < δ a and d c (a) < d < δ d the equation H(u, v; a, d) = 0 has no solutions on the set {|u| + |v| < }.
(iii) For every 0 < a < δ a we have the estimate Our strategy is to use the identity H 2 = 0 to eliminate v and then recast H 1 = 0 into the normal form of a saddle-node bifurcation.
Lemma 3.7. Pick δ > 0 sufficiently small. Then there exist constants > 0 and K ≥ 1 together with functions that satisfy the following properties.
(iii) For every (u, a, d) ∈ (−δ, δ) 3 we have the bounds Proof. Substituting the Ansatz (3.32) into H 2 , we obtain the fixed point problems together with (3.39) These fixed point problems can be successively solved for small a, d and u, along with their differentiated counterparts. The desired estimates can subsequently be obtained in a standard fashion by computing Taylor expansions.
In order to eliminate v from H 1 , we define the function which forces us to write β 0 (a, d) = 2d + 2dα 0 (a, d), (3.41) By applying a shift to u the linear term in (3.40) can be removed, transforming (3.40) into a normal form for saddle node bifurcations.
Lemma 3.8. Pick δ > 0 sufficiently small. Then there exist constants > 0 and K ≥ 1 together with functions that satisfy the following properties.
Proof. We first introduce the notation . This allows us to compute (3.50) On account of (3.36) we have 2 and hence also Setting γ 1 = 0 leads to the fixed point problem which has a unique small solution that we write for the moment as In a standard fashion one obtains the bound Differentiating (3.53) we obtain which yields the estimate and finally the desired estimates follow by writing and computing ∂ d ζ 0 directly.
Proof of Proposition 3.6. We note first that the map In order to find d c it hence suffices to solve ζ 0 (a, d c ) = 0, which gives the fixed-point problem Our estimate on ∂ d S ζ0 guarantees the existence of a unique small solution for small a. The estimate (iii) now follows in a standard fashion. Writing u c = u * a, d c (a) and v c = v * u * (a, d c (a)); a, d c (a) the expansions in (iv) follow directly by substitution.

The cusp bifurcation around a = 1/2
Our goal here is to unfold the structure of the solution-set to G(u, v; a, d) near the critical point (a, d) = (1/2, 1/24) where the branches (u # , u # ) with # ∈ {A, B, C, D} all collide in a cusp bifurcation; see Figures 1 and 6. In particular, we introduce the cusp location together with the function and set out to determine the zeroes of H cp for which (u, v, a, d) are small. Proposition 3.9. There exist constants δ a > 0, δ d > 0 and > 0 together with a function a c : (−δ d , 0) → (0, δ a ) and a constant K ≥ 1 so that the following holds true.
(v) For any d ∈ (−δ d , 0) we have the estimate In order to recast our equation into an efficient form, we introduce two functions (h 1 , h 2 ) by writing which can be evaluated as Since h 2 contains a term that is linear in q, we set out to eliminate this variable by demanding h 2 = 0.
that satisfy the following properties.
(i) For every (p, a, d) ∈ (−δ, δ) 3 the equation h 2 (p, q; a, d) = 0 has a unique solution q = q * in the set {|q| < }. This solution is given by (iii) For every (p, a, d) ∈ (−δ, δ) 3 we have the bounds Proof. Substituting the Ansatz (3.70) into h 2 , we obtain the fixed point problems (3.76) These fixed point problems can be successively solved for small a, d and p, which yields the desired estimates.
In order to eliminate q from h 1 , we define the function and obtain the following representation.
Corollary 3.11. Pick δ > 0 sufficiently small. Then there exist a constant K ≥ 1, together with functions that satisfy the following properties.
By applying a (small) shift to p the undesired quadratic term in (3.79) can be eliminated. The bifurcation curve in (a, d) space can subsequently be found by determing when the remaining equation has roots of order two or higher. and that satisfy the following properties.
Setting γ 2 (a, d, p * ) = 0 leads to a fixed-point equation for p * , which has a unique small solution p * (a, d) that admits the bounds The results hence follow by writing ζ 0 (a, d) = γ 0 a, d, p * (a, d) , Here we use (3.97) to get bounds on R ζ;4 and its derivatives.
Proof of Proposition 3.9. In order to find roots of order two or higher we need to solve the system Solving the second equation we find two branches Plugging this into the first line of (3.102) we obtain from which the desired expression for a c follows.

Geometry of the cubic
Our strategy to establish Propositions 3.1-3.3 hinges upon geometric properties of the cubic g(u; a). As a preparation, we introduce the notation and note that g u min (a); a = 0, g u infl (a); a = 0, g u max (a); a = 0. (3.108) In addition, we note that g (u; a) > 0 for u < u infl (a) and g (u; a) < 0 for u > u infl (a). When 0 < a < 1 2 we have the ordering 0 < u min (a) < a < u infl (a) < u max (a) < 1. Figure 7: The functions v ± defined in Lemmata 3.13-3.14 for a = 0.45 (left) and a = 0.5 (right). In the latter case the derivatives have a discontinuity at u min ( 1 2 ).
Recalling the definition (2.6), one readily sees that In order to exploit the fact that this identity does not depend on d, we first use basic properties of the cubic to parametrize solutions to g(u; a) + g(v; a) = 0. Restricting ourselves to a ∈ [0, 1 2 ], it is possible to construct two solution curves v ± (u) that are defined for u ∈ [0, a]; see Figure 7. for all 0 ≤ u ≤ a.
To obtain (v), we take u min (a) < u < a and recall a < v − (u) < u max . If v − (u) ≤ u infl (a) then clearly g (v − (u); a) > g (u; a) > 0, as desired. In order to hande the remaining case v − (u) > u infl (a), we introduce the reflection u refl = 2u infl (a) − u. Exploiting the point symmetry of the graph of g(·; a) around its inflection point u infl (a), g(u infl (a); a) , the inequality g(u infl (a); a) > 0 implies that (3.114) Since g (ũ; a) < 0 forũ > u infl (a), we obtain g v − (u); a > g (u refl ; a) = g (u; a) > 0, (3.115) which implies v − (u) > −1. Proof. Items (i) and (iii) follow directly from the symmetry of g(·; a), while (ii) follows from the implicit function theorem. To obtain (iv), we first compute g(u min + u; a) = g(u min ; a) + 1 2 In particular, the identity g(u min + u; a) = −g(u max + v; a) (3.122) can be rewritten as v 2 = u 2 − 2 3 which can be interpreted as a fixed point problem for v 2 upon assuming that v and u have the same sign. For small |u| this problem has a solution that can be expanded as The desired limits follow directly from this expansion.

Tangencies
Let us now fix a ∈ [0, 1 2 ]. In order to find solutions to G(u, v; a, d) = 0 with d > 0, we introduce the function v d (u) = u − g(u; a) 2d (3.126) and note that the results above show that it suffices to find u ∈ [0, a] for which one of the equations holds.
Our goal here is to show that non-transverse intersections of this type can only occur at local minima of v ± − v d . Together with the strict monotonicity that holds for u ∈ (0, a), this will allow us to obtain global results in §3.5. For intersections with u ∈ (0, a) such explicit computations are significantly harder to carry out, which is why we pursue a more indirect approach here. As a preparation, we compute In addition, for any κ ≤ g (u infl (a); a) = 1 3 (a 2 − a + 1), we introduce the expressions

(3.134)
It is easy to verify that u l (κ) and u r (κ) are the two solutions to the quadratic equation g (u; a) = κ. for some # ∈ {−, +} and 0 ≤ u ≤ a, with u = u min (a) in case a = 1 2 . Then the following statements hold. ∞). On the other hand, the inclusion β ∈ (0, 1) holds if # = +.

Items (iv) and (v) now follow directly, remembering that
(3.144) The desired identity in (i) hence follows directly from (3.133). In order to conclude the proof, it suffices to note that the arguments above remain valid when a = 1 2 . Indeed, the critical cases g (u; a) = 0 and g (v ± ; a) = 0 are excluded by the requirement that u = u min (a).  (−1, 0). In addition, the inequalities v − (u) > u and g < 0 imply that g (v − (u); a) ≤ g (u; a). (3.147) in which we used g (u; a) > 0.
Intersections with v ± (u) > 0 are more delicate to analyze. Items (i), (iii) and (iv) of Lemma 3.15 suggest that it is worthwhile to consider the two functions Proof. Observe first that for β > 0 we have which implies that h l (β) and h r (β) are well-defined. A little algebra yields
Proof. An easy computation yields We introduce the critical value and remark that u c = 0 when a = 1 2 . This allows us to define the value which yields a contradiction. In particular, if (3.155) holds then we must have u ≥ min{u min (a), u c }. If u min (a) ≤ u < a, Lemma 3.16 shows that (a) must hold. On the other hand, if u c ≤ u < u min (a), then we can define β = v d (u) = v − (u) ≥ 0 and conclude as above that β > 1. In addition, we have v − (u) ≥ u infl (a), which allows us to use h r (β) > 0 and item (iv) of Lemma 3.15 to show that (a) must hold. In the final case u = a, the remarks at the start of this section together with a direct computation of v d (a) and v − (a) imply the identities in (b).
In the remainder of this section we collect several consequences of these computations. In each case, we either rule out non-transverse intersections of v ± with v d or show that they must occur at local minima In particular, we see that for all sufficiently small > 0. If a = 1 2 , the conclusion now follows from (3.154) together with (a) from Lemma 3.18.
For a = 1 2 , one also needs to use the identities together with the limits in item (iv) of Lemma 3.14 to conclude that for all sufficiently small > 0. The arguments above allow us to extend this to ∈ (0, u min (a)]. In addition, we have v − (ũ) = −1 < v d (ũ) for u min (a) <ũ < a.
for u ∈ [0, u min (a)). for all sufficiently small > 0. Corollary 3.19 allows us to extend this conclusion to the desired interval ∈ 0, a − u min (a) .

Structure
We are now ready to analyze the global structure of the solution set to G(u, v; a, d) = 0. Our first two results fix a ∈ (0, 1 2 ] and track the intersections of the curves v ± that were introduced in §3.3 with the curve v d introduced in §3.4. These intersections disappear as the parameter d is increased; see Figure 8. (iv) Consider any 0 < d < g (a;a) 4 for which (a, d) = ( 1 2 , 1 24 ). Then we have the inequality ).
On the other hand, (3.166) implies that h d has at least two zeroes for 0 < d < g (a;a) 4 . Furthermore, we claim that h d has precisely two zeroes for 0 < d ≤ d 1 upon choosing d 1 > 0 to be sufficiently small. Indeed, for any u ∈ (0, a) we have v d (u) < 0 and we can enforce by restricting the size of d > 0. We now pick > 0 in such a way that the function h d is strictly decreasing on [0, u min (a) + ] for all . This is possible because Corollary 3.22 allows us to enforce h d < 0 on this interval, with the exception of the single point u = u min (a) when a = 1 2 . Let us now define the critical value and assume for the moment that d c < g (a; a)/4. The preparations above show that there exists u min (a) < u < a with h dc (u) = 0 and h dc (u) = 0. Corollary 3.20 now implies h dc (u) > 0. As a consequence of the monotonicity ∂ d v d (u) < 0, this means that for all sufficiently small δ > 0, the function h d with d = d c − δ must have at least three zeroes. This yields a contradiction, which implies that d c = g (a; a)/4. We may hence define u AD (d) ∈ [0, a] to be the left-most root of h d (u) = 0 for 0 < d ≤ d c . The statements (i)-(v) follow readily from the observations above together with the monotonicity ∂ d v d (u) < 0 that holds for u ∈ (0, a). (ii) The function u B is strictly increasing, while the function u C is strictly decreasing.
, we observe first that h d is strictly decreasing on [0, u min (a)] because v d > 0 and v + < 0 on the interior of this interval.
In addition, in the special case a = 1 2 we can use Corollary 3.23 to conclude that h d is strictly increasing on [u min (a), a]. Since h d u min (a) = 0 occurs precisely when d = 1 24 , all the desired statements can be easily verified.
Throughout the remainder of this proof we therefore assume that 0 < a < 1 2 . Arguing as in the proof of Lemma 3.24, we may pick d 1 > 0 in such a way that h d has precisely two zeroes on [0, a] for every 0 < d ≤ d 1 . This allows us to define the critical value d − = sup{d 1 ≤ d : h d = 0 has two distinct solutions on [0, a]}. (3.189) . Let us assume for the moment that h d− has two or more zeroes on [0, a]. This implies that there exists at least one u ∈ (0, a) for which h d− (u) = 0 and h d− (u) = 0. Using Corollary 3.20 it follows that h d− (u) > 0 must hold for all such zeroes. As a consequence of the monotonicity ∂ d v d (u) < 0, this means that for all sufficiently small δ > 0, the function h d with d = d − − δ must have at least three zeroes. This yields a contradiction, which by continuity shows that h d− (u) = 0 has precisely one root on [0, a]. Upon defining u B (d) and u C (d) to be the left-most respectively right-most root of h d (u) = 0, the desired properties (i) -(vii) can be easily verified.
Recalling the function G introduced in (2.6), we see that In order to study the stability and parameter-dependence of the roots constructed in Lemmata 3.24-3.25, it is crucial to understand when the determinant of D 1,2 G vanishes. The result below states that this happens at tangential intersections of v ± and v d . Proof. We first consider the case v # (u) = 0, which occurs when g (u; a) = 0 and hence u = u min (a). Using 0 < a < 1 2 we see that v − (u) < u max (a) < v + (u). (3.192) Writing In particular, we find (3.197) For any 0 < a ≤ 1 2 , we use the functions defined in Lemma 3.25 to introduce the notation  Proof. As a preparation, we note that det D 1,2 G(a, a; a, d) = 0 if and only if d = g (a;a) 4 . In addition, it is easy to check that det D 1,2 G(0, 0; a, d) > 0 and det D 1,2 G(1, 1; a, d) > 0 for all d ≥ 0.
The implication (b) → (a) can now be verified directly using Lemma 3.26 and (3.188). In addition, we only need to establish the reverse implication under the additional assumption that v > u.
(3.205) Expanding the subdeterminants of (3.204) and reusing (3.203), we also find   , we may use Corollary 3.27, together with the first inequality in (3.202) to apply the implicit function and establish the C ∞ -smoothness of ω.
Since g (a; a)/4 ↓ 0 as a ↓ 0, a squeezing argument shows that lim a↓0 ω(a) = (0, 1, 0). Exploiting To obtain the strict ordering in (iii), we fix 0 < a ≤ 1 2 and 0 < d < d − (a). We observe that v + (u) ≥ v − (u) for u ∈ [0, a], with equality only at u = u min (a) for a = 1 2 . Since v d (0) = 0 this implies that u A (a, d) < u B (a, d) and v A (a, d) < v B (a, d). By construction, we also have u B (a, d) < u C (a, d). In addition, it cannot be the case that v B (a,d) = v C (a,d) for any 0 <d < d − (a) since then G 2 = 0 implies that also u B (a,d) = u C (a,d). Using the general identity which yields

Travelling waves
Our goal here is to establish the existence of bichromatic wave solutions to the Nagumo LDE (2.1) and to obtain detailed results concerning their speed. In particular, we establish Theorems 2.2 and 2.3, which are the main results of this paper. Upon introducing the standard discrete Laplacian together with the off-diagonal matrix the travelling wave system (2.5) can be rewritten as For any (a, d) ∈ Ω − , we set out to seek solutions to (4.3) that satisfy the boundary conditions The existence of such solutions will be established in §4.1, where we also show c ≥ 0 and establish Theorem 2.2. In §4.2 we subsequently set out to derive criteria that distinguish between the two cases c = 0 and c > 0. We verify these criteria in §4.3 for parameters (a, d) ∈ Ω − that are close to the cusp ( 1 2 , 1 24 ) and the corner (1,0). This allows us to establish our main results contained in Theorem 2.3.

Existence of waves
Our preparatory work in §3 allows us to invoke the theory developed in [11] to establish a general existence result for bichromatic waves. Indeed, the equilibria (0, 0) and (u B , v B ) are both stable under the dynamics of (u,v) = G(u, v; a, d) and all intermediate equilibria are unstable. Using a straightforward estimate based on the ordering (3.11), the wavespeeds can be shown to be non-negative. Proof. These statements follow directly from the main results in [11]. Proof. Suppose that c = 0. We estimate Since 0 < u B (a, d) < a < v B (a, d), there exists ξ * ∈ R for which Φ v (ξ * ) = u B (a, d). This yields − cΦ v (ξ * ) ≤ g(u; a) < 0, (4.6) from which we conclude c > 0.
Writing c(a, d) and Φ(a, d) for the wavespeed and waveprofile defined in Lemma 4.1, we introduce the set which corresponds to the set T low used in §2. In addition, for any (a, d) ∈ T we introduce the linear operators L a,d : , d); a, d)ψ. (4.9) The results in [28, §8] These ingredients allow us to use the implicit function theorem to show that the pair (c, Φ) depends smoothly on the parameters (a, d) ∈ T . In addition, we obtain a sign on ∂ a c.

Characterization of waves
We now set out to derive conditions that guarantee either c(a, d) = 0 or c(a, d) > 0. To this end, we introduce the functions , v a,d (u) = u − g(u; a) 2d (4.18) and note that for each # ∈ {A, B, C} and (a, d) ∈ Ω − . The local extrema of these functions are located at the critical points γ c;± (a, d) (4.20) One can verify that the functions u a,d and v a,d are strictly decreasing on [γ c;− (a, d), γ c;+ (a, d)] and strictly increasing outside this interval. The following ordering result exploits this characterization and will allow us to establish c(a, d) = 0 for a significant portion of the parameter set Ω − .  Proof. Let us first fix (a, d) ∈ Ω − . We note that 2d < 2d + (a) = g (a;a) 2 ≤ g (u infl (a);a) 2 , which implies that γ c;± (a, d) are well-defined. In addition, this allows us to compute which means that a ∈ γ c;− (a, d), γ c;+ (a, d) .
We now set out to derive a (non-sharp) condition that guarantees c(a, d) > 0. The strategy will be to rule out the existence of standing bichromatic waves. Let us therefore pick any (a, d) ∈ Ω − \ T , which implies c(a, d) = 0. Writing (Φ u , Φ v ) = Φ(a, d) for the corresponding profile, we introduce the sequence Figure 9: Illustration of the two reflections described by the system (4.30). The values have been modified for illustrative purposes, since the real regions are minuscule.
which satisfies the limits together with the difference equation (4.28) Applying a shift to the first equation, we obtain the implicit system which can be written as In particular, we can obtain (u i+1 , v i+1 ) by first reflecting (u i , v i ) horizontally through the curve u = u a,d (v) and then vertically through the curve v = v a,d (u). Based on this geometric intuition, we set out to construct a rectangle (4.31) that must contain (u i0+1 , v i0+1 ) for some critical i 0 , together with a rectangle that must contain the intermediate point (u i0+1 , v i0 ); see Figure 9. that satisfy the inequalities together with the identities for each (a, d) ∈ Ω − . Furthermore, these functions can be continuously extended to Ω − in such a way that (4.35) holds whenever d > 0.    Proof. For any i ∈ Z we have the inequalities which implies that we must have u i ≤ u a,d (v i ) ≤ u i+1 . In particular, we see that u a,d (v i ) ≤ u B (a, d), (4.41) which implies that v i / ∈ v bot (a, d), v top (a, d) . In addition, we have d).  The limits (4.27) imply that there exists M 1 so that (4.44) holds for all i ≤ −M and (4.45) holds for all i ≥ M . In particular, there must be jump between these two sets.
For any (a, d) ∈ Ω − with d > 0, the inequalities (4.34) imply that the function u a,d is strictly increasing on [0, v bot (a, d)]. This allows us to define an inverse  holds. Then we have c(a, d) > 0.
Proof. Suppose the contrary that c(a, d) = 0 and consider the sequence {(u i , v i )} defined by (4.26), together with the critical value i 0 ∈ Z that appears in Lemma 4.8. Since the sequence {u i } is nondecreasing, we have u i0+1 ≥ u i0 which implies that u i0+1 ≥ u a,d (v i0 ). Combining this with (4.52), we see that u i0+1 ≤ u bot (a, d). However, (4.38) implies that u i0+1 ≥ u top (a, d), which yields the desired contradiction.

Verification of u top > u bot
In Figure 10 we show where one may numerically verify that the scalar inequality (4.50) holds, which ensures that c(a, d) > 0. We also plot the curve d = (1−a) 2

8
, below which we have established that c(a, d) = 0. Taken together, we feel that these results cover a reasonable portion of the parameter space Ω − .
Our final task is to analytically verify (without resorting to any numerics) that c(a, d) > 0 near the cusp (a, d) = 1 2 , 1 24 and the corner (a, d) = (1, 0). As a preparation, we construct a simplified but weaker version of the condition u top > u bot by exploiting the monotonicity of v a,d;r .   Figure 10: The darkest region contains all pairs (a, d) where the assumption u bot (a, d) < u top (a, d) was verified numerically. We also include the boundaries of the sets T and Ω − as computed numerically by the procedure described in §2.
Proof. We note first that v a,d (u) = 1 − g (u;a) 2d > 0 for u ∈ [0, u min (a)]. In particular, we only have to consider the case u B (a, d) > u min (a), which cannot occur for a = 1 2 . Let us first assume that 0 < a < Proof. This follows from the uniform bound u inv a,d ≤ v bot (a, d) and the fact that v a,d;r is strictly increasing.
We now set out to verify the explicit condition (4.56) for the boundary points a, d − (a) with a ∼ 1 and a = 1 2 . Using the continuity of u top and u bot , this means that c(a, d) > 0 for all (a, d) ∈ Ω − that are sufficiently close to these critical boundary points. The desired expansions hence follow from Proposition 3.6. Proof. These expansions can be found by substition of (4.57) into the definitions (4.37) and (4.47).