Micropteron traveling waves in diatomic Fermi-Pasta-Ulam-Tsingou lattices under the equal mass limit

The diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice is an infinite chain of alternating particles connected by identical nonlinear springs. We prove the existence of micropteron traveling waves in the diatomic FPUT lattice in the limit as the ratio of the two alternating masses approaches 1, at which point the diatomic lattice reduces to the well-understood monatomic FPUT lattice. These are traveling waves whose profiles asymptote to a small periodic oscillation at infinity, instead of vanishing like the classical solitary wave. We produce these micropteron waves using a functional analytic method, originally due to Beale, that was successfully deployed in the related long wave and small mass diatomic problems. Unlike the long wave and small mass problems, this equal mass problem is not singularly perturbed, and so the amplitude of the micropteron's oscillation is not necessarily small beyond all orders (i.e., the traveling wave that we find is not necessarily a nanopteron). The central challenge of this equal mass problem hinges on a hidden solvability condition in the traveling wave equations, which manifests itself in the existence and fine properties of asymptotically sinusoidal solutions (Jost solutions) to an auxiliary advance-delay differential equation.

1. Introduction 1.1. The diatomic FPUT lattice. A diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice is an infinite one-dimensional chain of particles of alternating masses connected by identical springs. These lattices, also called mass dimers, are a material generalization of the finite lattice of identical particles studied numerically by Fermi, Pasta, and Ulam [FPU55] and Tsingou [Dau08]; such lattices are valued in applications as models of wave propagation in discrete and granular materials [Bri53,Kev11].
We index the particles and their masses by j ∈ Z and let u j denote the position of the jth particle. After a routine nondimensionalization, we may assume that the jth particle has mass (1.1.1) m j = 1, j is odd m, j is even and that each spring exerts the force F (r) = r + r 2 when stretched a distance r from its equilibrium length. Newton's law then implies that the position functions u j satisfy the We sketch a diatomic FPUT lattice in Figure 1. We are interested in the existence and properties of traveling waves in diatomic lattices in the limit as the ratio of the two alternating masses approaches 1. Specifically, we define the relative displacement between the jth and the (j + 1)st particle to be r j := u j+1 − u j , and then we make the traveling wave ansatz (1.1.3) r j (t) = p 1 (j − ct), j is even p 2 (j − ct), j is odd.
Here p 1 and p 2 are the traveling wave profiles and c ∈ R is the wave speed. When the masses are identical, the lattice reduces to a monatomic lattice, which, due to the work of Friesecke and Wattis [FW94] and Friesecke and Pego [FP99,FP02,FP04a,FP04b] is known to bear solitary traveling waves, i.e., waves whose profiles vanish exponentially fast at spatial infinity; see also Pankov [Pan05] for a comprehensive overview of monatomic traveling waves. Our interest, then, is to determine how the monatomic solitary traveling wave changes when the mass ratio is close to 1.
1.2. Parameter regimes. We derive our motivation for this equal mass situation from two recent papers studying traveling waves in diatomic lattices under different limits. Faver and Wright [FW18] fix the mass ratio 1 and consider the long wave limit, i.e., they look for traveling waves where the wave speed is close to a special m-dependent threshold called the "speed of sound" and where the traveling wave profile is close to a certain KdV sech 2 -type soliton. Hoffman and Wright [HW17] fix the wave speed and consider the small mass limit, in which the ratio of the alternating masses approaches zero, thereby reducing the lattice from diatomic to monatomic 2 . Figure 3 sketches the bands of long wave (in yellow) and 1 They work with w := 1/m > 1; after rescaling and relabeling the lattice, we may equivalently think of their results for m ∈ (0, 1).
2 This is not the same monatomic lattice that results from the equal mass limit; the springs in this small mass limiting lattice are double the length of the original springs in the diatomic lattice, and so they exert twice the original force. This factor of 2 ends up affecting the wave speed of the traveling waves that Hoffman and Wright construct: theirs have speed close to √ 2.
small mass (in red) traveling waves and indicates, roughly, how they depend on wave speeds and mass ratios. In both problems, the solitary wave that exists in the limiting case perturbs into a traveling wave whose profile asymptotes to a small amplitude periodic oscillation or "ripple." That is, the wave is not "localized" in the "core" of the classical solitary wave, and so, per Boyd [Boy98], it is a nonlocal solitary wave. Moreover, the long wave and small mass problems are singularly perturbed, which causes the amplitude of their periodic ripples to be small beyond all orders of the long wave/small mass parameter. So, these nonlocal traveling waves are, in Boyd's parlance, nanopterons; see [Boy98] for an overview of the nanopteron's many incarnations in applied mathematics and nature.
The question of the equal mass limit then follows naturally from the success of these two studies. It was raised in the conclusion of [FW18], where the authors wondered if the diatomic long wave profiles would converge to those found by Friesecke and Pego in the monatomic long wave limit [FP99], and appears as far back as Brillouin's book [Bri53], which examines both the small and equal mass limits for lattices with linear spring forces.
The articles [FW18] and [HW17] both derive their nanopteron traveling waves via a method due to Beale [Bea91] for a capillary-gravity water wave problem. Beale's method was later adapted by Amick and Toland [AT92] for a model singularly perturbed KdV-type fourth order equation. More recently, Faver [Fav,Fav18] used Beale's method to study the long wave problem in spring dimer lattices (FPUT lattices with alternating spring forces but constant masses), and Johnson and Wright [JW] adapted it for a singularly perturbed Whitham equation.
Although the equal mass problem is not, ultimately, singularly perturbed, its structure has enough in common with these predecessors that we are able to adapt Beale's method to this situation, too. We discover that the monatomic traveling wave perturbs into a "micropteron" traveling wave as the mass ratio hovers around 1. Boyd uses this term to refer to a nonlocal solitary wave whose ripples are only algebraically small in the relevant small parameter, not small beyond all algebraic orders. We do not find such small beyond all orders estimates in our equal mass problem, and so we consciously avoid using the term "nanopteron" for our profile.
We sketch this micropteron wave in Figure 2 and provide in Figure 3 an informal, but evocative, cartoon comparing the three families of nonlocal solitary waves (long wave, small mass, equal mass) that now exist for the diatomic FPUT lattice. We state our main result below in Theorem 1.2.
Beyond these nanopteron problems, wave propagation in diatomic and more generally heterogeneous lattices has received considerable recent attention; we mention, among others, the papers [CBCPS12, BP13, Qin15, VSWP16, SV17, Wat19, Lus] for theoretical and numerical examples of waves in FPUT lattices under different material regimes. See [GMWZ14] for a discussion of long wave KdV approximations in polyatomic FPUT lattices and [GSWW19] for a further discussion of the metastability of these waves in diatomic lattices. The paper [HMSZ13] studies a regular perturbation problem in monatomic FPUT lattices in which the spring force is perturbed from a known piecewise quadratic potential; the resulting solutions are asymptotic, like ours, to a sinusoid whose amplitude is algebraically small. Lattice differential equations abound in contexts beyond the FPUT model that we study here; see, for example, [HMSSVV] for a survey of traveling wave results for the Nagumo lattice equation and related models. One of the micropteron profiles (p 1 or p 2 ) sketched close to the monatomic solitary wave ς c . In a nanopteron, the periodic ripple would be so small as to be invisible relative to the monatomic profile; this is not the case in the equal mass limit.
We prove these symmetries in Appendix B. We say that a function f is "mean-zero" if f (0) = 0; here f is the Fourier transform of f , and our conventions for the Fourier transform are outlined in Appendix A.1.  Figure 3. A comparison of nonlocal solitary waves across the different diatomic FPUT problems. In this paper, for |c| 1 fixed and m ≈ 1, we find micropterons close to a Friesecke-Pego solitary wave profile ς c . Under suitable hypotheses on the existence of a monatomic solitary wave for |c| ≫ 1, we may still find micropterons close to that profile. For |c| √ 2 fixed and m ≈ 0, Hoffman and Wright found nanopterons close to a (different) Friesecke-Pego solitary wave ζ c . For m ∈ (0, 1) fixed and |c| close to an m-dependent threshold called the "speed of sound," Faver and Wright found nanopterons close to a KdV sech 2 -type profile σ m . That the bands collapse as |c| → 1 + or |c| → √ 2 + is intentional; see (the proofs of) Lemma C.7 in [FW18], Lemma 7.1 in [HW17], and Lemma C.1 in this paper. This graphic elides the interesting, but difficult, question of how the different families of waves interact; for example, for |c| 1 and m ≈ 1, how do the long wave nanopterons and equal mass micropterons compare? See Section 7 in [FW18] for a further discussion of the challenges involved in addressing these questions.

1.4.
Linearizing at the Friesecke-Pego solution. If we set ς c := (ς c , 0), we see that G c (ς c , 0) = 0. Then for µ small, we are interested in solutions ρ to G c (ρ, µ) that are close to the Friesecke-Pego solution ς c . In order to perturb from ς c , we first define Sobolev spaces of exponentially localized functions.
. and E r q := H r q ∩ {even functions} and O r q := H r q ∩ {odd functions}. For a function f = (f 1 , f 2 ) ∈ H r q × H r q , we set f r,q := f 1 r,q + f 2 r,q .
Under this notation, we have ς c ∈ ∩ ∞ r=0 E r q for q sufficiently small. We set ρ = ς c + ̺, where ̺ = (̺ 1 , ̺ 2 ) ∈ E 2 q × O 2 q , and compute that G c (ς c + ̺, µ) = 0 if and only if ̺ 1 and ̺ 2 satisfy the system The right side R c (̺, µ) is "small" in the sense that it consists, roughly, of linear combinations of terms of the form µ, µ̺, and ̺ .2 . The first component of this system has the form The operator H c is the linearization of the monatomic traveling wave problem at ς c . Proposition 3.1 from [HW17] tells us that, for q sufficiently small, H c is invertible from E r+2 q to E r q,0 for any r ≥ 0, where E r q,0 := f ∈ E r q f (0) = 0 .
In some sense, this can be seen as a spectral stability result for the monatomic wave; see also Lemma 4.2 in [FP04a] and Lemma 6 in [HM17] , which is a fixed point equation for ̺ 1 on the function space E r q . We now attempt to construct a similar fixed point equation for ̺ 2 ; our failure in this attempt will be quite instructive.
1.5. The operator L c . The second component of (1.4.2) is The operator L c is the sum of a constant-coefficient second-order advance-delay differential operator and a nonlocal term, which we write more explicitly as The minus sign on Σ c is purely for convenience. If L c were invertible from O r+2 q to O r q , then (1.5.1) would rearrange to a fixed point equation for ̺ 2 , which we could combine with (1.4.4) to get a fixed point problem for ̺. In this attempt we fail: L c is injective but not surjective.
1.5.1. Injectivity. We first sketch how, at least in the case |c| 1, the operator L c is injective from O r+2 q to O r q . We begin with the constant-coefficient part B c . This operator is a Fourier multiplier with symbol That is, if f ∈ L 2 or f ∈ L 2 per , then Next, when |c| 1, the Friesecke-Pego solution ς c is small in the sense that ς c L ∞ = O((c − 1) 2 ); see part (v) of Proposition D.2. Consequently, the operator Σ c is also small. However, Σ c does not map O r+2 q to D r q,c ; otherwise, we could use the Neumann series to invert B c − Σ c . Nonetheless, one can parley the smallness of Σ c and the invertibility of B c into a coercive estimate of the form to O r q , and so the Fredholm index of L c = B c + Σ c equals the index of B c The Fourier analysis in Section 1.5.1 shows that the index of B c is −1, so L c also has index −1. Since L c is injective, we conclude that L c has a one-dimensional cokernel in O r q and thus is not surjective. However, we can characterize the range of L c in O r q more precisely. Classical functional analysis tells us there is a nontrivial bounded linear functional z c on O 0 q such that (1.5.5) The Riesz representation theorem then furnishes an nonzero function z c ∈ Z ⋆ q such that It follows that L * c z c = 0, where (1.5.6) L * c g := B c g + 2ς c (x)(2 + A)g −Σ * c g is the L 2 -adjoint of L c . We conclude from (1.5.5) the "solvability condition" It appears, then, that our attempt to solve the traveling wave problem G c (ρ, µ) = 0 by perturbing from the Friesecke-Pego solution ς c will fail, since L c is not surjective. Moreover, although we can characterize the range of L c precisely via (1.5.7), all we know about z c is that L * c z c = 0 and z c ∈ Z ⋆ q . The kernel of L * c in Z ⋆ q must be one-dimensional, as otherwise, L * c would have Fredholm index −2 or lower, and so z c is unique up to scalar multiplication. But this function space Z ⋆ q is quite large -it contains, for example, all odd functions in L 2 and L ∞ -and so further features of z c are not immediately apparent.
To determine our next steps, we look back to the work of our predecessors in the long wave [FW18] and small mass [HW17] problems. In each of these problems, an operator similar to L c appears; each of these operators on O r+2 q has a one-dimensional cokernel in O r q because of a solvability condition like (1.5.7). Moreover, the authors were able to construct odd solutions in W 2,∞ to their versions of L * c g = 0 that asymptote to a sinusoid. We refer to such solutions as "Jost solutions," due to their similarity to the classical Jost solutions for the Schrodinger equation [RS79]. Seeing how the Jost solutions in the long wave and small mass problems are determined both guides us to the features of the Jost solution that we seek for L * c and help us appreciate what is intrinsically different about L * c when compared to its analogues in the prior nanopteron problems.
1.5.3. Jost solutions in the long wave limit [FW18]. The analogue of L c in this problem is, roughly, the operator where ε ≈ 0 and M ε is a Fourier multiplier with the real-valued symbol M(ε·), i.e., M ε f (k) = M(εk) f (k). The wave speed is intrinsically linked to the small long wave parameter ε, and so c does not appear here. Since W ε is a Fourier multiplier with real-valued symbol W ε (k) := −(1 + ε 2 )ε 2 k 2 + M(εk), it is self-adjoint in L 2 . An intermediate value theorem argument similar to the one referenced in Section 1.5.1 gives the existence of a unique Ω ε > 0 such that W ε (K) = 0 if and only if K = ±Ω ε , and so W ε sin(Ω ε ·) = W ε cos(Ω ε ·) = 0. That is, the Jost solutions are exactly sinusoidal.
1.5.4. Jost solutions in the small mass limit [HW17]. Here the analogue of L c is, roughly, We take m ≈ 0, where m is the mass ratio of the diatomic lattice, per (1.1.1). The operators M m and J m are Fourier multipliers that are O(1) in m, so the perturbation terms mM m and mJ m are indeed small, but Hoffman and Wright found it essential to retain these terms rather than absorb them into their analogues of our right side R c,2 from (1.5.1). Here the exponentially localized function ζ c is the Friesecke-Pego solitary wave profile corresponding to the monatomic lattice formed by taking m = 0 in (1.1.1).
Hoffman and Wright construct a nontrivial solution to T c (m) * g = 0 as follows. First, they show that S c (m) vanishes on certain asymptotically sinusoidal functions, i.e., there exists for some critical frequency Ω m c and phase shift θ m c . This solution j m c is indeed a classical Jost solution for the Schrodinger operator S c (m). The proof of the asymptotics in (1.5.8) uses a polar coordinate decomposition that closely relies on the structure of S c (m) as a second-order linear differential operator.
Second, it is clear that T c (m) * is a small nonlocal perturbation of S c (m). Using these facts, Hoffman and Wright perform an intricate variation of parameters argument (which again relies on the differential operator structure of S c (m)) in an asymptotically sinusoidal subspace of W 2,∞ to construct a function γ m c ∈ W 2,∞ with the properties that for some (new) phase shift ϑ m c , which is a small perturbation of θ m c . A corollary to this analysis is the frequency-phase shift "resonance" relation sin(Ω m c ϑ m c ) = 0 for almost all values of m close to zero, which turns out, in a subtle and surprising way, to be critical for their subsequent analysis.
1.5.5. Toward Jost solutions for the equal mass operator L * c . Unlike the long wave operator W ε , the operator L * c is not a "pure" Fourier multiplier as it has the variable-coefficient piece Σ * c . And unlike the small mass operator T c (m) * , we cannot decompose L * c as the sum of a classical differential operator and a perturbation term that is small in µ. So, we cannot directly import prior results to produce the Jost solutions of L * c . Our approach is to take advantage of two particular aspects of the structure of L * c . First, the constant-coefficient part B c is an advance-delay operator formed by a simple linear combination of shift operators. The Fredholm properties of such operators have received significant attention from Mallet-Paret [MP99]. Next, the variable-coefficient piece Σ * c is both exponentially localized and small for |c| 1. These facts are sufficient to solve the equation (B c − Σ * c )f = 0 in a class of "one-sided" exponentially weighted Sobolev spaces, whose features we specify below in Definition 2.2.
In broad strokes, then, we first use an adaptation of Mallet-Paret's theory due to Hupkes and Verduyn-Lunel [HL07] to invert B c on these one-sided spaces and, moreover, obtain a precise formula for its inverse. Next, it turns out that Σ * c does map between these one-sided spaces 3 and so, since Σ * c is still "small," we are able to invert B c − Σ * c with the Neumann series. This procedure yields a function γ c ∈ W 2,∞ that satisfies L * c γ c = 0 and is asymptotic to a phase-shifted sinusoid of frequency ω c at ∞, where ω c is the "critical frequency" of B c that appeared in Section 1.5.1. The exact formulas that we enjoy for the inverses of B c and then B c − Σ * c permit us to calculate an asymptotic expansion for the phase shift.
3 Unlike its failure to map between O r+2 q and D r c,q , as we saw in Section 1.5.1).

1.6.
Main result for the case |c| 1. Now that we understand precisely the range of L c , as well as its lack of surjectivity, we can confront again the traveling wave problem G c (ρ, µ) = 0 from (1.3.1), which we seek to solve for ρ ≈ ς c and µ ≈ 0. The general structure of this problem, most especially the solvability condition (1.5.7), puts us in enough concert with our predecessors in [FW18] and [HW17] that we may follow their modifications of a method due to Beale [Bea91] for solving problems with such a solvability condition. Specifically, we replace the perturbation ansatz ρ = ς c + ̺ with the nonlocal solitary wave anzatz ρ = ς c + aφ µ c [a] + η, where η = (η 1 , η 2 ) is exponentially localized and aφ µ c [a] is a periodic solution to the traveling wave problem with amplitude roughly a and frequency (very) roughly ω c . Of course, proving the existence of periodic traveling wave solutions is a fundamental part of our analysis.
Under Beale's ansatz, one easily finds a fixed point equation for η 1 , similar to how we converted (1.4.3) into (1.4.4). Then we can extract from the solvability condition a fixed point equation for a, and, in turn, an equation for η 2 . We carry out this construction in Section 5, where we prove (as Theorem 5.4) a more technical version of our main theorem below. When µ = 0 and our lattice is monatomic, the micropteron in Theorem 1.2 reduces to the Friesecke-Pego solitary wave. We remark that the stability of these micropteron traveling wave solutions for the equal mass limit, as well as the stability of the long wave and small mass nanopterons, is an intriguing open problem; see [JW] for some initial forays into this arena.
1.7. Toward the case |c| ≫ 1. Throughout this introduction, we have assumed that |c| is close to 1. This first allows us to summon a Friesecke-Pego solitary wave solution ς c for the monatomic problem and next shows, through the coercive estimate (1.5.4), that L c is injective from O r+2 q to O r q . Third, taking |c| close to 1 is fundamental for the Neumann series argument that allows us to invert B c − Σ * c in the one-sided spaces. Ostensibly, then, it appears that we are working with two small parameters, µ and |c| − 1, in our equal mass problem.
This turns out, from a certain point of view, to be superfluous. In Section 2, we present four simple and natural hypotheses that, if satisfied for an arbitrary wave speed |c| > 1 guarantee a micropteron solution in the equal mass limit. All of these hypotheses are satisfied for |c| 1. Among these hypotheses is the existence of an exponentially localized and spectrally stable traveling wave solution for the monatomic problem with wave speed possibly much greater 1; this is the origin of the dashed blue line in Figure 3. We emphasize that the existence of the Jost solutions to L * c g = 0 is not one of these hypotheses; their construction, for an arbitrary |c| > 1, follows from a weaker hypothesis. This approach has several advantages and justifications over working only in the near-sonic regime |c| 1. First, it decouples the starring small parameter µ from the deuteragonist 4 |c| − 1; after all, we are interested in the equal mass limit, not the near-sonic limit. Next, it frees us from overreliance on the Friesecke-Pego traveling wave and leaves our results open to interpretation and invocation in the case of high-speed monatomic traveling waves. For example, Herrmann and Matthies [HM15,HM17,HM19] have developed a number of results on the asymptotics, uniqueness, and stability of solitary traveling wave solutions to the monatomic FPUT problem (albeit with different spring forces from ours) in the "high-energy limit," which inherently assumes a large wave speed. Additionally, the original monatomic solitary traveling wave of Friesecke and Wattis [FW94] does not come with a near-sonic restriction on its speed, and so, in principle, that wave could have speed much greater than 1. Further study of the existence and properties of solutions to the monatomic traveling wave problem in the high wave speed regime remains an interesting open problem, and we are eager to see how our hypotheses may exist in concert with future solutions to that problem.
1.8. Remarks on notation. We define some basic notation that will be used throughout the paper.
• The letter C will always denote a positive, finite constant; if C depends on some other quantities, say, q and r, we will write this dependence in function notation, i.e., C = C(q, r). Frequently C will depend on the wave speed c, in which case we write C(c).
• We employ the usual big-O notation: if x ǫ ∈ C for ǫ ∈ R, then we write • Next, we need a c-dependent notion of big-O notation. Suppose that x µ c ∈ C for c, µ ∈ R. Then we write x µ c = O c (µ p ) if there are constants µ c , C(c, p) > 0 such that |µ| ≤ µ c =⇒ |x µ c | ≤ C(c, p)|µ| p . • If X and Y are normed spaces, then B(X , Y) is the space of bounded linear operators from X to Y and B(X ) := B(X , X ).
1.9. Outline of the remainder of the paper. Here we briefly discuss the structure of the rest of the paper.
• Section 2 contains the precise statements of the four hypotheses under which we will work.
• Section 3 constructs a family of small-amplitude periodic traveling wave solutions to the traveling wave problem (1.3.1) that possess a number of uniform estimates in the small parameter µ. These periodic traveling waves exist for arbitrary |c| > 1.
• Section 4 characterizes the range of L c as an operator from O r+2 q to O r q by constructing a Jost solution for L * c g = 0.
4 In fact, a third small parameter also lurks in nonlocal solitary wave problems: the precise decay rate q of the exponentially localized spaces E r q and O r q . Our hypotheses make explicit the decay rates that we employ and highlight the relationships between the rates at different stages of the problem.
• Section 5 converts the nonlocal solitary wave equations of Section 5.1 into a fixed point problem, which we then solve with a modified contraction mapping argument.
• Section 6 shows that the four main hypotheses of Section 2 are valid in the case |c| 1.
• The appendices contain various technical proofs and ancillary background material.

The Traveling Wave Problem for Arbitrary Wave Speeds
In this section we discuss the four hypotheses that are sufficient to guarantee micropteron traveling wave solutions in the equal mass limit for arbitrary wave speeds. Throughout, we fix c ∈ R with |c| > 1. Since |c| may not be close to 1, we are not guaranteed a monatomic traveling wave solution from Friesecke and Pego, and so we make its existence our first hypothesis.
Hypothesis 1. There exist q ς (c) > 0 and a real-valued function ς c ∈ E 2 qς (c) such that That is, ς c is an exponentially localized traveling wave profile with wave speed c for the monatomic FPUT equations of motion. With ς c satisfying (2.0.1), it follows that if ς c := (ς c , 0), then G c (ς c , 0) = 0. A straightforward bootstrapping argument, discussed in Appendix F.1, endows arbitrary regularity to ς c .
Proposition 2.1. The monatomic traveling wave solution ς c from Hypothesis 1 also satisfies ς c ∈ ∩ ∞ r=1 E r qς (c) . Next, we add the invertibility of the linearization of the monatomic traveling wave problem at ς c as a hypothesis for the situation when |c| is not necessarily close to 1.
Hypothesis 2. There exists q H (c) ∈ (0, min{1, q ς (c)}) such that the operator H c from (1.4.3) is invertible from E 2 q H (c) to E 0 q H (c),0 . In Section 1.5.5, we claimed that the key to characterizing the range of L c , defined in (1.5.1), was the invertibility of its L 2 -adjoint, L * c , defined in (1.5.6), on a class of one-sided exponentially weighted spaces. Now we make precise the definitions of these spaces and the weights for which L * c needs to be invertible. Definition 2.2. For q ∈ R and m, r ∈ N, we define When m = 1, we write just W r,∞ q and L ∞ q . We set f W r,∞ q (R,C m ) := e −q· f W r,∞ (R,C m ) . Hypothesis 3. There exists q L (c) ∈ (0, min{q H (c), q ς (c)/2, 1}) such that the operator L * c is invertible from W 2,∞ −q L (c) to L ∞ −q L (c) . Assuming this hypothesis we obtain a precise characterization of the range of L c . We prove this theorem in Section 4. Proposition 2.3. There is a nonzero odd function γ c ∈ W 2,∞ (a "Jost solution" to L * c g = 0) such that for f ∈ O r+2 q and g ∈ O r q , with r ≥ 0 and q ∈ [q L (c), 1), we have Moreover, γ c is asymptotically sinusoidal in the sense that, for some θ c ∈ R, Last, like Hoffman and Wright, we need one more condition on the interaction between the critical frequency ω c from Section 1.5.1 and the phase shift ϑ c of the Jost solution to L * c f = 0 from Theorem 2.3. This condition arises in practice much later for quite a technical reason; see Section 5.3.
All four hypothesis hold when |c| 1; we give the proof of the next theorem in Section 6. The verification of Hypotheses 1 and 2 is merely a matter of quoting results from [FP99] and [HW17], On the other hand, verifying Hypotheses 3 and 4 relies on results from [HL07], and the proof of their validity is among the central technical results of this paper.
Under these hypotheses we retain the existence of micropterons in the equal mass limit. We prove the next theorem, our central result, as Theorem 5.4.

Periodic Solutions
In this section we state our existence result for small-amplitude periodic solutions to the traveling wave problem G c (ρ, µ) = 0 from (1.3.1). We emphasize that the results in this section hold for all |c| > 1; we do not need to invoke any of the c-dependent hypotheses here.
We work in the periodic Sobolev spaces These are Hilbert spaces with the inner product When m = 1, we write just H r per . We will continue to exploit the symmetries of our traveling wave problem and work on the subspaces and ω ∈ R. Under this scaling, the problem G c (ρ, µ) = 0 becomes As in the proofs of periodic solutions for the long wave problems (Theorems 4.1 in [FW18] and 3.1 in [Fav]) and the small mass problem (Theorem 5.1 in [HW17]), we first look for solutions to the linear problem, which is Thus Γ µ c [ω]φ = 0 with φ nonzero if and only if, for some k ∈ Z, φ(k) = 0 is an eigenvector of the matrix D µ [ωk] ∈ C 2×2 corresponding to the eigenvalue c 2 ω 2 k 2 .
We compute that the eigenvalues of D µ [K] are λ ± µ (K), where (3.0.7) λ ± µ (K) := 2 + µ ± µ 2 + 4(1 + µ) cos 2 (K). These are the same eigenvalues studied in [FW18], [HW17], and [Fav]. See Figure 4 for a sketch of the curves λ ± µ (K) against We will prove below in Proposition C.2 that if |c| > 1, then the first equality above can never hold, at least over an appropriate c-dependent range of µ, while the second holds precisely at ω = ω µ c /k, k ∈ Z \ {0}, for a certain "critical frequency" ω µ c > 0, which is an O c (µ) perturbation of the frequency ω c from Section 1.5.1. We elect to take ω = ω µ c (i.e., k = ±1) so that the frequency of our periodic solutions is close to ω c ; this is important in our construction of the micropterons in Section 5.
We now highlight several important properties of ω µ c , both contained in and proved as part of Lemma C.1. Figure 4. Sketches of the graphs of the eigencurves λ ± µ (K) and the parabola c 2 K 2 for different cases on |c| and |µ|. When |c| < 1, λ ± µ (K) may have more intersections with c 2 K 2 than just K = ±ω µ c . Note that when µ = 0, the curves λ ± 0 intersect at odd multiples of π/2, but this does not affect the eigenvalue analysis, since the critical frequencies ω µ c are contained in (−π/2, π/2).
Remark 3.2. The restriction |c| ∈ (1, √ 2] will appear in several technical estimates throughout the rest of the paper. This is merely to ensure the convenient lower bound ω µ c > 1, which will be useful when we verify the four hypotheses for |c| 1. Following the methods of our predecessors, we will use this critical frequency ω µ c in a modified Crandall-Rabinowitz-Zeidler bifurcation from a simple eigenvalue argument [CR71,Zei95] to construct the exact periodic solutions to the full problem G c (ρ, µ) = 0. Here is our result, proved in Appendix C.
For later use, we isolate two additional estimates on the periodic solutions and their frequencies; the proof follows directly from Proposition 3.3.
Corollary 3.4. Under the notation of Proposition 3.3, we have We emphasize that our periodic solutions persist for µ = 0, i.e., for the monatomic lattice. Such persistence at the zero limit of the small parameter was impossible in the long wave and small mass limits, as there the analogue of the critical frequency diverged to +∞ as the small parameter approached zero. This cannot happen in our problem, due to the bounds on ω µ c in part (ii) in Proposition 3.1. We also note that the existence of periodic solutions to the monatomic traveling wave problems has already been established by Friesecke and Mikikits-Leitner [FML15] in the long wave limit using a construction inspired by [FP99]. These periodic traveling waves are close to a KdV cnoidal profile, whereas when µ = 0, the expansion (3.0.8) says that the periodic solutions from Proposition 3.3 are, to leading order in the amplitude parameter a, close to (0, sin(ω c ·)).

4.
Analysis of the Operators L c and L * c Proposition 2.3 characterized the range of L c , defined in (1.5.2), as an operator from O r+2 q to O r q . In this section we prove that theorem, which we restate below with some additional details.
Moreover, L * c γ c = 0, where L * c is the L 2 -adjoint of L c and was defined in (1.5.6).
(iii) The function γ c satisfies the limits Throughout the proof of this theorem, we will use a number of properties of the symbol B c of the operator B c , which are proved in Appendix D.1.  has the following properties.
where ω c ∈ ( √ 2/|c|, π/2) was previously studied in Section 1.5.1. The zeros at z = ±ω c are simple. Additionally, (ii) The function 1/ B c is meromorphic on the strip S 3q B . The only poles of 1/ B c in S 3q B are z = ±ω c , and each of these poles is simple. Moreover, Last, we mention two useful properties of the one-sided exponentially weighted spaces from Definition 2.2. First, if q > 0 and f ∈ W r,∞ q (R, C m ), then f vanishes at −∞, and if f ∈ W r,∞ −q (R, C m ), then f vanishes at ∞. Next, the Sobolev embedding and some calculus tell us that the two-sided exponentially weighted spaces H r q , defined in (1.4.1), are continuously embedded in W r−1,∞ ±q for r ≥ 1. Now we are ready to prove Theorem 4.1. We distribute the proof over the remainder of this section.
. We do this by considering L c and L * c as unbounded operators on L 2 with domain O 2 q L (c) . For convenience, we recall that . Hypothesis 3 then implies that f = 0, and so L * c must be injective. That is, 0 is not an eigenvalue of L * c , and so 0 is also not an eigenvalue of To obtain the forward implication in (4.0.1), it suffices to find a function γ c with L * c γ c = 0. We will construct a very particular γ c that will give us the asymptotics (4.0.2). We do this in three steps. First, we find a function f c satisfying L * c f c = 0 using methods derived from [MP99]. Then the structure of the operator We use the definition of Σ * c in (4.1.1) and the property e qς (c) Now we need the asymptotics of f c at −∞. For this, we turn to the methods of Mallet-Paret in [MP99]. Specifically, we follow the proof of his Proposition 6.1. We rewrite (4.2.1) as is therefore defined and analytic on Re(z) < −q L (c); the definition and some essential properties of this Laplace transform are given in Appendix A.2. The bounds on q L (c) from Hypothesis 3 ensure that we may find q γ (c) > 0 with Elementary properties of the Laplace transform give the bounds Next, we use the formula (A.2.2) for the inverse Laplace transform to write Then we apply the Laplace transform L − to our advance-delay problem (4.3.2) and find where the remainder term R c arises from the identities (A.2.3) and (A.2.5) for the Laplace transform under derivatives and shifts, respectively. More precisely, we have and we note that R c is entire. Part (i) of Lemma 4.2 tells us that B c (iz) = 0 for 0 < | Re(z)| < q B and so B c (iz) = 0 for 0 ≤ | Re(z)| < q γ (c) and z = ±iω c . We can therefore solve for L − [g c ](z) in (4.3.5) and find For x < 0, the formula (4.3.4) for g c then becomes This integrand is meromorphic on an open set containing | Re(z)| < q γ (c); it has simple poles at z = ±iω c and is analytic elsewhere. Moreover, the quadratic decay of B c (z) for |z| large from part (iii) of Lemma 4.2, the estimate (4.3.3), and the definition of R c in (4.3.6) imply We can therefore shift the contour of the integral in (4.3.4) from Re(z) = q γ (c) to Re(z) = −q γ (c) and obtain from the residue theorem that Since B c has simple zeros at z = ±iω c by part (i) of Lemma 4.2, and since I c (x, ·) and e x· are analytic on | Re(z)| ≤ q γ (c), these residues are The strategy of the remainder of Mallet-Paret's proof of his Proposition 6.1 in [MP99] shows that both r c (x) and Remark 4.3. The limitation of this approach is that we do not have an explicit formula for g c , and so we cannot calculate further the residues in (4.3.9) and (4.3.10). When |c| 1, we can use the Neumann series and results from [HL07] to produce an explicit formula for (B c − Σ * c ) −1 ; we do this in Sections 6.3.2 and 6.3.3. In turn, this does give a formula for g c and, ultimately, α c and β c .
On the other hand, we point out that our proof here does not use the fact that ς c solves the monatomic traveling wave problem; instead, we need only the decay property e qς (c)|·| ς c ∈ L ∞ . The methods of this section could therefore be applied to much more general advance-delay operators than L * c ; indeed, Mallet-Paret's results in [MP99] are phrased for a very broad class of such operators.
we have This follows by estimating and using the limits (4.3.1) and (4.3.11). Now we claim We prove this claim in Section 4.
Then L * cf c = 0 and, from (4.4.2), we have Now we need the identity does not vanish identically, at least one of the coefficients (2 + α c,r − β c,r ), (α c,i + β c,i ) is nonzero. We then apply the identity (4.4.6) directly to (4.4.5) and conclude that Since L * c γ c = 0, we have the forward implication of (4.0.1) in part (ii) of Theorem 4.1, and also (4.4.7) implies the first limit in the asymptotics (4.0.2) from part (iii). Proving the second limit in (4.0.2) is essentially a matter of establishing the limit (4. ]. The validity of this limit with derivatives, in turn, is a consequence of the two representations for f c : first, per (4.2.2), as f c (x) = e iωcx + g c (x), where g c and g ′ c vanish at ∞, and, next, with g c replaced by its expression in (4.3.8), in which r ′ c decays at −∞. Last, since f c and f ′ c are asymptotic to bounded functions at ±∞ by (4.3.1) and (4.3.11), it follows that γ c , γ ′ c ∈ L ∞ , which implies (4.0.3). This proves part (iv).

4.5.
Characterization of the range of L c as an operator from O r+2 Here we prove the reverse implication in (4.0.1). In Section 1.5.2, we argued 5 that the cokernel of L c in O r q was one-dimensional, and we characterized the range of L c via (4.5.1) for some odd function z c ∈ L 1 loc with sech q (·) z c ∈ L 2 . On the other hand, we constructed in Section 4.4 an odd nontrivial function γ c ∈ W 2,∞ such that q . The functions γ c and z c must be linearly dependent, as otherwise the functionals ι c and z c would be linearly independent and L c would have a cokernel of dimension at least 2. The characterization (4.5.1) therefore implies (4.0.1). 4.6. Proof of the claim (4.4.3). Fix q ∈ [q L (c), q B ) and suppose the claim is false, so that E c (x) = 0 for all x. Then (4.4.2) and the exponential decay of g c imply that where f c and its complex conjugate f c are linearly independent since they asymptote, respectively, to e iωcx and e −iωcx at ∞. The functionals defined for g ∈ H 0 q , are therefore also linearly independent. Moreover, The methods of Section 1.5.2 can be adapted to show that L c has a two-dimensional cokernel in H 0 q when considered as an operator from H 2 q to H 0 q . Consequently, if z is any functional on H 0 q such that z[L c f ] = 0 for all f ∈ H 2 q , then z must be a linear combination of z c,1 and z c,2 . On the other hand, with z c and z c from (4.5.1), we already have z c [L c f ] = 0 for all f ∈ H 2 q , and so z c must be a linear combination of z c,1 and z c,2 . But then the odd function z c must be a linear combination of f c and f c , and so z c is even, a contradiction.
• η ∈ E r q × O r q with r ≥ 2 and q > 0 to be specified later.
We expand G c (ς c + aϕ µ c [a] + η, µ) using the bilinearity (B.0.4) of Q and the following decomposition of D µ from (1.3.2) as the sum of a diagonal operator and a small perturbation term: Next, we cancel a number of terms with the existing solutions G c (ς c , 0) = 0 and G c (aϕ µ c [a], µ) = 0. After some further rearrangements, we find that G c (ς c + aϕ µ c [a] + η) = 0 is equivalent to (5.1.1) where the multitude of terms on the right side are There are, indeed, many terms here, but the salient features are that h µ c,k (η, a) ∈ E r q,0 and ℓ µ c,k (η, a) ∈ O r q for η ∈ E r q × O r q and a ∈ R and that most of these terms are "small." For example, h µ c,1 is O c (µ), h µ c,2 is, very roughly, of the form µη, and h µ c,4 and h µ c,5 are quadratic in η and a. The terms h µ c,3 and ℓ µ c,3 are more complicated and merit more precise analysis later. We will use the explicit algebraic structure of the terms and their smallness frequently in our subsequent proofs.

5.2.
Construction of the equation for η 1 . We first extend Hypothesis 2 to allow H c to be invertible over a broader range of exponentially localized spaces. The proof of the next proposition is in Appendix F.2.

5.3.
Construction of the fixed point equations for a. From the system (5.1.2), the unknowns η 2 and a must satisfy We formally differentiate the right side of (5.3.1) to isolate a term containing a factor of a: We calculate this formal derivative by recalling the definitions of ℓ µ c,1 , . . . , ℓ µ c,5 in (5.1.3) and observing that all the terms except ℓ µ c,3 are either constant in a or quadratic in η and a, in which case their derivatives with respect to a at η = 0 and a = 0 are zero. We expect, therefore, that the term ℓ µ c,3 (η, a) + aχ µ c will be roughly quadratic in a. We could then write ℓ µ c,k (η) + ℓ µ c,3 (η, a) + aχ µ c these terms are "small" .
However, it turns out to be more convenient not to work with χ µ c but instead with (5.3.3) ) · e 2 = (2 + A)(ς c sin(ω c ·)). We now set Then we apply the functional ι c , defined in (4.0.1), to both sides of (5.3.5) to find The asymptotics (4.0.2) on γ c give us an explicit formula for ι c [χ c ], which we prove in Appendix F.3: Since |c| > 1, we have If we set It is obvious from the definition of χ c in (5.3.3) that χ c ∈ ∩ ∞ r=0 O r qς (c) ⊆ ∩ ∞ r=0 O r q , and so P c f ∈ O r q for any f ∈ O r q . Also, a straightforward calculation shows that ι c [P c f ] = 0. Now, recall that in Section 5.2 we specified q so that q ∈ (q L (c), 1). Then part (ii) of Theorem 4.1 implies that P c f is in the range of L c . Conversely, if L c f = g, then ι c [g] = 0 and so P c g = g. That is, The injectivity of L c on O r q for r ≥ 1, established as part (i) of Theorem 4.1, then implies that L c is bijective from O r+2 q to P c [O r q ]. The functional ι c is continuous on O r q by part (iv) of Theorem 4.1, and the subspace q is a bounded operator, which we denote by L −1 c . We are now able to rewrite (5.4.3) as a fixed point equation for η 2 : With our previously constructed equations (5.2.1) for η 1 and (5.3.9) for a, this gives us a fixed point problem for our three unknowns from Beale's ansatz: (5.4.5) (η, a) = (N µ c,1 (η, a), N µ c,2 (η, a), N µ c,3 (η, a)) =: N µ c (η, a). 5.5. Solution of the full fixed point problem. We will solve the fixed point problem (5.4.5) using the following lemma, which was stated and proved as Lemma 4.10 in [JW].
Lemma 5.2. Let X 0 and X 1 be reflexive Banach spaces with X 1 ⊆ X 0 . For r > 0, let B(r) := {x ∈ X 1 | x X 1 ≤ r}. Suppose that for some r 0 > 0, there is a map N : B(r 0 ) → X 1 with the following properties.
The spaces X r are Hilbert spaces and therefore they are reflexive, and the sets U r τ,µ are the balls of radius τ |µ| centered at the origin in X r .
The next proposition shows that N µ c satisfies the estimates from Lemma 5.2. Its proof is in Appendix G.2.
Lemma 5.2 therefore applies to produce a unique (η µ c , a µ c ) ∈ U 1 τc,µ such that (η µ c , a µ c ) = N µ c (η µ c , a µ c ). We then bootstrap with part (iii) of Proposition 5.3 to conclude that η µ c is smooth, and so we obtain our main result.
Theorem 5.4. Assume that |c| > 1 satisfies Hypotheses 1, 2, 3, and 4 and take µ ⋆ (c) and τ c from Proposition 5.3. Then for each |µ| ≤ µ ⋆ (c), there exists a unique (η and, for each r ≥ 0, there is C(c, r) > 0 such that (5.5.1) η µ c r,q⋆(c) + |a µ c | ≤ C(c, r)|µ|. Remark 5.5. The estimate (5.5.1) shows that the amplitude of the ripple is a µ c = O c (µ), which is not necessarily small beyond all orders of µ. Recall from (5.3.9) that, roughly, . The analogues of ι c in the lattice nanopteron problems [FW18], [HW17], and [Fav] all depended on µ and, if we denote one of these functionals by ι µ , roughly had an estimate of the form | ι µ [f ]| ≤ C(q, r)|µ| r f r,q for f ∈ H r q . The proof of this estimate parallels the Riemann-Lebesgue lemma and closely relied on the fact that the analogue in those problems of the critical frequency ω µ c was roughly O(µ −1 ). In turn, this "high frequency" estimate hinged on the existence of a singular perturbation in the problem. Our equal mass problem is not singularly perturbed, our critical frequency ω µ c remains bounded as µ → 0, and our ripple is not necessarily small beyond all orders of µ.
6. The Proof of Theorem 2.4: Verification of the Hypotheses for |c| 1 6.1. Verification of Hypothesis 1. We extract the following result from Theorem 1.1 in [FP99].
(i) If (6.1.1) c ǫ := 1 + ǫ 2 24 then for each r ≥ 0, there is a constant C(r) > 0 such that We set (6.1.4) q ς (c ǫ ) := ǫq FP in Hypothesis 1. We remark that Friesecke and Pego label the decay rate for their profile ς c as b 0 (c); part (c) of their Theorem 1.1 and their detailed proof in Lemma 3.1 reveal that, with c ǫ from (6.1.1), they have b 0 (c ǫ ) = O(ǫ). It is convenient for us to make explicit this order-ǫ dependence and rewrite this decay rate as b 0 (c ǫ ) = ǫq FP .
6.2. Verification of Hypothesis 2. As we mentioned in Section 1.4 when we met the operator H c , the invertibility of H c for |c| ≈ 1 arises from Proposition 3.1 in [HW17]. Here is that proposition.
Proposition 6.2 (Hoffman & Wright). There exists ǫ HWr ∈ (0, ǫ FP ] such that for 0 < ǫ < ǫ HWr , r ≥ 0, and 0 < q < ǫq FP , the operator H cǫ , defined in (1.4.3), is invertible from E r+2 q to E r q,0 . For the precise value of the decay rate in Hypothesis 2, we will set Recalling the definition of q ς (c ǫ ) in (6.1.4) and that we took q FP < 1 in Theorem 6.1, we have q H (c ǫ ) < min{1, q ς (c ǫ )}. We wish to point out that the crux of the proof of Proposition 6.2 by Hoffman and Wright is a clever factorization of H cǫ as a product of two operators, one of which is invertible from E r+2 q to E r q,0 due to Fourier multiplier theory proved by Beale in [Bea80] (which later appears as Lemma A.3 in this paper), and the other of which is ultimately a small perturbation of an operator that Friesecke and Pego [FP99] prove is invertible from E r to E r . Hoffman and Wright apply operator conjugation (cf. Appendix D of [Fav18]) to extend the invertibility of this second operator from E r q to E r q . It is interesting to note that this perturbation argument is the only time that Hoffman and Wright need to assume that their wave speed is particularly close to 1; for the rest of their paper, the small parameter |c| − 1 does not play an explicit role, as it does for us in the concrete verification of our four main hypotheses.
We also note that we are writing ǫq FP for the decay rate of our Friesecke-Pego solitary wave profile ς cǫ ; Hoffman and Wright use the notation b c for the decay rate of their Friesecke-Pego profile, which they denote by σ c . In turn, this b c is equal to what Friesecke and Pego call b 0 (c).
6.3. Verification of Hypothesis 3 in the case |c| 1. 6.3.1. Preliminary remarks. We will chose our decay rates q to depend in a very precise way on ǫ. First, let (6.3.1) b We convert the problem B cǫ f = g with f ∈ W 2,∞ −qǫ , g ∈ L ∞ −qǫ to an equivalent first-order system. Let Then B cǫ f = g is equivalent to and observe that (6.3.9) det(∆ c (z)) = z 2 + 1 c 2 2 + 2 cosh(z) = 1 where B c is defined in (4.0.4).
As above, given g ∈ L ∞ ±qǫ , the function [B ± cǫ ] −1 g is the unique element of W 1,∞ ±qǫ to satisfy Last, we prove the operator norm estimate (6.3.5). Part (i) of Theorem E.1 and some matrix-vector arithmetic imply the existence of a function G ǫ ∈ ∩ ∞ p=1 L p such that We have We calculate and so The last inequality comes from (4.0.8).
Since the image of [B + cǫ ] −1 consists of functions that decay to zero at −∞, we have, as in (4.3.11), (6.4.10) lim It is possible to obtain very precise estimates on the coefficients on e ±iωc ǫ x in (6.4.10); the proof is in Appendix E.3.

Appendix A. Transform Analysis
A.1. Fourier analysis. If f ∈ L 1 (R, C m ), we set If f ∈ L 2 per (R, C m ), we define and we have In either case, with (S d f)(x) := f (x + d), we have the identity A.1.1. Fourier multipliers. We first work on the periodic Sobolev spaces H r per (R, C m ) from (3.0.1). Take r, s ≥ 0 and suppose that M : R → C m×m is measurable with Then Fourier multiplier M with symbol M, defined by i.e., by is a bounded operator from H r per (R, C m ) to H s per (R, C m ). We will need some calculus on "scaled" Fourier multipliers. Let M be the Fourier multiplier with symbol M. For ω ∈ R, define M[ω] to be the Fourier multiplier with symbol M(ωk), i.e., Last, for a function g : R → C m , let The methods of Appendix D.3 of [Fav18] prove the next lemma.
Lemma A.1. Let M : R → C m×m be measurable. Then . Next, we discuss some properties of the adjoint of a Fourier multiplier on periodic Sobolev spaces.  Mf, g H s per (R,C m ) = f, M * g H r per (R,C m ) , f ∈ H r per (R, C m ), g ∈ H s per (R, C m ), is given by is a Fourier multiplier with symbol M : R → C m×m . Suppose as well that M has a one-dimensional kernel spanned by ν with ν(k) = 0 for all but finitely many k and that M(k) is self-adjoint for all k ∈ R. Then the adjoint M * from part (i) has a one-dimensional kernel in H r per (R, C m ) spanned by ν * , whose Fourier coefficients are Last, we state a slight generalization of a result for Fourier multipliers on the exponentially weighted spaces H r q from (1.4.1). In this case, a Fourier multiplier on H r q (or H r ) is, of course, defined as before by (A.1.1). For 0 < q <q, write (Beale). Let 0 < q 0 ≤ q 1 < q 2 and suppose that M : R → C is a measurable function with the following properties.
(M1) The function M is analytic on the strips S 0,q 1 and S q 1 ,q 2 .
(M2) The function M has finitely many zeros in R, all of which are simple. Denote the collection of these zeros by P M .
(M3) There exist C, z 0 > 0 and s ≥ 0 such that if z ∈ S 0,q 1 ∪ S q 1 ,q 2 with |z| ≥ z 0 , then Now let M be the Fourier multiplier with symbol M. There exist q ⋆ , q ⋆⋆ > 0 with q 1 ≤ q ⋆ < q ⋆⋆ ≤ q 2 such that if q ∈ [q ⋆ , q ⋆⋆ ], then, for any r ≥ 0, M is invertible from H r+s q to the subspace and, for f ∈ D r M,q , There are two additional special cases.
(i) If q 1 = q 2 , then the result above is true for all 0 < q ≤ q 2 .
We note that (B.0.2) is the same system that was derived in equation (2.4) in [FW18]. Set and so Q is indeed symmetric and bilinear. Last, we discuss the even-odd symmetries of G c . First, observe that if f is even, then Af is even and δf is odd, while if f is odd, then Af is odd and δf is even. So, if ρ 1 is even and ρ 2 is odd and ρ = (ρ 1 , ρ 2 ), then (D µ ρ) · e 1 is even and (D µ ρ) · e 2 is odd, where e 1 = (1, 0) and e 2 = (0, 1). Likewise, if ρ 1 andρ 1 are even and ρ 2 andρ 2 are odd, then Q(ρ,ρ) · e 1 is even and Q(ρ,ρ) · e 2 is odd. We conclude that G c (ρ, µ) · e 1 is even and G c (ρ, µ) · e 2 is odd when ρ 1 is even and ρ 2 is odd.
Next, if f is integrable on R or 2P -periodic and integrable on [−P, P ], then (2 − A)f and δf are "mean-zero" in the sense that Thanks to the structure of D µ , all terms in G c (ρ, µ) · e 1 contain either a factor of 2 − A or δ, and so G c (ρ, µ) · e 1 is always mean-zero. We conclude that the symmetries in (1.3.4) hold.
Appendix C. Existence of Periodic Solutions C.1. Linear analysis. We begin with two propositions that study the linearization Γ µ c [ω] defined in (3.0.4). The first of these contains all the technical details needed to prove Proposition 3.1.
Lemma C.1. For each |c| > 1, there is M(c) > 0 with the following properties.

Now we estimate
This last inequality holds if c 2 −(1+µ) > 0, and we can ensure this by taking |µ| < (c 2 −1)/4. So, we take our threshold for µ to be and assume in the following that |µ| ≤ M(c). Note that M(c) → 0 + as |c| → 1 + , and so the µ-interval over which we work necessarily shrinks as |c| → 1 + .
(v) Now we show that ω µ c − ω c = O c (µ). If ω c = ω µ c , then there is nothing to prove, so assume ω c = ω µ c . Recall that ω c satisfies c 2 ω 2 c − (2 + 2 cos(ω c )) = 0 and ω µ c satisfies c 2 (ω µ c ) 2 − λ + µ (ω µ c ) = 0. Subtracting these two equalities and using the definition of λ + µ in (3.0.7), we obtain c 2 (ω c − ω µ c ) + µ − 2 cos(ω c ) + µ 2 + 4(1 + µ) cos 2 (ω µ c ) = 0. Taylor-expanding the square root and using the uniform bound 0 < A c < ω µ c < B c from part (iv), this rearranges to Since the cosine has Lipschitz constant equal to 1, we have From part (iv), specifically (C.1.11), we have √ 2/|c| ≤ ω µ c . Likewise, since B c (ω c ) = 0 with B c defined in (1.5.3), one can extract the inequaliy √ 2/|c| < ω c . Specifically, the proof is in Appendix D.1 in the context of the proof of part (i) of Proposition 4.2. Thus whenever |c| > 1, and so we have inf |µ|≤M(c) We conclude from (C.1.14) that The next lemma contains the remaining details that we need for the quantitative bifurcation argument that we will carry out in the following sections. To phrase this lemma, we need the definitions of the periodic Sobolev spaces H r per , E r per , and O r per from the start of Section 3. In this appendix only, we abbreviate We also need to recall the definition of the operator Γ µ c [γ µ c ] from (3.0.4). Lemma C.2. For each |c| > 1, there is µ per (c) ∈ (0, M(c)] such that the following hold.
Here we have used the assumption that φ 1 is even and φ 2 is odd. We know that φ(1) is an eigenvector of D µ [ω µ c ] corresponding to the eigenvalue λ + µ (ω µ c ). Using the definition of this matrix in (3.0.6), we conclude there is a ∈ C such that , provided that the term in the denominator of υ µ c is nonzero. It is, in fact, bounded below away from zero. If we assume . It follows that for |µ| small, say, |µ| ≤ µ per (c) ≤ M 3 (c), we have C.2. Bifurcation from a simple eigenvalue. We are now ready to solve the periodic traveling wave problem Φ µ c (φ, ω) = 0 posed in (3.0.3). We follow [HW17] and [CR71] and make the revised ansatz φ = aν µ c + aψ and ω = ω µ c + ξ. where a, ξ ∈ R and ν µ c , ψ 0 = 0. Then our problem Φ µ c (φ, ω) = 0 becomes Φ µ c (aν µ c + aψ, ω µ c + ξ) = 0. After some considerable rearranging, we find that ψ and a must satisfy . This is roughly the same system that Hoffman and Wright study when they construct periodic solutions for the small mass problem; specifically, its analogue appears in equation (B.9) in [HW17]. Our goal, like theirs, is to rewrite (C.2.1) as a fixed point argument in the unknowns ψ and ξ on the space E 2 per,0 × O 2 per × R with µ as a small parameter and c fixed. We intend to use the following lemma, proved in [FW18] as Lemma C.1. Lemma C.3. Let X be a Banach space and for r > 0 let B(r) = {x ∈ X : x ≤ r}.
(ii) Suppose as well that the maps F µ (·, a) are Lipschitz on B(r 0 ) uniformly in a and µ, i.e., there is L 1 > 0 such that The intermediate value theorem furnishes ω c ∈ (0, π/2) such that B c (ω c ) = 0. Since 0 < ω c < π/2, we can rewrite the relation B c (ω c ) = 0 as and so we have the refined bounds √ 2/|c| < ω c < π/2. Next, for k ∈ R, we calculate since c 2 > 1. This shows that the zeros at z = ±ω c are simple and, moreover, unique in R.
Now we prove that z = ±ω c are the only zeros of B c on a suitably narrow strip in C. For k, q ∈ R, we compute B c (k + iq) = −c 2 (k 2 − q 2 ) + 2 + (e −q + e q ) cos(k) +i −2c 2 kq + (e −q − e q ) sin(k) .
We claim for a suitable q B > 0, if |q| ≤ 3q B , then I c (k, q) = 0 if and only if k = 0. But R c (0, q) = q 2 + e q + e −q > 0 for all q ∈ R, and so, if our claim is true, then B c (z) = 0 for | Im(z)| ≤ 3q B .
So, we just need to prove the claim about I c . Observe that, for q = 0, I c (k, q) = 0 if and only if (D.1.1) 2c 2 q e −q − e q = sinc(k) := sin(k) k .
We claim that If this estimate holds, then (D.1.5) will establish the desired estimate (4.0.8). We first prove (D.1.7) for the L 1 (R − ) case and then comment briefly on how to proceed with the similar L 1 (R + ) estimate. Observe that R c vanishes as | Re(z)| → ∞, since the other three functions in (D.1.2), where R c is defined implicitly, all vanish as | Re(z)| → ∞. Moreover, R c is analytic on the strip S 3q B . We can therefore shift the integration contour in (D.1.6) from Im(z) = q to Im(z) = 2q B and obtain The integral ∞ −∞ e ikx R c (−k + 2iq B ) dk in (D.1.9) is the inverse Fourier transform of R c (−· +2iq B ), which we may estimate using the implicit definition of R c in (D.1.2): We have The quantities II and III are uniformly bounded for 1 < |c| ≤ √ 2 due to the calculations of the specific Fourier transforms in (D.1.3) and (D.1.4) and the bounds (4.0.5). We obtain a uniform bound for I over 1 < |c| ≤ √ 2 by estimating the integral To study F −1 [R c (− · +iq) L 1 (R + ) , we repeat the work above, except we shift the contour in (D.1.6) to Im(z) = −2q B , so that Then we obtain an estimate analogous to (D.1.9), which in turn proves (D.1.7) for L 1 (R + ). D.2. Additional estimates on the Friesecke-Pego solitary wave ς cǫ . We deduce the following lemma from Lemmas 3.1 and 3.2 in [HW08].
Lemma D.1 (Hoffman & Wayne). There exist ǫ HWa ∈ (0, ǫ FP ] and a, C > 0 such that Now we prove a bevy of estimates on ς cǫ , all of which say either that ς cǫ is "small" in a certain norm or that ς cǫ and ǫ 2 σ(ǫ·), where σ was defined in (6.1.2), are "close" in some norm.
Proposition D.2. There is C > 0 such that the following estimates hold for all ǫ ∈ (0, ǫ HWa ).
Proof. (i) We use the evenness of the integrand and substitute u = ǫx to find e a|·| (ς cǫ −ǫ 2 σ(ǫ·)) 2 (ii) The Cauchy-Schwarz inequality implies The first integral on the second line is just a constant (depending on a), and the second integral is O(ǫ 7/2 ) by part (i).
(iv) The Sobolev embedding and the Friesecke-Pego estimate (6.1.3) imply and then rescaling gives the desired estimate.
Appendix E. Some Proofs for the Verification of Hypotheses 3 and 4 in the case |c| 1 E.1. Background from the theory of modified functional differential equations.
We extract the content of the following theorem from Theorem 3.2 and Proposition 3.4 in [HL07].
Theorem E.1 (Hupkes & Verduyn Lunel). Let A 0 , A 1 , . . . , A n ∈ C m×m and let d 0 < d 1 < · · · < d n be real numbers. For f ∈ W 1,∞ , define where 1 is the m × m identity matrix. We call ∆ the characteristic matrix corresponding to the system (E.1.1). Suppose that q ∈ R with det[∆(z)] = 0 for Re(z) = q. Then Λ is an isomorphism between W 1,∞ q (R, C m ) and L ∞ q (R, C m ), where these spaces were defined in Definition 2.2. We denote its inverse by Λ −1 q . Moreover, we have two formulas for Λ −1 q .
(i) There is a function G q ∈ ∩ ∞ p=1 L p (R, C m×m ) such that for g ∈ L ∞ q (R, C m×m ), Moreover, (ii) Given δ ∈ (0, |q|) and g ∈ L ∞ q (R, C m ), we have (E.1.5) where L ± are the Laplace transforms defined in Appendix A.2.
Combining all these estimates, we conclude and so the line integral in (E.2.9) over ̺ + ǫ,2 (R) is To establish (E.2.9), it suffices, of course, to show that The methods of Section E.2.3 can be adapted to show that |L + [ς cǫ g](z ǫ (t, R))| is bounded above by a constant independent of t or R (but dependent on g and ǫ, although that does not matter here). Next, we refer to the definition of K ǫ in (6.3.12) to estimate The numerator on the right side above is independent of R and bounded in t. We can infer from the quadratic estimates on B cǫ in (4.0.6) that the denominator is O(R 2 ) uniformly in t, and so the whole expression above vanishes as R → ∞.
The estimates on the functionals α ǫ and β ǫ in (6.4.7) then follow from the estimates on the Laplace transforms in Section E.2.3, particularly (E.2.8).
E.3.1. Refined estimates on α ǫ . The functional α ǫ was defined in (6.4.5). Using the expression for g ǫ from (6.4.8), we have Since Me iωc ǫ · = M(ω cǫ )e iωc ǫ · , per (6.3.14), we can calculate the first term just by computing The Laplace transforms are Here we needed the result from Theorem 6.1 that ς cǫ is positive. Thus and we note that θ + ǫ,0 is real. The uniform boundedness of ω cǫ , the estimate (4.0.5), the definition of M in (6.3.14), and part (iii) of Proposition D.2 tell us there are C 1 , C 2 > 0 such that This holds if there exists C = O(1) such that In that case, we have Then we set (E.3.5) ǫ res,1 := min ǫ B , 1 2C , in which case the geometric series in (E.3.4) converges for 7 0 < ǫ < ǫ res,1 . We conclude We are, admittedly, being rather cavalier about what C is. In Section 1.8, we agreed that C would denote any constant that is O(1) in ǫ; now we are restricting our range of ǫ based on one of these C. To be more precise, we could trace the lineage of the C defining ǫ res,1 in (E.3.5) back to three sources: the estimates in Proposition 6.3 on [B − cǫ ] −1 , in (6.4.7) on α ǫ , and in Proposition D.2 on ς cǫ . by part (v) of Proposition D.2. Otherwise, if h ∈ L ∞ , we can use the estimate e qǫ· ς cǫ L ∞ ≤ Cǫ 2 from part (vi) of that proposition to bound ς cǫ h L ∞ −qǫ ≤ Cǫ 2 h L ∞ . Next, we estimate (E.3.8) by Proposition 6.3 and (E.3.7). Last, we use the estimate (6.4.7) on α ǫ to bound then we can induct on k and use (E.3.9) to obtain our desired estimate (E.3.3).
We have another formula, easily established through direct calculation and u-substitution: We γ c (x) sin(ω c (x + 1)) dx .
We can evaluate exactly each of the four terms above. Trigonometric identities and the asymptotics of γ c from (4.0.2) give lim R→∞ γ ′ c (R) sin(ω c R) − ω c γ c (R) cos(ω c R) = ω c sin(ω c ϑ c ).
Next, since B c e ±iωc· = 0, we have lim R→∞ R 0 γ c (x)(B c sin(ω c ·))(x) dx = 0, and so the second term is zero. For the third term, we rewrite the integrals using the addition formula for sine and then use the evenness of γ c sin(ω c ·) and the oddness of γ c cos(ω c ·) to show that the resulting integrals all add up to zero. Last, we rewrite the first integral in the fourth term as sin(ω c (x + ϑ c )) sin(ω c (x + 1)) dx = − sin(ω c ϑ c ) sin(ω c ).
All together, we have ι c [χ c ] = 2c 2 ω c − sin(ω c ) sin(ω c ϑ c ), and this is exactly the formula (5.3.7). The preceding calculation of ι c [χ c ] is similar to the work in Appendix D of [HW17], except there the small parameter µ appeared throughout the calculations as well, which allowed Hoffman and Wright to ignore some terms analogous to those in (F.3.4).
We refer to parts (iv) and (v) as "decay borrowing" estimates, as they permit us to achieve a Lipschitz estimate on a product starting in a space of lower decay by "borrowing" from the decay rates of one of the faster-decaying functions in the product. A version of part (iv) in particular was stated and proved as Lemma A.2 in [FW18]. From this proposition we immediately deduce two estimates for our quadratic nonlinearity Q from (1.3.3).

Next, we have
G.2. Proof of Proposition 5.3. We rely on the following lemma, which we prove in the next three sections. We inherit the general techniques from the myriad nanopteron estimates in [FW18], [HW17], [Fav], and [JW].
Lemma G.4. For all r ≥ 1, there are constants C(c, r), C(c) > 0 such that the following estimates hold for any |µ| ≤ µ per (c).
We do this in the following sections; throughout, we recall that these h µ c,k and ℓ µ c,k terms were defined in (5.1.3) and (5.3.4).
G.4.6. Lipschitz estimates for h µ c,5 and ℓ µ c,5 . These estimates follow immediately from the quadratic estimate (G.1.6) for Q. The individual mapping estimates above show that each of these eleven terms on the right is bounded by C(c, r)R µ map ( η r,q⋆(c) , a), and so we conclude that (G.2.3) also holds.