1.1. The travelling wave problem

A dimer Fermi–Pasta–Ulam–Tsingou (FPUT) lattice is a chain of infinitely many particles coupled to their nearest neighbours by springs, with motion restricted to the horizontal direction, and with at least one of the following material heterogeneities: either the particle masses alternate, or the spring potentials alternate, or both alternate. A dimer with alternating particles and identical springs is called a mass dimer; one with alternating springs and identical masses is a spring dimer. See Figure 1. Dimers are among the simplest nontrivial generalizations of the classical monatomic FPUT lattice, in which all of the particles have the same mass and all of the springs have the same potential [Reference Dauxois12, Reference Fermi, Pasta and Ulam21, Reference Pankov36, Reference Vainchtein39]. These lattices, and their many variants and generalizations, are prototypical models of wave dynamics in granular media [Reference Chong and Kevrekidis9, Reference Chong, Porter, Kevrekidis and Daraio10].

Figure 1.

The symmetric mass and spring dimers. (a) A mass dimer with alternating masses m₁ and m₂ and identical springs. (b) A spring dimer with alternating springs and identical masses m.

Figure 1(a) and (b) suggests that the mass and spring dimers possess certain physical ‘symmetries’ that a ‘general’ dimer, in which both masses and springs alternate, does not. We sketch such a general dimer, along with some notation for future use, in Figure 2. Physically, the mass dimer is the same when ‘reflected’ about a particle, as is the spring dimer when reflected about a spring. Such symmetries manifest themselves mathematically in a variety of useful ways, as we elaborate in Section 5, and these manifestations have been key to multiple prior analyses of mass and spring dimer dynamics. Here we consider the general dimer, and one of the main novelties of our techniques is that we do not rely at all on symmetry.

Figure 2.

A general dimer with alternating masses and springs.

Specifically, we construct nontrivial periodic travelling waves for general dimers with wave speed greater than a certain critical threshold called the lattice’s ‘speed of sound’ – that is, supersonic periodic travelling waves. We state our precise results below in Theorem 1.1 and discuss the connection of these travelling waves to the long wave problem in dimers, and related problems, in Section 1.2.

Here is our problem. Index the particles by integers $j \in \mathbb{Z}$ and let u_j be the displacement of the jth particle from its equilibrium position, let m_j be the mass of the jth particle, and let $\mathcal{V}_j$ be the potential of the spring connecting the jth and $(j+1)$st particles. To ensure a dimer structure, we assume

\begin{equation*} m_{j+2} = m_j \qquad\text{and }\qquad \mathcal{V}_{j+2} = \mathcal{V}_j \end{equation*}

for all j. More precisely, after nondimensionalization [Reference Faver15, Sec. 1.3, App. F.5], we take

(1.1)\begin{equation} m_j = \begin{cases} 1, \ j \text{ is odd} \\ m, \ j \text{ is even} \end{cases} \qquad\text{and }\qquad \mathcal{V}_j'(r) = \begin{cases} r + r^2 + \mathcal{O}(r^3), \ j \text{ is odd} \\ \kappa{r} + \beta{r}^2 + \mathcal{O}(r^3), \ j \text{ is even}. \end{cases} \end{equation}

The minimum regularity required for our proofs is that $\mathcal{V}_j \in \mathcal{C}^7(\mathbb{R})$ for precise technical reasons detailed in Appendix A.5, but for broader applications to FPUT travelling wave problems, we may as well assume $\mathcal{V}_j \in \mathcal{C}^{\infty}(\mathbb{R})$. Our methods require that the heterogeneity appear at the linear level, so we will always assume

(1.2)\begin{equation} \frac{1}{m} \gt 1 \qquad\text{or }\qquad \kappa \gt 1. \end{equation}

We emphasize that m and κ, and indeed all of the material data, are fixed throughout our analysis and that virtually all operators, quantities and thresholds depend on at least these quantities; we do not indicate such dependence in our notation. Additionally, beyond the regularity requirements on $\mathcal{V}_j$, the nonlinear terms, even the quadratic ones, play no important role.

Newton’s second law requires that the displacements u_j satisfy

(1.3)\begin{equation} m_j\ddot{u}_j = \mathcal{V}_j'(u_{j+1}-u_j)-\mathcal{V}_{j-1}'(u_j-u_{j-1}). \end{equation}

Under the travelling wave ansatz

(1.4)\begin{equation} u_j(t) = \begin{cases} p_1(j-ct), \ j \text{ is odd} \\ p_2(j-ct), \ j \text{ is even}, \end{cases} \qquad \mathbf{p}(X) := \begin{pmatrix} p_1(X) \\ p_2(X) \end{pmatrix}, \ X = j-ct, \end{equation}

these equations of motion become the advance-delay problem

(1.5)\begin{equation} \begin{cases} c^2p_1'' = \mathcal{V}_1'(S^1p_2-p_1) - \mathcal{V}_2'(p_1-S^{-1}p_2) \\ c^2mp_2'' = \mathcal{V}_2'(S^1p_1-p_2)-\mathcal{V}_1'(p_2-S^{-1}p_1). \end{cases} \end{equation}

Here, for $\theta \in \mathbb{R}$, S ^θ is the shift operator

\begin{equation*} (S^{\theta}p)(X) := p(X+\theta). \end{equation*}

The following is our main result for (1.5). We use the notation for periodic Sobolev spaces developed in Appendix A.2.

Theorem 1.1. Suppose that the lattice’s material data m_j and $\mathcal{V}_j$ satisfy the dimer condition (1.1) and the linear heterogeneity condition (1.2). Let the wave speed c in the ansatz (1.4) satisfy $|c| \gt c_{\star}$, where the lattice’s ‘speed of sound’ $c_{\star}$ is defined in (2.8). Then there exists $a_c \gt 0$ such that for $|a| \le a_c$, there is a travelling wave solution $\mathbf{p}_c^a$ to (1.5) of the form

(1.6)\begin{equation} \mathbf{p}_c^a(X) = \boldsymbol{\phi}_c^a(\omega_c^aX), \qquad \boldsymbol{\phi}_c^a(x) = a\boldsymbol{\nu}_1^c(x)+a^2\boldsymbol{\psi}_c^a(x). \end{equation}

The smooth, 2π-periodic profile term $\boldsymbol{\phi}_c^a$ and the frequency $\omega_c^a \in \mathbb{R}$ have the following properties.

1. (i) The leading order term $\boldsymbol{\nu}_1^c$ has an exact formula given below by (2.12).

2. (ii) The remainder term $\boldsymbol{\psi}_c^a$ is orthogonal to $\boldsymbol{\nu}_1^c$ and uniformly bounded in a in the sense that

   \begin{equation*} \langle\boldsymbol{\nu}_1^c,\boldsymbol{\psi}_c^a\rangle_{L_{\operatorname{per}}^2} = 0 \qquad\text{and }\qquad \sup_{|a| \le a_c} \|\boldsymbol{\psi}_c^a\|_{H_{\operatorname{per}}^r} \lt \infty, \ r \ge 0, \end{equation*}

   where the periodic Sobolev spaces $L_{\operatorname{per}}^2$ and $H_{\operatorname{per}}^r$ are defined in Appendix A.2.

3. (iii) The frequency $\omega_c^a$ has the expansion

   \begin{equation*} \omega_c^a = \omega_c + a\xi_c^a, \end{equation*}

   where $\omega_c \gt 0$ is the lattice’s ‘critical frequency’, as developed in Theorem 2.1, and

   \begin{equation*} \sup_{|a| \le a_c} |\xi_c^a| \lt \infty. \end{equation*}

These solutions are locally unique up to shifts and translations in the following sense. If $\mathbf{p}(X) = \boldsymbol{\phi}(\omega{X})$ solves (1.5) with $\|\boldsymbol{\phi}\|_{H_{\operatorname{per}}^2}$ and $|\omega-\omega_c|$ both sufficiently small, then there exist α, $\theta \in \mathbb{R}$ and $|a| \le a_c$ such that $\boldsymbol{\phi}(x) = \alpha\boldsymbol{\nu}_0 + \boldsymbol{\phi}_c^a(x+\theta)$, where $\boldsymbol{\nu}_0$ has the exact formula given by (2.11).

We approach this theorem from multiple points of view. Specifically, Sections 3 and 4 give proofs inspired by the techniques of Wright and Scheel [Reference Wright and Scheel40] for constructing asymmetric solitary wave solutions to a system of coupled KdV equations; the lack of symmetry in their problem manifests itself mathematically in a complication very close to ours, as we discuss below in Remark 3.4. The proof of local uniqueness up to translations follows from Corollary 3.3, and the proof of local uniqueness up to shifts appears in Section 3.5. Section 5 gives proofs in the special cases of mass and spring dimers when symmetry is present; this offers fresh perspectives on the prior results from [Reference Faver16, Reference Faver and Wright20]. And Section 6 develops precise quantitative estimates for the solution components from Theorem 1.1 relative to the wave speed c in the special case that $|c|$ is close to the speed of sound $c_{\star}$ (rather than just greater than $c_{\star}$ as in the theorem); we have excluded these estimates from the theorem above, as they are extremely technical. In particular, the exact, but general, hypotheses of Theorem 6.2 subsume all prior constructions of dimer periodics into one quantitative result. For brevity, Theorem 1.1 does not contain our results in the long wave scaling, which we discuss instead in Section 6.2.

We finally state the actual periodic travelling wave problem that we solve to prove Theorem 1.1; the following notation was not strictly necessary above, but all of our subsequent work depends on it. Since we are interested in periodic travelling waves, we adjust the original travelling wave ansatz (1.4) by decoupling the profile and frequency via the additional ansatz

\begin{equation*} \mathbf{p}(X) = \boldsymbol{\phi}(\omega{X}), \qquad \boldsymbol{\phi} := \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix}. \end{equation*}

The new profiles ϕ ₁ and ϕ ₂ are now 2π-periodic and $\omega \in \mathbb{R}$. We emphasize that the parameter ω is now the wavenumber and does not denote a dispersion relation; this parameter ω will serve as the crucial bifurcation parameter in our analysis.

The travelling wave equations (1.5) then become

(1.7)\begin{equation} \begin{cases} c^2\omega^2\phi_1'' = \mathcal{V}_1'(S^{\omega}\phi_2-\phi_1) - \mathcal{V}_2'(\phi_1-S^{-\omega}\phi_2) \\ mc^2\omega^2\phi_2'' = \mathcal{V}_2'(S^{\omega}\phi_1-\phi_2) - \mathcal{V}_1'(\phi_2-S^{-\omega}\phi_1). \end{cases} \end{equation}

We compress (1.7) in the form

(1.8)\begin{equation} \boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0, \end{equation}

where

(1.9)\begin{equation} \boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) := c^2\omega^2M\boldsymbol{\phi}'' + \begin{pmatrix} \mathcal{V}_2'(\phi_1-S^{-\omega}\phi_2) - \mathcal{V}_1'(S^{\omega}\phi_2-\phi_1) \\ \mathcal{V}_1'(\phi_2-S^{-\omega}\phi_1) - \mathcal{V}_2'(S^{\omega}\phi_1-\phi_2) \end{pmatrix} \end{equation}

and

(1.10)\begin{equation} M := \begin{bmatrix} 1 &0 \\ 0 &m \end{bmatrix}. \end{equation}

The primary challenge that we confront is that the linearization $D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_c(0,\omega_c)$ has a three-dimensional kernel and cokernel. Translation invariance allows us to reduce both dimensions to two (and by restricting the domain to the right subspace, we could lower the dimension of the kernel even further), but, in the absence of symmetry, we cannot get below that. Specifically, we must contend with a two-dimensional cokernel, which adds two solvability conditions to our problem without giving us extra variables to help meet them. We now discuss the broader relevance of this periodic travelling wave problem and, in the process, why this dimension counting is so important.

1.2. Motivation, context and connections other FPUT travelling wave problems

Our primary motivation in constructing these particular periodics is the long wave problem for dimers. This limit looks for travelling waves whose profiles are close to a suitably scaled $\operatorname{sech}^2$-type solution of a KdV equation that acts as the ‘continuum limit’ for the lattice. More precisely, one posits $\mathbf{p}(X) = \epsilon^2\mathbf{h}(\epsilon{X})$ and $c^2 = c_{\star}^2 + \epsilon^2$, with $c_{\star}$ given by (2.8) and ϵ > 0 small. The solutions from Theorem 1.1 are not strictly long-wave solutions, as they lack this scaling and are valid for all wave speeds $|c| \gt c_{\star}$. We discuss this further in Section 6.2. The relevance of this ansatz is that in a ‘polyatomic’ FPUT lattice, for which the material data repeats with some arbitrary period, long wave-scaled solutions to certain KdV equations (whose coefficients depend on the lattice’s material data) are very good approximations to solutions to the equations of motion over very long time scales [Reference Chirilus-Bruckner, Chong, Prill and Schneider8, Reference Gaison, Moskow, Wright and Zhang25, Reference Schneider and Wayne38].

Faver and Wright constructed long wave solutions for the mass dimer [Reference Faver and Wright20] and Faver treated the spring dimer [Reference Faver16]. Faver and Hupkes produced a different development of mass and spring dimer nanopterons via spatial dynamics in [Reference Faver and Hupkes19] and obtained results for equilibrium displacement coordinates as we do; Deng and Sun [Reference Deng and Sun13] performed a related spatial dynamics analysis to yield similar results. These dimer travelling waves were not solitary waves, as Friesecke and Pego found for the monatomic lattice [Reference Friesecke and Pego23], but rather nanopterons [Reference Boyd7]: the superposition of a leading-order localized (here, $\operatorname{sech}^2$-type) term, a higher-order localized remainder, and a high-frequency periodic ‘ripple’ of amplitude small beyond all algebraic orders of the long wave parameter. Both constructions relied on lattice symmetries in two critical steps to adapt functional analytic techniques from Beale’s work on capillary gravity water waves [Reference Beale5] and its later deployment by Amick and Toland [Reference Amick and Toland3] for a model fourth-order KdV equation.

Firstly, as mentioned above, the periodics in [Reference Faver16, Reference Faver and Wright20] were constructed with a modified ‘bifurcation from a simple eigenvalue’ argument in the style of Crandall and Rabinowitz and Zeidler [Reference Crandall and Rabinowitz11, Reference Zeidler41]. We adapt further this bifurcation analysis in our arguments, and our preferred reference is [Reference Kielhöfer31, Thm. 1.5.1]. Symmetry permitted the restriction of the travelling wave problem $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ from (1.8) and (1.9) to function spaces on which the linearization $D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_c(0,\omega)$ at $\boldsymbol{\phi} = 0$ and $\omega = \omega_c$, with ω_c as the ‘critical frequency’ from Theorem 2.1, had a one-dimensional kernel and cokernel. This was the key to the modified bifurcation from a simple eigenvalue argument.

Up to a useful linear change of coordinates that diagonalized the travelling wave problem and the long wave scaling, the long wave periodics in [Reference Faver16, Reference Faver and Wright20] have the same structure as ours from Theorem 1.1. However, the main technical accomplishment of our results here is that we manage a two-dimensional kernel and cokernel in the absence of symmetry via other inherent properties of the lattice – namely, the special ‘orthogonality condition’ that $\langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\phi}'\rangle_{L_{\operatorname{per}}^2} = 0$, proved in Corollary 3.2 and Lemma 4.1. While there certainly exist other results on bifurcations with two-dimensional kernels, their hypotheses are inappropriate for our problem. For example, [Reference Krömer, Healey and Kielhöfer32] and [Reference Liu, Shi and Wang34] assume certain ‘nondegeneracy’ conditions on what for us would be the second derivative $D_{\boldsymbol{\phi}\boldsymbol{\phi}}^2\boldsymbol{\Phi}_c(0,\omega_c)$, over which we expect to have no control (beyond its existence), while [Reference Akers, Ambrose and Sulon1] and [Reference Baldi and Toland4] assume some (non)resonance conditions among their critical frequencies. Indeed, the two-dimensional kernel is far less of a problem than the two-dimensional cokernel, as we cannot make the latter smaller by restricting the domain to a better-behaved subspace.

The second use of symmetry in the full nanopteron constructions of [Reference Faver16, Reference Faver and Wright20] was somewhat subtler and involved the actual need for periodics in the first place. The obstacle was that attempting to solve the travelling wave problem (1.5) by a merely localized perturbation from the $\operatorname{sech}^2$-type continuum limit – and thereby construct solitary travelling waves in dimers – resulted in an overdetermined system with two unknowns (the two components of the localized perturbation) but four equations. These are the expected two components from (1.5) and a surprising ‘solvability condition’: the vanishing of the Fourier transform of a certain related operator at $\pm\omega_c$. Symmetry ensured that the vanishing at ω_c implied the vanishing at −ω_c, reducing the overdetermined problem to only three equations, which were managed by adding a third variable via the periodic amplitude – which is exactly why we seek periodics in Theorem 1.1 that are parametrized in amplitude. While the full nanopteron problem in the general dimer without symmetry remains challenging and beyond the scope of our work here to address, the periodic solutions constructed here will be a fundamental component of the nanopteron ansatz for the general dimer’s long wave problem.

Periodic travelling waves for lattices have been constructed in several other ‘material limit’ regimes in addition to the long wave limit. These include the small mass limit for mass dimers by Hoffman and Wright [Reference Hoffman and Wright27, Thm. 5.1], the equal mass limit for mass dimers by Faver and Hupkes [Reference Faver and Hupkes18, Prop. 3.3], and the small mass limit for the mass-in-mass variant of monatomic FPUT by Faver [Reference Faver14, Thm. 2]. Each of these limits views the heterogeneous lattice as a small material perturbation of a monatomic FPUT lattice, and the nanopteron is a small nonlocal perturbation of a monatomic solitary wave [Reference Friesecke and Pego23, Reference Friesecke and Wattis24]. While each limit has a nontrivially different solvability condition that makes the travelling wave problem overdetermined, all of the periodic constructions are fundamentally alike and can be deduced from Theorem 6.2.

In all of these nanopteron problems, it is not enough to have a family of periodic solutions whose amplitude can serve as an extra variable to close the overdetermined travelling wave problem. Additionally, one also needs exact, uniform, quantitative estimates on how these periodic solutions behave with respect to the overall problem’s natural small parameter (the long wave problem ϵ, the small mass ratio, etc.) To that end, a result like Theorem 1.1, which does not uniformly depend on the wave speed c, is not enough. This is the motivation for the results in Section 6, which are too cumbersome to be included in Theorem 1.1.

These are not the only methods for producing periodic travelling waves for FPUT, and we give a brief, selected overview of others here for both monatomic lattices and dimers, in various limiting regimes and for various kinds of material data. Friesecke and Mikikits-Leitner [Reference Friesecke and Mikikits-Leitner22] adapted the perturbative approach for monatomic solitary waves from [Reference Friesecke and Pego23] to prove the existence of long wave periodics in the monatomic lattice that were small perturbations of a KdV cnoidal profile. Pankov constructed periodics in the monatomic lattice using variational methods [Reference Pankov36], as did Qin for mass dimers [Reference Qin37] with spring force given by the FPUT β-model, i.e., roughly of the form $\mathcal{V}'(r) = r+\mathcal{O}(r^3)$; while these proofs do not give information on amplitude, the constructed periodics do exist for arbitrary wavenumbers and frequencies. Herrmann constructed both solitary waves and periodic travelling waves in monatomic FPUT with convex spring potentials via variational methods [Reference Herrmann26]; these periodics can be constructed to have arbitrary mean value, which in turn determines the speed of the travelling wave. Iooss used spatial dynamics and centre manifold theory to capture all small travelling waves in monatomic FPUT, including solitary waves and periodics [Reference Iooss29]. Betti and Pelinovsky used an implicit function theorem argument to produce periodics in mass dimers with Hertzian spring potentials [Reference Betti and Pelinovsky6], and we note with interest that their proofs also relied on symmetry to reduce the dimension of a key linearization’s kernel. James also used implicit function theory to construct periodics parametrized by amplitude in monatomic FPUT with Hertzian potential [Reference James30]. Finally, we mention that Lombardi’s spatial dynamics method for nanopterons under very general hypotheses includes the full development of periodics from that point of view [Reference Lombardi35], with the more stringent requirement that the spring potentials be real analytic.

1.3. Notation

We summarize several aspects of notation that we will use without further comment.

• • If $\mathcal{X}$ is a vector space, then $\mathcal{I}_{\mathcal{X}}$ is the identity operator on $\mathcal{X}$.

• • If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ are normed spaces and $f \colon \mathcal{U} \subseteq \mathcal{X} \times \mathcal{Y} \to \mathcal{Z}$ is differentiable at some $(x_0,y_0) \in \mathcal{U}$, then we denote its partial derivative at $(x_0,y_0)$ with respect to x by $D_xf(x_0,y_0)$. Likewise, $D_yf(x_0,y_0)$ is the partial derivative with respect to y. We reserve the notation $f' = \partial_xf$ for a function $f \colon I \subseteq \mathbb{R} \to \mathbb{R}$.

• • If $\mathcal{X}$ and $\mathcal{Y}$ are sets and $f \colon \mathcal{U} \subseteq \mathcal{X} \to \mathcal{Y}$ is a function, then for any $\mathcal{U}_0 \subseteq \mathcal{U}$ we denote by ${\left. f \vphantom{\big|} \right|_{\mathcal{U}_0} }$ the restriction of f to $\mathcal{U}_0$.

• • If $\mathcal{X}$ and $\mathcal{Y}$ are Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_{\mathcal{X}}$ and $\langle\cdot,\cdot\rangle_{\mathcal{Y}}$, respectively, then the adjoint of a bounded linear operator $\mathcal{T} \colon \mathcal{X} \to \mathcal{Y}$ is the bounded linear operator $\mathcal{T}^* \colon \mathcal{Y} \to \mathcal{X}$ satisfying $\langle\mathcal{T}{x},y\rangle_{\mathcal{Y}} = \langle x,\mathcal{T}^*y\rangle_{\mathcal{X}}$ for all $x \in \mathcal{X}$ and $y \in \mathcal{Y}$. If the range $\mathcal{T}(\mathcal{X})$ of $\mathcal{T}$ is closed, then $\mathcal{T}(\mathcal{X}) = \ker(\mathcal{T}^*)^{\perp}$, where $\mathcal{U}^{\perp}$ is the orthogonal complement of $\mathcal{U} \subseteq \mathcal{Y}$.

• • If $\mathcal{X}$ is a normed space, $x \in \mathcal{X}$, and r > 0, then $\mathcal{B}(x;r)$ is the open ball

  \begin{equation*} \mathcal{B}(x;r) := \!\left\{y \in \mathcal{X} \ \big| \ \|x-y\|_{\mathcal{X}} \lt r\right\}. \end{equation*}

• • If $\mathcal{X}$ and $\mathcal{Y}$ are normed spaces, then $\mathbf{B}(\mathcal{X},\mathcal{Y})$ is the space of bounded linear operators from $\mathcal{X}$ to $\mathcal{Y}$ with operator norm $\|\mathcal{T}\|_{\mathcal{X} \to \mathcal{Y}}$.

3.1. The gradient structure of the travelling wave problem

We rewrite the travelling wave operator $\boldsymbol{\Phi}_c$ from (1.9) as the $L_{\operatorname{per}}^2$-gradient of a certain ‘kinetic + potential energy’ functional on $H_{\operatorname{per}}^2(\mathbb{R}^2)$. This formulation yields transparent proofs of certain properties of $\boldsymbol{\Phi}_c$ from shift invariance, and from these properties follow our first existence proof in Sections 3.3 and 3.4.

Firstly, we need some new notation; all of the consequences below of this notation are straightforward calculations, which we omit. For $\omega \in \mathbb{R}$, put

(3.1)\begin{equation} \Delta_+(\omega) := \begin{bmatrix} -1 &S^{\omega} \\ 1 &-S^{-\omega} \end{bmatrix} \qquad\text{and }\qquad \Delta_-(\omega) = \begin{bmatrix} 1 &-1 \\ -S^{-\omega} &S^{\omega} \end{bmatrix}. \end{equation}

We then have the adjoint relationship

(3.2)\begin{equation} \langle\Delta_+(\omega)\boldsymbol{\phi},\boldsymbol{\eta}\rangle = -\langle\boldsymbol{\phi},\Delta_-(\omega)\boldsymbol{\eta}\rangle \end{equation}

for any ϕ, $\boldsymbol{\eta} \in L_{\operatorname{per}}^2(\mathbb{R}^2)$.

Next, let

\begin{equation*} \boldsymbol{\mathcal{V}}(\mathbf{p}) := \begin{pmatrix} \mathcal{V}_1(p_1) \\ \mathcal{V}_2(p_2) \end{pmatrix} \qquad\text{and }\qquad \boldsymbol{\mathcal{V}}'(\mathbf{p}) := \begin{pmatrix} \mathcal{V}_1'(p_1) \\ \mathcal{V}_2'(p_2) \end{pmatrix}, \end{equation*}

where $\mathcal{V}_1$ and $\mathcal{V}_2$ are the spring potentials from (1.1), and $\mathbf{p} = (p_1,p_2) \in L_{\operatorname{per}}^2(\mathbb{R}^2)$. For $\mathbf{v} = (v_1,v_2)$, $\mathbf{w} = (v_1,v_2) \in \mathbb{R}^2$, define componentwise multiplication as

\begin{equation*} \mathbf{v}.\mathbf{w} := \begin{pmatrix} v_1v_1 \\ v_2v_2 \end{pmatrix}. \end{equation*}

We then have the derivative formula

(3.3)\begin{equation} D_{\mathbf{p}}\boldsymbol{\mathcal{V}}(\mathbf{p})\grave{\mathbf{p}} = \boldsymbol{\mathcal{V}}'(\mathbf{p}).\grave{\mathbf{p}} \end{equation}

for any $\grave{\mathbf{p}} \in L_{\operatorname{per}}^2(\mathbb{R}^2)$. Define $\boldsymbol{1}(x) := (1,1)$; then since v, $\mathbf{w} \in \mathbb{R}^2$ have real entries, the useful identity

(3.4)\begin{equation} \langle\mathbf{v}.\mathbf{w},\boldsymbol{1}\rangle = \langle\mathbf{v},\mathbf{w}\rangle \end{equation}

is true.

Last, we have

\begin{equation*} \Delta_-(\omega)\boldsymbol{\mathcal{V}}'(\Delta_+(\omega)\boldsymbol{\phi}) = \begin{pmatrix} \mathcal{V}_1'(S^{\omega}\phi_2-\phi_1)-\mathcal{V}_2'(\phi_1-S^{-\omega}\phi_2) \\ \mathcal{V}_2'(S^{\omega}\phi_1-\phi_2)-\mathcal{V}_1'(\phi_2-S^{-\omega}\phi_1) \end{pmatrix}. \end{equation*}

Comparing this to the second term in $\boldsymbol{\Phi}_c$ from (1.9), we conclude

(3.5)\begin{equation} \boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = c^2\omega^2M\boldsymbol{\phi}''-\Delta_-(\omega)\boldsymbol{\mathcal{V}}'(\Delta_+(\omega)\boldsymbol{\phi}). \end{equation}

This version of $\boldsymbol{\Phi}_c$ allows us to recognize it as a gradient; similar calculations for the monatomic lattice appear in [Reference Pankov36, Prop. 3.2] and for mass dimers with the FPUT β-model in [Reference Qin37, Lem. 3.1].

Theorem 3.1. Let $c \in \mathbb{R}$. Define

(3.6)\begin{equation} \mathcal{T} \colon H_{\operatorname{per}}^2(\mathbb{R}^2) \times \mathbb{R} \to \mathbb{R} \colon (\boldsymbol{\phi},\omega) \mapsto \frac{\omega^2}{2}\langle M\boldsymbol{\phi}'',\boldsymbol{\phi}\rangle \end{equation}

and, with $\boldsymbol{1}(x) := (1,1)$,

(3.7)\begin{equation} \mathcal{P} \colon H_{\operatorname{per}}^2(\mathbb{R}^2) \times \mathbb{R} \to \mathbb{R} \colon (\boldsymbol{\phi},\omega) \mapsto \langle\boldsymbol{\mathcal{V}}(\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{1}\rangle \end{equation}

Put

(3.8)\begin{equation} \mathcal{G}_c := c^2\mathcal{T} + \mathcal{P}. \end{equation}

Then

\begin{equation*} \boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = \nabla\mathcal{G}_c(\boldsymbol{\phi},\omega) \end{equation*}

in the sense that

(3.9)\begin{equation} D_{\boldsymbol{\phi}}\mathcal{G}_c(\boldsymbol{\phi},\omega)\boldsymbol{\eta} = \langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\eta}\rangle \end{equation}

for all ϕ, $\boldsymbol{\eta} \in H_{\operatorname{per}}^2(\mathbb{R}^2)$ and $\omega \in \mathbb{R}$.

Proof. The proof is just a careful calculation using the definition of the derivative and the inner product $\langle\cdot,\cdot\rangle$ and the various identities stated above. More precisely, we compute the following.

Firstly, for ϕ, $\boldsymbol{\eta} \in H_{\operatorname{per}}^2(\mathbb{R}^2)$ and ω, $h \in \mathbb{R}$, we use the definition of $\mathcal{T}$ in (3.6) to compute

\begin{equation*} \mathcal{T}(\boldsymbol{\phi} + h\boldsymbol{\eta},\omega)-\mathcal{T}(\boldsymbol{\phi},\omega) = h\langle\omega^2M\boldsymbol{\phi}'',\boldsymbol{\eta}\rangle + h^2\frac{\omega^2\langle M\boldsymbol{\eta}'',\boldsymbol{\eta}\rangle}{2}. \end{equation*}

This uses two applications of the integration by parts identity (A.3) to compute $\langle\boldsymbol{\phi}'',\boldsymbol{\eta}\rangle = \langle\boldsymbol{\phi},\boldsymbol{\eta}''\rangle$, the symmetry of M, and the assumption that ϕ and η are $\mathbb{R}^2$-valued. It follows that

\begin{equation*} D_{\boldsymbol{\phi}}\mathcal{T}(\boldsymbol{\phi},\omega)\boldsymbol{\eta} = \langle\omega^2M\boldsymbol{\phi}'',\boldsymbol{\eta}\rangle. \end{equation*}

Next, we use the definition of $\mathcal{P}$ in (3.7) to compute

\begin{align*} D_{\boldsymbol{\phi}}\mathcal{P}(\boldsymbol{\phi},\omega)\boldsymbol{\eta} &= \langle D_{\boldsymbol{\phi}}\boldsymbol{\mathcal{V}}(\Delta_+(\omega)\boldsymbol{\phi})\Delta_+(\omega)\boldsymbol{\eta},\boldsymbol{1}\rangle \text{ by the chain rule} \\ &= \langle\boldsymbol{\mathcal{V}}'(\Delta_+(\omega)\boldsymbol{\phi}).\Delta_+(\omega)\boldsymbol{\eta},\boldsymbol{1}\rangle \text{ by (3.3)} \\ &= \langle\boldsymbol{\mathcal{V}}'(\Delta_+(\omega)\boldsymbol{\phi},\Delta_+(\omega)\boldsymbol{\eta}\rangle \text{ by (3.4)} \\ &= -\langle\Delta_-(\omega)\boldsymbol{\mathcal{V}}'(\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{\eta}\rangle \text{ by (3.2)}. \end{align*}

All together, we have

\begin{equation*} D_{\boldsymbol{\phi}}\mathcal{G}_c(\boldsymbol{\phi},\omega)\boldsymbol{\eta} = D_{\boldsymbol{\phi}}\mathcal{T}(\boldsymbol{\phi},\omega)\boldsymbol{\eta} + D_{\boldsymbol{\phi}}\mathcal{P}(\boldsymbol{\phi},\omega)\boldsymbol{\eta} = \langle c^2\omega^2M\boldsymbol{\phi}'',\boldsymbol{\eta}\rangle - \langle\Delta_-(\omega)\boldsymbol{\mathcal{V}}'(\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{\eta}\rangle. \end{equation*}

By our rewritten formula for $\boldsymbol{\Phi}_c$ in (3.5), this proves (3.9).

We collect two families of properties of $\mathcal{G}_c$ and $\boldsymbol{\Phi}_c$. The proofs for $\mathcal{G}_c$ are easy consequences of its definition in Theorem 3.1, while those for $\boldsymbol{\Phi}_c$ are also straightforward and could in fact be done (somewhat more laboriously) using just the definition of $\boldsymbol{\Phi}_c$ in (1.9). However, we present proofs for $\boldsymbol{\Phi}_c$ here as a consequence of the gradient formulation to emphasize the utility and efficiency of this formulation.

An immediate consequence of the translation invariance of $\boldsymbol{\Phi}_c$ from (3.14) is that solutions ϕ to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ are only unique up to translation by $\boldsymbol{\nu}_0$, as claimed in Theorem 1.1.

3.2. Function spaces and projection operators

The translation invariance identities (3.14) and (3.15) mean that we can effectively ignore the contributions of $\boldsymbol{\nu}_0$ to the problem $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$. So, we put

(3.16)\begin{equation} \mathcal{Y} := \!\left\{\boldsymbol{\eta} \in L_{\operatorname{per}}^2(\mathbb{R}^2) \ \big| \ \langle\boldsymbol{\eta},\boldsymbol{\nu}_0\rangle = 0\right\}, \end{equation}

(3.17)\begin{equation} \mathcal{X} := H_{\operatorname{per}}^2(\mathbb{R}^2) \cap \mathcal{Y}, \end{equation}

and

(3.18)\begin{equation} \mathcal{Z}_c := \operatorname{span}(\boldsymbol{\nu}_1^c,\boldsymbol{\nu}_2^c). \end{equation}

It follows that $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) \in \mathcal{Y}$ for all $\boldsymbol{\phi} \in \mathcal{X}$ and $\omega \in \mathbb{R}$, and also

\begin{equation*} \mathcal{Z}_c = \ker(\mathcal{L}_c[\omega_c]) \cap \mathcal{X} = \ker(\mathcal{L}_c[\omega_c]^*) \cap \mathcal{Y}. \end{equation*}

Define

(3.19)\begin{equation} \Pi_c \colon \mathcal{Y} \to \mathcal{Z}_c \colon \boldsymbol{\phi} \mapsto \langle\boldsymbol{\phi},\boldsymbol{\nu}_1^c\rangle\boldsymbol{\nu}_1^c + \langle\boldsymbol{\phi},\boldsymbol{\nu}_2^c\rangle\boldsymbol{\nu}_2^c. \end{equation}

Since $\boldsymbol{\nu}_1^c$ and $\boldsymbol{\nu}_2^c$ are orthogonal, per Corollary 2.2, the operator $\Pi_c$ is the orthogonal projection of $\mathcal{Y}$ (and $\mathcal{X}$) onto $\mathcal{Z}_c$. In particular,

(3.20)\begin{equation} \langle\Pi_c\boldsymbol{\phi},\boldsymbol{\eta}\rangle = \langle\boldsymbol{\phi},\Pi_c\boldsymbol{\eta}\rangle \end{equation}

for all ϕ, $\boldsymbol{\psi} \in \mathcal{Y}$.

It turns out to be quite useful for us that the projection $\Pi_c$ and the first derivative $\partial_x$ commute.

Lemma 3.5. Let $\boldsymbol{\phi} \in H_{\operatorname{per}}^1(\mathbb{R}^2)$. Then

(3.21)\begin{equation} \Pi_c\partial_x\boldsymbol{\phi} = \partial_x\Pi_c\boldsymbol{\phi}. \end{equation}

Proof. We use the integration by parts identity $\langle\boldsymbol{\phi}',\boldsymbol{\eta}\rangle = -\langle\boldsymbol{\phi},\boldsymbol{\eta}'\rangle$ from (A.3) and the derivative identities (2.15) to compute

\begin{align*} \Pi_c\partial_x\boldsymbol{\phi} &= \langle\boldsymbol{\phi}',\boldsymbol{\nu}_1^c\rangle\boldsymbol{\nu}_1^c + \langle\boldsymbol{\phi}',\boldsymbol{\nu}_2^c\rangle\boldsymbol{\nu}_2^c \\ &= -\langle\boldsymbol{\phi},\partial_x\boldsymbol{\nu}_1^c\rangle\boldsymbol{\nu}_1^c - \langle\boldsymbol{\phi},\partial_x\boldsymbol{\nu}_2^c\rangle\boldsymbol{\nu}_2^c \\ &= \langle\boldsymbol{\phi},\boldsymbol{\nu}_2^c\rangle\boldsymbol{\nu}_1^c - \langle\boldsymbol{\phi},\boldsymbol{\nu}_1^c\rangle\boldsymbol{\nu}_2^c \\ &= \langle\boldsymbol{\phi},\boldsymbol{\nu}_2^c\rangle\partial_x\boldsymbol{\nu}_2^c + \langle\boldsymbol{\phi},\boldsymbol{\nu}_1^c\rangle\partial_x\boldsymbol{\nu}_1^c \\ &= \partial_x\Pi_c\boldsymbol{\phi}. \end{align*}

Last, we state precisely the regularity of $\boldsymbol{\Phi}_c$ and some of its derivatives on periodic Sobolev spaces. The technical challenge here is that $\boldsymbol{\Phi}_c$ is infinitely differentiable from $H_{\operatorname{per}}^2(\mathbb{R}^2)$ to $L_{\operatorname{per}}^2(\mathbb{R}^2)$ with respect to ϕ, but any order derivative with respect to ϕ is only once continuously differentiable with respect to ω. This is ultimately a consequence of the limited differentiability of shift operators between periodic Sobolev spaces, as we discuss in Appendix A.4. We prove the next lemma in Appendix A.4.

3.3. The Lyapunov–Schmidt decomposition: infinite-dimensional analysis

The approach here is classical and follows, for example, the proof of the Crandall–Rabinowitz–Zeidler theorem in [Reference Kielhöfer31, Thm. 1.5.1]. The difference appears in the following section, when we manage the two-dimensional kernel.

We use the projection operator $\Pi_c$ from (3.19) to make a Lyapunov–Schmidt decomposition for our problem $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$. Firstly, with the spaces $\mathcal{X}$ and $\mathcal{Y}$ defined in (3.17) and (3.16), let

(3.22)\begin{equation} \mathcal{X}_c^{\infty} := (\mathcal{I}_{\mathcal{X}}-\Pi_c)(\mathcal{X}) \qquad\text{and }\qquad \mathcal{Y}_c^{\infty} := (\mathcal{I}_{\mathcal{Y}} -\Pi_c)(\mathcal{Y}), \end{equation}

where $\mathcal{I}_{\mathcal{X}}$ and $\mathcal{I}_{\mathcal{Y}}$ are the identity operators on $\mathcal{X}$ and $\mathcal{Y}$, respectively. Consequently,

(3.23)\begin{equation} \mathcal{Z}_c \cap \mathcal{X}_c^{\infty} = \mathcal{Z}_c \cap \mathcal{Y}_c^{\infty} = \{0\}. \end{equation}

Next, write $\boldsymbol{\phi} = \boldsymbol{\nu}+\boldsymbol{\psi}$, where $\boldsymbol{\nu} \in \mathcal{Z}_c$ and $\boldsymbol{\psi} \in \mathcal{X}_c^{\infty}$. Then $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ if and only if

(3.24)\begin{equation} \quad\qquad\qquad\qquad\qquad\qquad\qquad\begin{cases} (\mathcal{I}_{\mathcal{Y}} -\Pi_c)\boldsymbol{\Phi}_c(\boldsymbol{\nu}+\boldsymbol{\psi},\omega) = 0 \qquad\qquad\qquad\qquad\qquad\qquad\quad(3.24a)\\ \Pi_c\boldsymbol{\Phi}_c(\boldsymbol{\nu}+\boldsymbol{\psi},\omega) = 0. \qquad\qquad\qquad\quad\qquad\quad\qquad\qquad\qquad(3.24b) \end{cases} \end{equation}

We solve (3.24a) quickly with a direct application of the implicit function theorem [Reference Kielhöfer31, Thm. I.1.1].

Define

(3.25)\begin{equation} \mathcal{F}_c^{\infty} \colon \mathcal{X}_c^{\infty} \times \mathcal{Z}_c \times \mathbb{R} \to \mathcal{Y}_c^{\infty} \colon (\boldsymbol{\psi},\boldsymbol{\nu},\omega) \mapsto (\mathcal{I}_{\mathcal{Y}} -\Pi_c)\boldsymbol{\Phi}_c(\boldsymbol{\nu}+\boldsymbol{\psi},\omega). \end{equation}

Certainly $\mathcal{F}_c^{\infty}(0,0,\omega) = 0$ for all ω, and we have

\begin{equation*} D_{\boldsymbol{\psi}}\mathcal{F}_c^{\infty}(0,0,\omega_c) = (\mathcal{I}_{\mathcal{Y}} -\Pi_c){\left. \mathcal{L}_c[\omega_c] \vphantom{\big|} \right|_{\mathcal{X}_c^{\infty}} }. \end{equation*}

This operator has trivial kernel in $\mathcal{X}_c^{\infty}$ and trivial cokernel in $\mathcal{Y}_c^{\infty}$ by (3.23), and so it is invertible. (More precisely, the closure of its range is the orthogonal complement of its cokernel, which is all of $\mathcal{Y}_c^{\infty}$. But the range is closed by the estimate in Corollary 2.4.) Since $\boldsymbol{\Phi}_c \in \mathcal{C}^1(H_{\operatorname{per}}^2(\mathbb{R}^2),L_{\operatorname{per}}^2(\mathbb{R}^2))$ by Lemma 3.6, with $\boldsymbol{\Phi}_c(0,\omega) = 0$ for all ω, the implicit function theorem yields δ_c, $\epsilon_c \gt 0$ and a map $\boldsymbol{\Psi}_c \in \mathcal{C}^1\big(\mathcal{B}_{\mathcal{Z}_c \times \mathbb{R}}((0,\omega_c);\delta_c), \mathcal{B}_{\mathcal{X}_c^{\infty}}(0;\epsilon_c)\big)$ such that

(3.26)\begin{equation} \mathcal{F}_c^{\infty}(\boldsymbol{\Psi}_c(\boldsymbol{\nu},\omega),\boldsymbol{\nu},\omega) = 0. \end{equation}

Moreover, if $\mathcal{F}_c^{\infty}(\boldsymbol{\psi},\boldsymbol{\nu},\omega) = 0$ for some $\boldsymbol{\psi} \in \mathcal{B}_{\mathcal{X}_c^{\infty}}(0;\epsilon_c)$ and $(\boldsymbol{\nu},\omega) \in \mathcal{B}_{\mathcal{Z}_c \times \mathbb{R}}((0,\omega_c);\delta_c)$, then $\boldsymbol{\psi} = \boldsymbol{\Psi}_c(\boldsymbol{\nu},\omega)$. (Recall that $\mathcal{B}_{\mathcal{X}}(x_0;r) = \!\left\{x \in \mathcal{X} \ \big| \ \|x-x_0\|_{\mathcal{X}} \lt r\right\}$ for $x_0 \in \mathcal{X}$ and r > 0.) We pause to collect some useful properties of this map $\boldsymbol{\Psi}_c$.

3.4. The Lyapunov–Schmidt decomposition: finite-dimensional analysis

Now we solve the second equation (3.24) in the Lyapunov–Schmidt decomposition with $\boldsymbol{\psi} = \boldsymbol{\Psi}_c(\boldsymbol{\nu},\omega)$. This amounts to solving the pair of equations

(3.31)\begin{equation} \begin{cases} \langle\boldsymbol{\Phi}_c(\boldsymbol{\nu}+\boldsymbol{\Psi}_c(\boldsymbol{\nu},\omega),\omega),\boldsymbol{\nu}_1^c\rangle = 0 \\ \langle\boldsymbol{\Phi}_c(\boldsymbol{\nu}+\boldsymbol{\Psi}_c(\boldsymbol{\nu},\omega),\omega),\boldsymbol{\nu}_2^c\rangle = 0, \end{cases} \end{equation}

where ν is a linear combination of the two linearly independent eigenfunctions $\boldsymbol{\nu}_1^c$ and $\boldsymbol{\nu}_2^c$. The apparent quandary is that we want solutions parametrized in amplitude, so formally this suggests $\boldsymbol{\nu} + \boldsymbol{\Psi}_c(\boldsymbol{\nu},\omega) = \mathcal{O}(a)$. This leads to our taking $\boldsymbol{\nu} = a\boldsymbol{\nu}_1^c$ below, which may appear to remove a degree of freedom from the ansatz. In turn, this could appear to be problematic, given that we have two equations to solve above in (3.31). Ostensibly, we could have stayed with ν as a combination of $\boldsymbol{\nu}_1^c$ and $\boldsymbol{\nu}_2^c$. None of this, however, is a problem, and we discuss at length in Section 3.5 why.

With the choice of

(3.32)\begin{equation} \boldsymbol{\nu} = a\boldsymbol{\nu}_1^c \end{equation}

for $a \in \mathbb{R}$ sufficiently small, we convert (3.31) to

(3.33)\begin{equation} \qquad\qquad\qquad\qquad\begin{cases} \langle\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega),\omega),\boldsymbol{\nu}_1^c\rangle = 0 \qquad\qquad\qquad\,\qquad\qquad\,\qquad\qquad\quad (3.33a)\\ \langle\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega),\omega),\boldsymbol{\nu}_2^c\rangle = 0. \qquad\qquad\qquad\,\qquad\qquad\,\qquad\qquad\quad (3.33b) \end{cases} \end{equation}

We claim that (3.33b) always holds. Indeed, for a = 0, it is trivially true, since $\boldsymbol{\Phi}_c(0,\omega) = 0$, while for $a \ne 0$ we have

\begin{align*} \langle\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega),\omega),\boldsymbol{\nu}_2^c\rangle &= a^{-1}\langle\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega),\omega),a\boldsymbol{\nu}_2^c\rangle \\ &= -a^{-1}\langle\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega),\omega),\partial_x[a\boldsymbol{\nu}_1^c]\rangle \text{ by (2.15)}\\ &= 0 \text{ by part (i) of Lemma 3.7}. \end{align*}

We emphasize that our success here traces back to the derivative orthogonality property (3.12).

We conclude by solving (3.33a) with another application of the implicit function theorem, and this, again, is effectively the remainder of the proof of the Crandall–Rabinowitz–Zeidler theorem [Reference Kielhöfer31, Thm. 1.5.1]. Define

\begin{equation*} \mathcal{F}_c^0 \colon \mathcal{B}_{\mathbb{R}^2}((\omega_c,0); \delta_c/2) \to \mathbb{R} \colon (\omega,a) \mapsto \langle\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega),\omega),\boldsymbol{\nu}_1^c\rangle. \end{equation*}

The threshold δ_c arose from the infinite-dimensional implicit function theorem argument in Section 3.3.

Since $\mathcal{F}_c^0(\omega,0) = 0$ for all ω, we have

\begin{equation*} \mathcal{F}_c^0(\omega,a) = a\mathcal{H}_c(\omega,a), \qquad \mathcal{H}_c(\omega,a) := \int_0^1 D_a\mathcal{F}_c^0(\omega,a\alpha) \ d\alpha. \end{equation*}

It therefore suffices to solve $\mathcal{H}_c(\omega,a) = 0$ by selecting ω as a function of a, and we do this by checking $\mathcal{H}_c(\omega_c,0) = 0$ and $D_{\omega}\mathcal{H}_c(\omega_c,0) \ne 0$.

Toward this end, we first differentiate

\begin{equation*} D_a\mathcal{F}_c^0(\omega,a) = \langle D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega))\big(\boldsymbol{\nu}_1^c+D_{\boldsymbol{\nu}}\boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega)\boldsymbol{\nu}_1^c\big),\boldsymbol{\nu}_1^c\rangle. \end{equation*}

We put a = 0 and use $\boldsymbol{\Psi}_c(0,\omega) = 0$ to find for any ω that

(3.34)\begin{equation} \mathcal{H}_c(\omega,0) = \int_0^1 D_a\mathcal{F}_c^0(\omega,0) \ d\alpha = D_a\mathcal{F}_c(\omega,0) = \langle\mathcal{L}_c[\omega]\big(\boldsymbol{\nu}_1^c+D_{\boldsymbol{\nu}}\boldsymbol{\Psi}_c(0,\omega)\boldsymbol{\nu}_1^c\big),\boldsymbol{\nu}_1^c\rangle. \end{equation}

In the special case of $\omega=\omega_c$, we can use either $D_{\boldsymbol{\nu}}\boldsymbol{\Psi}_c(0,\omega) = 0$ from part (iii) of Lemma 3.7 or the condition $\mathcal{L}_c[\omega_c]^*\boldsymbol{\nu}_1^c = 0$ to reduce (3.34) to

\begin{equation*} \mathcal{H}_c(\omega_c,0) = \langle\mathcal{L}_c[\omega_c]\boldsymbol{\nu}_1^c,\boldsymbol{\nu}_1^c\rangle = 0. \end{equation*}

Next, with ω arbitrary, we differentiate (3.34) with respect to ω and use the product rule to find

\begin{equation*} D_{\omega}\mathcal{H}_c(\omega,0) = \langle\mathcal{L}_c'[\omega]\boldsymbol{\nu}_1^c,\boldsymbol{\nu}_1^c\rangle + \langle\mathcal{L}_c'[\omega]D_{\boldsymbol{\nu}}\boldsymbol{\Psi}_c(0,\omega)\boldsymbol{\nu}_1^c,\boldsymbol{\nu}_1^c\rangle + \langle\mathcal{L}_c[\omega]D_{\boldsymbol{\nu}\omega}\boldsymbol{\Psi}_c(0,\omega)\boldsymbol{\nu}_1^c,\boldsymbol{\nu}_1^c\rangle. \end{equation*}

Here we are using the shorter notation from (2.18) of $\mathcal{L}_c'[\omega] = D_{\boldsymbol{\phi}\omega}\boldsymbol{\Phi}_c(0,\omega)$. Taking $\omega = \omega_c$, we use $D_{\boldsymbol{\nu}}\boldsymbol{\Psi}_c(0,\omega_c) = 0$ to find that the second term is 0. At ω_c, the third term is 0 since $\mathcal{L}_c[\omega_c]^*\boldsymbol{\nu}_1^c = 0$. And so

\begin{equation*} D_{\omega}\mathcal{H}_c(\omega_c,0) = \langle\mathcal{L}_c'[\omega_c]\boldsymbol{\nu}_1^c,\boldsymbol{\nu}_1^c\rangle \ne 0, \end{equation*}

by Corollary 2.3.

We are now in position to invoke the implicit function theorem once more, and we find a_c, $b_c \gt 0$ and a map $\Omega_c \colon (-a_c,a_c) \to \mathbb{R}$ such that

\begin{equation*} \mathcal{H}_c(\Omega_c(a),a) = 0 \end{equation*}

for $|a| \lt a_c$, while if $|\omega-\omega_c| \lt b_c$ and $|a| \lt a_c$ and $\mathcal{H}_c(\omega,a) = 0$, then $\omega = \Omega_c(a)$. In particular, $\Omega_c(0) = \omega_c$.

In short, taking

\begin{equation*} \boldsymbol{\phi}_c^a := a\boldsymbol{\nu}_1 + \boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1,\Omega_c(a)) \qquad\text{and }\qquad \omega_c^a := \Omega_c(a) \end{equation*}

solves our original problem $\boldsymbol{\Phi}_c(\boldsymbol{\phi}_c^a,\omega_c^a) = 0$. We can expose uniformly the ‘amplitude’ parameter of a in $\boldsymbol{\phi}_c^a$ by setting

\begin{equation*} \boldsymbol{\psi}_c(a) := \boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1,\Omega_c(a)), \end{equation*}

and computing

\begin{equation*} \boldsymbol{\psi}_c(0) = \boldsymbol{\Psi}_c(0,\omega_c) = 0, \quad \boldsymbol{\psi}_c(a) = a\int_0^1 D_a\boldsymbol{\psi}_c(a\alpha) \ d\alpha, \quad\text{and}\quad \boldsymbol{\phi} = a\left(\boldsymbol{\nu}_1+\int_0^1 D_a\boldsymbol{\psi}_c(a\alpha) \ d\alpha\right). \end{equation*}

Likewise, we can write

\begin{equation*} \omega_c^a = \omega_c + a\xi_c^a \qquad\text{and }\qquad \xi_c^a := \int_0^1 D_a\Omega_c(a\alpha) \ d\alpha. \end{equation*}

This concludes our first proof of Theorem 1.1.

3.5. Proof of local uniqueness up to shifts and translations

We discuss our decision at the start of Section 3.4 to specialize the finite-dimensional component ν to $\boldsymbol{\nu} = a\boldsymbol{\nu}_1^c$. We consider two aspects of this choice to allay any concerns about its peculiarity or restrictiveness.

Firstly, up to a shift, any solution ϕ to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ has this form $\boldsymbol{\phi} = a\boldsymbol{\nu}_1^c + \boldsymbol{\psi}$, with $a \in \mathbb{R}$ and ψ orthogonal to $\boldsymbol{\nu}_0$, $\boldsymbol{\nu}_1^c$, and $\boldsymbol{\nu}_2^c$.

Lemma 3.8. Let $\omega \in \mathbb{R}$. If $\boldsymbol{\phi} \in \mathcal{X}$ solves $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$, then there exist a, $\theta \in \mathbb{R}$ and $\boldsymbol{\psi} \in \mathcal{X}_c^{\infty}$ such that

(3.35)\begin{equation} \boldsymbol{\phi} = S^{\theta}(a\boldsymbol{\nu}_1^c + \boldsymbol{\psi}). \end{equation}

In particular, $\boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\psi},\omega) = 0$, as well.

Proof. We prove the last sentence first. If a solution ϕ to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ has the form (3.35), then the shift invariance of $\boldsymbol{\Phi}_c$ from (3.11) implies

\begin{equation*} \boldsymbol{\Phi}_c(a\boldsymbol{\nu}_1^c+\boldsymbol{\psi},\omega) = S^{-\theta}\boldsymbol{\Phi}_c(S^{\theta}(a\boldsymbol{\nu}_1^c+\boldsymbol{\psi}),\omega) = S^{-\theta}\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0. \end{equation*}

It remains to prove the decomposition (3.35). We can always write

(3.36)\begin{equation} \boldsymbol{\phi} = a_1\boldsymbol{\nu}_1^c + a_2\boldsymbol{\nu}_2^c + \widetilde{\boldsymbol{\psi}} \end{equation}

for some a ₁, $a_2 \in \mathbb{R}$ and $\widetilde{\boldsymbol{\psi}} \in \mathcal{X}_c^{\infty}$. Write

\begin{equation*} a_1-ia_2 = ae^{i\theta} \end{equation*}

in polar coordinates, where a, $\theta \in \mathbb{R}$. (If $a_1 = a_2 = 0$, just take a = 0 and $\boldsymbol{\psi} = \widetilde{\boldsymbol{\psi}}$.) The identities $\boldsymbol{\nu}_1^c(x) = 2\operatorname{Re}[e^{ix}\widehat{\boldsymbol{\nu}_1^c}(1)]$ from (2.12) and $\boldsymbol{\nu}_2^c = S^{-\pi/2}\boldsymbol{\nu}_1^c$ from (2.16) then imply

\begin{equation*} a_1\boldsymbol{\nu}_1^c(x) +a_2\boldsymbol{\nu}_2^c(x) = a_1\boldsymbol{\nu}_1^c(x)+a_2S^{-\pi/2}\boldsymbol{\nu}_1^c(x) = 2\operatorname{Re}[(a_1-ia_2)e^{ix}\widehat{\boldsymbol{\nu}_1^c}(1)] = a(S^{\theta}\boldsymbol{\nu}_1^c)(x). \end{equation*}

Returning to (3.36), we have

\begin{equation*} \boldsymbol{\phi} = S^{\theta}(a\boldsymbol{\nu}_1^c+\boldsymbol{\psi}), \qquad \boldsymbol{\psi} := S^{-\theta}\widetilde{\boldsymbol{\psi}}. \end{equation*}

We conclude by checking that if $\widetilde{\boldsymbol{\psi}} \in \mathcal{X}_c^{\infty}$, then $\boldsymbol{\psi} = S^{-\theta}\widetilde{\boldsymbol{\psi}} \in \mathcal{X}_c^{\infty}$. That is, we assume

(3.37)\begin{equation} \langle\widetilde{\boldsymbol{\psi}},\boldsymbol{\nu}_1^c\rangle = \langle\widetilde{\boldsymbol{\psi}},\boldsymbol{\nu}_2^c\rangle = 0, \end{equation}

and we want to show

(3.38)\begin{equation} \langle S^{-\theta}\widetilde{\boldsymbol{\psi}},\boldsymbol{\nu}_1^c\rangle = \langle S^{-\theta}\widetilde{\boldsymbol{\psi}},\boldsymbol{\nu}_2^c\rangle = 0. \end{equation}

By the orthogonality condition (2.17), our assumption (3.37) is equivalent to

(3.39)\begin{equation} \widehat{\widetilde{\boldsymbol{\psi}}}(1)\cdot\widehat{\boldsymbol{\nu}_1^c}(1) = 0, \end{equation}

and our desired conclusion (3.38) is equivalent to

(3.40)\begin{equation} (e^{-i\theta}\widehat{\widetilde{\boldsymbol{\psi}}}(1))\cdot\widehat{\boldsymbol{\nu}_1^c}(1) = 0. \end{equation}

Certainly (3.39) implies (3.40).

This lemma provides the local uniqueness of our solutions up to shifts, which combines with the local uniqueness up to translations (as discussed after the proof of Corollary 3.3) to give the statement at the end of Theorem 1.1. Specifically, let ϕ and ω solve $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ with $\|\boldsymbol{\phi}\|_{H_{\operatorname{per}}^2}$ and $|\omega-\omega_c|$ sufficiently small. Then $\boldsymbol{\phi} = S^{\theta}(a\boldsymbol{\nu}_1^c+\boldsymbol{\psi})$ for some $a \in \mathbb{R}$ and $\boldsymbol{\psi} \in \mathcal{X}_c^{\infty}$, and $\|a\boldsymbol{\nu}_1^c+\boldsymbol{\psi}\|_{H_{\operatorname{per}}^2} = \|S^{\theta}(a\boldsymbol{\nu}_1^c+\boldsymbol{\psi})\|$. By orthogonality, this ensures that $|a|$ and $\|\boldsymbol{\psi}\|_{H_{\operatorname{per}}^2}$ are sufficiently small. The uniqueness result from Section 3.3 implies $\boldsymbol{\psi} = \boldsymbol{\Psi}_c(a\boldsymbol{\nu}_1^c,\omega)$, and then the uniqueness result from Section 3.4 implies $\omega = \Omega_c(a)$.

Another consequence of this lemma is that it shows why trying an ansatz of the form $\boldsymbol{\phi} = a\boldsymbol{\nu}_1^c + b\boldsymbol{\nu}_2^c + \boldsymbol{\psi}$ in the hope that a and b would be enough to manage the two equations in (3.31) will not be effective. Informally, the problem simply does not ‘see’ the two unknowns a and b simultaneously. And such an ansatz would not expose the single uniform amplitude parameter that we desire, anyway.

Additionally, there is nothing special about $\boldsymbol{\nu}_1^c$ here, and we could just as easily show that any solution to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega)$ is a shifted version of a solution of the form $a\boldsymbol{\nu}_2^c + \boldsymbol{\psi}$. In fact, we could have run the bifurcation argument above using $\boldsymbol{\nu}_2^c$ throughout in place of $\boldsymbol{\nu}_1^c$. This hinges on expressing the transversality inequality of Corollary 2.3 in terms of $\boldsymbol{\nu}_2^c$, which is possible because of the calculation

(3.41)\begin{align} \langle\mathcal{L}_c'[\omega_c]\boldsymbol{\nu}_2^c,\boldsymbol{\nu}_2^c\rangle = \langle\mathcal{L}_c'[\omega_c]\partial_x\boldsymbol{\nu}_1^c,\partial_x\boldsymbol{\nu}_1^c\rangle = \langle\partial_x\mathcal{L}_c'[\omega_c]\boldsymbol{\nu}_1^c,\partial_x\boldsymbol{\nu}_1^c\rangle = -\langle\mathcal{L}_c'[\omega_c]\boldsymbol{\nu}_1^c,\partial_x^2\boldsymbol{\nu}_1^c\rangle = \langle\mathcal{L}_c'[\omega_c]\boldsymbol{\nu}_1^c,\boldsymbol{\nu}_1^c\rangle.\nonumber\\ \end{align}

The second equality relies on the commutativity of the Fourier multipliers $\partial_x$ and $\mathcal{L}_c'[\omega_c]$ on $H_{\operatorname{per}}^3(\mathbb{R}^2)$. However, since we can write any solution in the form

\begin{equation*} S^{\theta}(a\boldsymbol{\nu}_1^c+\boldsymbol{\psi}) = S^{\theta}(aS^{\pi/2}\boldsymbol{\nu}_2^c+\boldsymbol{\psi}) = S^{\theta+\pi/2}(a\boldsymbol{\nu}_2^c+S^{-\pi/2}\boldsymbol{\psi}) \end{equation*}

with ψ, and thus (by the end of the proof of Lemma 3.8) $S^{-\pi/2}\boldsymbol{\psi}$, orthogonal to $\boldsymbol{\nu}_1^c$ and $\boldsymbol{\nu}_2^c$, there is not much point to this line of inquiry.

Next, we consider further the special form of solutions to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ as given in Theorem 1.1: they are $\boldsymbol{\phi} = a(\boldsymbol{\nu}_1^c + \boldsymbol{\psi})$ with ψ again orthogonal to $\boldsymbol{\nu}_1^c$ and $\boldsymbol{\nu}_2^c$. This may appear to be less general than the result of Lemma 3.8, which says that, up to a shift, any solution has the form $a\boldsymbol{\nu}_1^c + \widetilde{\boldsymbol{\psi}}$, with $\widetilde{\boldsymbol{\psi}}$ satisfying the perennial orthogonality conditions. Of course, if $a \ne 0$, then this solution factors as $a(\boldsymbol{\nu}_1^c + a^{-1}\widetilde{\boldsymbol{\psi}})$, and that has the form given by Theorem 1.1. It turns out that all small nontrivial solutions to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ have this special factored form (again, up to a shift from Lemma 3.8).

We prove a negative version of this result, which says that if the shifted solution from Lemma 3.8 has the form $\boldsymbol{\phi} = \widetilde{\boldsymbol{\psi}}$ alone, i.e., if a = 0, and if this solution is sufficiently small, then it is trivial.

Proof. Define

\begin{equation*} \widetilde{\mathcal{F}}_c^{\infty} \colon \mathcal{X}_c^{\infty} \times \mathbb{R} \to \mathcal{Y}_c^{\infty} \colon (\boldsymbol{\psi},\omega) \mapsto (\mathcal{I}_{\mathcal{Y}} -\Pi_c)\boldsymbol{\Phi}_c(\boldsymbol{\psi},\omega). \end{equation*}

The notation and structure of this map are intentionally similar to those of $\mathcal{F}_c^{\infty}$ in (3.25), and the spaces $\mathcal{X}_c^{\infty}$ and $\mathcal{Y}_c^{\infty}$ are defined in (3.22). Then $\widetilde{\mathcal{F}}_c^{\infty}(0,\omega) = 0$ for all ω and

\begin{equation*} D_{\boldsymbol{\psi}}\widetilde{\mathcal{F}}_c^{\infty}(0,\omega_c) = (\mathcal{I}_{\mathcal{Y}} -\Pi_c){\left.\mathcal{L}_c[\omega_c] \vphantom{\big|} \right|_{\mathcal{X}_c^{\infty}} } \end{equation*}

As with the analogous linearization in Section 3.3, this operator has trivial kernel and cokernel and therefore is invertible. The implicit function theorem gives $\delta_c^{\infty}$, $\epsilon_c^{\infty} \gt 0$ and a map

\begin{equation*} \boldsymbol{\Psi}_c^{\infty} \colon (\omega_c-\delta_c^{\infty},\omega_c+\delta_c^{\infty}) \to \mathcal{X}_c^{\infty} \end{equation*}

such that $\widetilde{\mathcal{F}}_c^{\infty}(\boldsymbol{\Psi}_c^{\infty}(\omega),\omega) = 0$ for $|\omega-\omega_c| \lt \delta_c^{\infty}$. Moreover, if $|\omega-\omega_c| \lt \delta_c^{\infty}$ and $\|\boldsymbol{\psi}\|_{H_{\operatorname{per}}^2} \lt \epsilon_c^{\infty}$, and if $\widetilde{\mathcal{F}}_c^{\infty}(\boldsymbol{\psi},\omega) = 0$, then $\boldsymbol{\psi} = \boldsymbol{\Psi}_c^{\infty}(\omega)$. But $\widetilde{\mathcal{F}}_c^{\infty}(0,\omega) = 0$ for all ω, and so we must have $\boldsymbol{\Psi}_c^{\infty}(\omega) = 0$ for all ω. Conversely, if $\boldsymbol{\Phi}_c(\boldsymbol{\psi},\omega) = 0$ then $\widetilde{\mathcal{F}}_c^{\infty}(\boldsymbol{\psi},\omega) = 0$, too, and so if $|\omega-\omega_c| \lt \delta_c^{\infty}$ and $\|\boldsymbol{\psi}\|_{H_{\operatorname{per}}^2} \lt \epsilon_c^{\infty}$, then $\boldsymbol{\psi} = \boldsymbol{\Psi}_c^{\infty}(\omega) = 0$.

Lemmas 3.8 and 3.9 together effectively tell us that the only worthwhile form of solutions to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ is $\boldsymbol{\phi} = a(\boldsymbol{\nu}_1^c+\boldsymbol{\psi})$. When we study this problem quantitatively in Section 6.1, we will start directly with an ansatz of this form.

5.1. Symmetry operators and their properties

We point out that while shift operators S^d do satisfy the invariance property (i) above, and while $S^{\pm\pi}$ also satisfies (ii), shifts in general do not meet (iii) and (iv). The symmetries that we construct will not rely on shift operators.

Here are some useful properties of symmetries for our problem.

Now we adapt the nonconstant eigenfunctions $\boldsymbol{\nu}_1^c$ and $\boldsymbol{\nu}_2^c$ from Corollary 2.2 so that they respect symmetry.

Lemma 5.3. Let $\mathcal{S}$ be a symmetry and define

\begin{equation*} \boldsymbol{\nu}_+^c := \begin{cases} \boldsymbol{\nu}_1^c, \ \mathcal{S}\boldsymbol{\nu}_1^c = \boldsymbol{\nu}_1^c \\ \boldsymbol{\nu}_2^c, \ \mathcal{S}\boldsymbol{\nu}_1^c = -\boldsymbol{\nu}_1^c \\ (\boldsymbol{\nu}_1^c + \mathcal{S}\boldsymbol{\nu}_1^c)/\|\boldsymbol{\nu}_1^c+\mathcal{S}\boldsymbol{\nu}_1^c\|_{L_{\operatorname{per}}^2}, \ \mathcal{S}\boldsymbol{\nu}_1^c \ne \pm\boldsymbol{\nu}_1^c \end{cases} \end{equation*}

and

\begin{equation*} \boldsymbol{\nu}_-^c := \begin{cases} \boldsymbol{\nu}_2^c, \ \mathcal{S}\boldsymbol{\nu}_1^c = \boldsymbol{\nu}_1^c \\ \boldsymbol{\nu}_1^c, \ \mathcal{S}\boldsymbol{\nu}_1^c = -\boldsymbol{\nu}_1^c \\ (\boldsymbol{\nu}_2^c - \mathcal{S}\boldsymbol{\nu}_2^c)/\|\boldsymbol{\nu}_2^c-\mathcal{S}\boldsymbol{\nu}_2^c\|_{L_{\operatorname{per}}^2}, \ \mathcal{S}\boldsymbol{\nu}_1^c \ne \pm\boldsymbol{\nu}_1^c. \end{cases} \end{equation*}

1. (i) $\mathcal{S}\boldsymbol{\nu}_+^c = \boldsymbol{\nu}_+^c$ and $\mathcal{S}\boldsymbol{\nu}_-^c = \boldsymbol{\nu}_-^c$.

2. (ii) The vectors $\boldsymbol{\nu}_+^c$ and $\boldsymbol{\nu}_-^c$ form an orthonormal basis for $\mathcal{Z}_c$ as defined in (3.18).

3. (iii) ${\mathop{\inf}\limits_{|c| \gt c_{\star}}} \langle\mathcal{L}_c'[\omega_c]\boldsymbol{\nu}_+^c,\boldsymbol{\nu}_+^c\rangle \gt 0$.

5.2. Bifurcation in the presence of symmetry

Let $\mathcal{S}$ be a symmetry and define

\begin{equation*} \mathcal{Y}_{\mathcal{S}} := \!\left\{\boldsymbol{\phi} \in L_{\operatorname{per}}^2(\mathbb{R}^2) \ \big| \ \langle\boldsymbol{\phi},\boldsymbol{\nu}_0\rangle = 0 \text{ and } \mathcal{S}\boldsymbol{\phi} = \boldsymbol{\phi}\right\} \qquad\text{and }\qquad \mathcal{X}_{\mathcal{S}} := H_{\operatorname{per}}^2(\mathbb{R}^2) \cap \mathcal{Y}_{\mathcal{S}}. \end{equation*}

Part (i) of Lemma 5.2 shows that $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) \in \mathcal{Y}_{\mathcal{S}}$ for each $\boldsymbol{\phi} \in \mathcal{X}_{\mathcal{S}}$ and $\omega \in \mathbb{R}$. The effect of restricting $\boldsymbol{\Phi}_c$ to map from $\mathcal{X}_{\mathcal{S}} \times \mathbb{R}$ to $\mathcal{Y}_{\mathcal{S}}$ is that the restriction ${\left.\mathcal{L}_c[\omega_c] \vphantom{\big|} \right|_{\mathcal{X}_{\mathcal{S}}} }$ now has a one-dimensional kernel and cokernel. This, along with the transversality condition from part (iii) of Lemma 5.3, puts us in a position to use the classical Crandall–Rabinowitz–Zeidler theorem directly, without the work in Sections 3.4 or 4 to manage the extra finite-dimensional equation.

More precisely, we know that the three vectors $\boldsymbol{\nu}_0$, $\boldsymbol{\nu}_+^c$, and $\boldsymbol{\nu}_-^c$ form an orthonormal basis for $\ker(\mathcal{L}_c[\omega_c])$ and $\ker(\mathcal{L}_c[\omega_c]^*)$; now suppose that $\boldsymbol{\phi} \in \mathcal{Y}_{\mathcal{S}} \cap \operatorname{span}(\boldsymbol{\nu}_0,\boldsymbol{\nu}_+^c,\boldsymbol{\nu}_-^c)$. Then by orthonormality

\begin{equation*} \boldsymbol{\phi} = \langle\boldsymbol{\phi},\boldsymbol{\nu}_0\rangle\boldsymbol{\nu}_0 + \langle\boldsymbol{\phi},\boldsymbol{\nu}_+^c\rangle\boldsymbol{\nu}_+^c + \langle\boldsymbol{\phi},\boldsymbol{\nu}_-^c\rangle\boldsymbol{\nu}_-^c. \end{equation*}

By definition of $\mathcal{Y}_{\mathcal{S}}$, we already have $\langle\boldsymbol{\phi},\boldsymbol{\nu}_0\rangle = 0$, and now we compute

(5.2)\begin{equation} \langle\boldsymbol{\phi},\boldsymbol{\nu}_-^c\rangle = \langle\mathcal{S}\boldsymbol{\phi},\boldsymbol{\nu}_-^c\rangle = \langle\boldsymbol{\phi},\mathcal{S}\boldsymbol{\nu}_-^c\rangle = -\langle\boldsymbol{\phi},\boldsymbol{\nu}_-^c\rangle. \end{equation}

Thus $\langle\boldsymbol{\phi},\boldsymbol{\nu}_-^c\rangle = 0$, and so $\boldsymbol{\phi} \in \operatorname{span}(\boldsymbol{\nu}_+^c)$. This proves our claim above that $\boldsymbol{\nu}_+^c$ spans both the kernel and cokernel of $\mathcal{L}_c[\omega_c]$.

Alternatively, we could follow the bifurcation argument in Sections 3.3 and 3.4 and replace $\boldsymbol{\nu}_1^c$ with $\boldsymbol{\nu}_+^c$ and $\boldsymbol{\nu}_2^c$ with $\boldsymbol{\nu}_-^c$. The only change would be the new version of the finite-dimensional problem (3.31)

\begin{equation} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\begin{cases} \langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\nu}_+^c\rangle = 0 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(5.3a)\\ \langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\nu}_-^c\rangle = 0.\qquad\qquad\qquad\qquad\,\qquad\,\qquad\,\qquad\quad\,(5.3b) \end{cases} \end{equation}

By a calculation similar to (5.2), we always have (5.3b). Specifically, for $\boldsymbol{\phi} \in \mathcal{X}_{\mathcal{S}}$, we have

\begin{equation*} \langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\nu}_-^c\rangle = \langle\boldsymbol{\Phi}_c(\mathcal{S}\boldsymbol{\phi},\omega),\boldsymbol{\nu}_-^c\rangle = \langle\mathcal{S}\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\nu}_-^c\rangle = \langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\mathcal{S}\boldsymbol{\nu}_-^c\rangle = -\langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\nu}_-^c\rangle, \end{equation*}

thus $\langle\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega),\boldsymbol{\nu}_-^c\rangle = 0$ regardless of the form of $\boldsymbol{\phi} \in \mathcal{X}_{\mathcal{S}}$. (This is actually a stronger result than our managing of the second finite-dimensional equation (3.33a) in Section 3.4, as here there are no restrictions on the form of ϕ.) Last, we can solve (5.3a) using the transversality condition from part (iii) of Lemma 5.3, exactly as we did (3.33a) in Section 3.4. The major difference in the results here is that the solutions ϕ now respect the symmetry.

5.3. Existence of symmetries for the mass and spring dimers

We will build the symmetries primarily on a ‘reflection’ operator and a ‘flip’ operator.

Lemma 5.5. The operator

(5.4)\begin{equation} (R\boldsymbol{\phi})(x) := \boldsymbol{\phi}(-x) \end{equation}

has the following properties.

1. (i) $R\partial_x = -\partial_xR$.

2. (ii) $RS^{\theta} = S^{-\theta}R$ for all $\theta \in \mathbb{R}$.

3. (iii) $\langle R\boldsymbol{\eta},R\boldsymbol{\phi}\rangle = \langle\boldsymbol{\phi},\boldsymbol{\eta}\rangle$ for all ϕ, $\boldsymbol{\eta} \in L_{\operatorname{per}}^2(\mathbb{R}^2)$.

We will also use the ‘flip’ operator

(5.5)\begin{equation} J := \begin{bmatrix} 0 &1 \\ 1 &0 \end{bmatrix}, \end{equation}

which commutes with R. These reflection and flip operators also appeared in the manifestation of symmetries for spatial dynamics coordinates [Reference Faver and Hupkes19 Sec. 3.2].

5.3.1. Symmetry in the mass dimer

The mass dimer symmetry is

\begin{equation*} \mathcal{S}_{\mathbf{M}}\boldsymbol{\phi} := -R\boldsymbol{\phi}. \end{equation*}

The subscript here is meant to emphasize the role of the mass ratio $m = w^{-1}$ in the mass dimer analysis.

We show that $\mathcal{S}_{\mathbf{M}}$ satisfies property (i) of Definition 5.1; all of the other properties of symmetries are quick and direct calculations. That is, we show

\begin{equation*} \mathcal{G}_c(\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi},\omega) = \mathcal{G}_c(\boldsymbol{\phi},\omega) \end{equation*}

with $\mathcal{G}_c = c^2\mathcal{T} + \mathcal{P}$ as defined in (3.8). The operator $\mathcal{T}$ is defined in (3.6) and $\mathcal{P}$ in (3.7), and it is important here that in (3.7) we are assuming $\mathcal{V}_1 = \mathcal{V}_2 =: \mathcal{V}$. In particular, we take κ = 1.

First we compute

\begin{equation*} \frac{2}{\omega^2}\mathcal{T}(\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi},\omega) = \langle-M(R\boldsymbol{\phi})'',-R\boldsymbol{\phi}\rangle = \langle RM\boldsymbol{\phi}'',R\boldsymbol{\phi}\rangle = \langle M\boldsymbol{\phi}'',\boldsymbol{\phi}\rangle = \frac{2}{\omega^2}\mathcal{T}(\boldsymbol{\phi},\omega). \end{equation*}

Here we have used $\partial_x^2R = R\partial_x^2$, which follows from part (i) of Lemma 5.5, and also part (iii) of that lemma to get the penultimate equality.

Next,

\begin{equation*} \mathcal{P}(\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi},\omega) = \langle\boldsymbol{\mathcal{V}}(\Delta_+(\omega)\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi}),\boldsymbol{1}\rangle, \end{equation*}

where $\boldsymbol{\mathcal{V}}(\mathbf{p}) = (\mathcal{V}(p_1),\mathcal{V}(p_2))$ for $\mathbf{p} = (p_1,p_2)$, $\boldsymbol{1} = (1,1)$ and $\Delta_+(\omega)$ is defined in (3.1). Since $S^{\pm\omega}R = RS^{\mp\omega}$ by part (ii) of Lemma 5.5, we have

(5.6)\begin{equation} \Delta_+(\omega)\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi} = -\Delta_+(\omega)R\boldsymbol{\phi} = -R\Delta_+(-\omega)\boldsymbol{\phi} = RJ\Delta_+(\omega)\boldsymbol{\phi}. \end{equation}

Here we used the property that

(5.7)\begin{equation} -\Delta_+(-\omega) = J\Delta_+(\omega) \end{equation}

with J from (5.5).

Thus

\begin{equation*} \mathcal{P}(\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi},\omega) = \langle\boldsymbol{\mathcal{V}}(RJ\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{1}\rangle = \langle R\boldsymbol{\mathcal{V}}(J\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{1}\rangle. \end{equation*}

Since 1 is constant, $\boldsymbol{1} = R\boldsymbol{1}$, and so part (iii) of Lemma 5.5 implies

\begin{equation*} \mathcal{P}(\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi},\omega) = \langle R\boldsymbol{\mathcal{V}}(J\Delta_+(\omega)\boldsymbol{\phi}),R\boldsymbol{1}\rangle = \langle\boldsymbol{\mathcal{V}}(J\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{1}\rangle. \end{equation*}

Last, since $\mathcal{V}(\mathbf{p}) = (\mathcal{V}(p_1),\mathcal{V}(p_2))$ for $\mathbf{p} = (p_1,p_2)$, we have

\begin{equation*} \langle\boldsymbol{\mathcal{V}}(J\mathbf{p}),\boldsymbol{1}\rangle = \int_{-\pi}^{\pi} \big(\mathcal{V}(p_2)+\mathcal{V}(p_1)\big) = \langle\mathcal{V}(\mathbf{p}),\boldsymbol{1}\rangle. \end{equation*}

We conclude

\begin{equation*} \mathcal{P}(\mathcal{S}_{\mathbf{M}}\boldsymbol{\phi},\omega) = \langle\boldsymbol{\mathcal{V}}(\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{1}\rangle = \mathcal{P}(\boldsymbol{\phi},\omega). \end{equation*}

Last, we use the definitions of $\boldsymbol{\nu}_1^c$ and $\boldsymbol{\nu}_2^c$ from (2.12) and (2.13) to compute, assuming κ = 1,

\begin{equation*} \boldsymbol{\nu}_1^c(x) = \frac{2\cos(x)}{\operatorname{N}_c}\begin{pmatrix} 2\cos(\omega_c) \\ 2-c^2\omega_c^2 \end{pmatrix} \qquad\text{and }\qquad \boldsymbol{\nu}_2^c(x) = \frac{2\sin(x)}{\operatorname{N}_c}\begin{pmatrix} 2\cos(\omega_c) \\ 2-c^2\omega_c^2 \end{pmatrix}. \end{equation*}

This shows $\mathcal{S}_{\mathbf{M}}\boldsymbol{\nu}_1^c = \boldsymbol{\nu}_1^c$ and $\mathcal{S}_{\mathbf{M}}\boldsymbol{\nu}_2^c = \boldsymbol{\nu}_2^c$ directly. Consequently, when we run the symmetric bifurcation argument for the mass dimer, we can just use the first case for $\boldsymbol{\nu}_{\pm}^c$ in Lemma 5.3.

5.3.2. Symmetry in the spring dimer

The spring dimer symmetry is

(5.8)\begin{equation} \mathcal{S}_{\mathbf{K}} := -RJ = -JR \end{equation}

with R defined in (5.4) and J defined in (5.5). The subscript is meant to emphasize the role of the linear spring coefficient ratio κ in the spring dimer analysis.

Again, we just check that $\mathcal{G}_c(\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi},\omega) = \mathcal{G}_c(\boldsymbol{\phi},\omega)$ in the case $w = m^{-1} = 1$, as the other symmetry properties from Definition 5.1 are evident. With $\mathcal{G}_c = c^2\mathcal{T} + \mathcal{P}$ and $\mathcal{T}$ defined in (3.6) and $\mathcal{P}$ in (3.7), we have

\begin{align*} \frac{2}{\omega^2}\mathcal{T}(\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi},\omega) = \langle-(RJ\boldsymbol{\phi})'',-RJ\boldsymbol{\phi}\rangle = \langle RJ\boldsymbol{\phi}'',RJ\boldsymbol{\phi}\rangle = \langle J\boldsymbol{\phi}'',J\boldsymbol{\phi}\rangle = \langle J^2\boldsymbol{\phi}'',\boldsymbol{\phi}\rangle = \langle\boldsymbol{\phi}'',\boldsymbol{\phi}\rangle \\ = \frac{2}{\omega^2}\mathcal{T}(\boldsymbol{\phi},\omega). \end{align*}

Here we have again used properties of R from Lemma 5.5 and also $J^*= J^{-1} = J$.

Next,

\begin{equation*} \mathcal{P}(\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi},\omega) = \langle\boldsymbol{\mathcal{V}}(\Delta_+(\omega)\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi}),\boldsymbol{1}\rangle, \end{equation*}

with $\boldsymbol{\mathcal{V}}(\mathbf{p}) = (\mathcal{V}_1(p_1),\mathcal{V}_2(p_2))$ for $\mathbf{p} = (p_1,p_2)$, $\boldsymbol{1} = (1,1)$, and $\Delta_+(\omega)$ defined in (3.1). We have

\begin{equation*} \Delta_+(\omega)\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi} = -\Delta_+(\omega)RJ = RJ\Delta_+(\omega)J \end{equation*}

by (5.6) and (5.7), and so

\begin{equation*} \mathcal{P}(\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi},\omega) = \langle\boldsymbol{\mathcal{V}}(RJ\Delta_+(\omega)J\boldsymbol{\phi}),\boldsymbol{1}\rangle = \langle\boldsymbol{\mathcal{V}}(J\Delta_+(\omega)J\boldsymbol{\phi}),\boldsymbol{1}\rangle. \end{equation*}

We compute

\begin{equation*} J\Delta_+(\omega)J = \Lambda(\omega)\Delta_+(\omega), \qquad \Lambda(\omega) := \begin{bmatrix} S^{\omega} &0 \\ 0 &S^{-\omega} \end{bmatrix}, \end{equation*}

and, for $\mathbf{p} = (p_1,p_2) \in L_{\operatorname{per}}^2(\mathbb{R}^2)$,

\begin{equation*} \langle\boldsymbol{\mathcal{V}}(\Lambda(\omega)\mathbf{p}),\boldsymbol{1}\rangle = \int_{-\pi}^{\pi} \mathcal{V}_1(S^{\omega}p_1) + \int_{-\pi}^{\pi} \mathcal{V}_2(S^{-\omega}p_2) = \int_{-\pi}^{\pi} \big(\mathcal{V}_1(p_1) + \mathcal{V}_2(p_2)\big) = \langle\boldsymbol{\mathcal{V}}(\mathbf{p}),\boldsymbol{1}\rangle. \end{equation*}

We conclude

\begin{equation*} \mathcal{P}(\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi},\omega) = \langle\boldsymbol{\mathcal{V}}(\Lambda(\omega)\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{1}\rangle = \langle\boldsymbol{\mathcal{V}}(\Delta_+(\omega)\boldsymbol{\phi}),\boldsymbol{1}\rangle = \mathcal{P}(\boldsymbol{\phi},\omega). \end{equation*}

Unlike in the mass dimer, it is not always the case that $\mathcal{S}_{\mathbf{K}}\boldsymbol{\nu}_1^c = \boldsymbol{\nu}_1^c$ for the spring dimer. Indeed, the situation is rather more complicated here, as we outline below. It is for this reason that we developed Lemma 5.3, which is unnecessarily elaborate for the mass dimer.

Lemma 5.6. Assume w = 1.

1. (i) $\mathcal{S}_{\mathbf{K}}\boldsymbol{\nu}_1^c = \boldsymbol{\nu}_1^c$ if and only if $\omega_c = j\pi$ for some even $j \in \mathbb{Z}$.

2. (ii) $\mathcal{S}_{\mathbf{K}}\boldsymbol{\nu}_1^c = -\boldsymbol{\nu}_1^c$ if and only if $\omega_c = j\pi$ for some odd $j \in \mathbb{Z}$.

We prove this lemma in Appendix B.6. A consequence is that outside the isolated situations $\omega_c = j\pi$ for some $j \in \mathbb{Z}$, we must use the third, more complicated case of Lemma 5.3 to obtain symmetric eigenfunctions for the spring dimer.

We conclude this discussion of symmetry by noting that not all solutions to the travelling wave problem are symmetric. Indeed, since the travelling wave problem is shift invariant (part (ii) of Corollary 3.2), any solution ϕ to $\boldsymbol{\Phi}_c(\boldsymbol{\phi},\omega) = 0$ generates other solutions $S^{\theta}\boldsymbol{\phi}$ for $\theta \in \mathbb{R}$. Still working in the spring dimer, suppose that ϕ is symmetric with respect to $\mathcal{S}_{\mathbf{K}}$, so $\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi} = \boldsymbol{\phi}$. We compute

\begin{equation*} \mathcal{S}_{\mathbf{K}}S^{\theta}\boldsymbol{\phi} = -JRS^{\theta}\boldsymbol{\phi} = -JS^{-\theta}R\boldsymbol{\phi} = S^{-\theta}(-JR)\boldsymbol{\phi} = S^{-\theta}\mathcal{S}_{\mathbf{K}}\boldsymbol{\phi} = S^{-\theta}\boldsymbol{\phi}. \end{equation*}

Typically $S^{-\theta}\boldsymbol{\phi} \ne \boldsymbol{\phi}$ unless θ is an even integer multiple of π. Thus the shifted solution need not be symmetric.

6.1. An abstract quantitative bifurcation theorem

We first prove a very abstract bifurcation result from which our quantitative result for lattice periodics follows easily. This result subsumes all of the existing quantitative periodic constructions for lattices – all of which, we emphasize, relied on symmetry to control (co)kernel dimensionality – and does not strictly depend on the long wave structure of the problem considered more broadly here. That is, we claim that any of the prior quantitative periodic proofs follows from Theorem 6.2 below.

We rely on the following fixed-point theorem, which was proved as [Reference Faver and Wright20, Lem. C.1].

Lemma 6.1. For $0 \lt \epsilon \lt \epsilon_0$, let $\mathcal{X}^{\rho}$ be a Banach space and let $\mathcal{F}_{\epsilon} \colon \mathcal{X}^{\rho} \times \mathbb{R} \to \mathcal{X}^{\rho}$ be a family of maps. Suppose that for some C ₀, a ₀, $b_0 \gt 0$, if x, $\grave{x} \in \mathcal{X}^{\rho}$ and $a \in \mathbb{R}$ with $\|x\|_{\mathcal{X}^{\rho}}$, $\|\grave{x}\|_{\mathcal{X}^{\rho}} \le b_0$ and $|a| \le a_0$, then

(6.1)\begin{equation} \|\mathcal{F}_{\epsilon}(x,a)\|_{\mathcal{X}^{\rho}} \le C_0\big(|a| + \|x\|_{\mathcal{X}^{\rho}}^2\big) \end{equation}

and

(6.2)\begin{equation} \|\mathcal{F}_{\epsilon}(x,a)-\mathcal{F}_{\epsilon}(\grave{x},a)\|_{\mathcal{X}^{\rho}} \le C_0\big(\|x\|_{\mathcal{X}^{\rho}} + \|\grave{x}\|_{\mathcal{X}^{\rho}} + |a|\big)\|x-\grave{x}\|_{\mathcal{X}^{\rho}} \end{equation}

for all $0 \lt \epsilon \lt \epsilon_0$. Then there exist a ₁, $r_1 \gt 0$ such that for each $|a| \le a_1$ and $0 \lt \epsilon \lt \epsilon_0$, there is a unique $x_{\epsilon}^a \in \mathcal{X}^{\rho}$ with $x_{\epsilon}^a = \mathcal{F}_{\epsilon}(x_{\epsilon}^a,a)$ and $\|x_{\epsilon}^a\|_{\mathcal{X}^{\rho}} \le r_1$.

Moreover, suppose that there is $L_0 \gt 0$ such that

(6.3)\begin{equation} \|\mathcal{F}_{\epsilon}(x,a)-\mathcal{F}_{\epsilon}(x,\grave{a})\|_{\mathcal{X}^{\rho}} \le L_0|a-\grave{a}| \end{equation}

for all $x \in \mathcal{X}^{\rho}$ with $\|x\|_{\mathcal{X}^{\rho}} \le b_0$, a, $\grave{a} \in \mathbb{R}$ with $|a|$, $|\grave{a}| \le a_0$, and $0 \lt \epsilon \lt \epsilon_0$. Then there is $L_1 \gt 0$ such that

(6.4)\begin{equation} \|x_{\epsilon}^a - x_{\epsilon}^{\grave{a}}\|_{\mathcal{X}^{\rho}} \le L_1|a-\grave{a}| \end{equation}

for all $0 \lt \epsilon \lt \epsilon_0$ and a, $\grave{a} \in \mathbb{R}$ with $|a|$, $|\grave{a}| \le a_1$.

Here is our primary abstract result. It is very technical. We discuss the application of this result to our long wave problem in Section 6.2 below, but for now we encourage the reader to think of the map $\boldsymbol{\Phi}_{\epsilon}$ in the theorem as $\boldsymbol{\Phi}_{c_{\epsilon}}$ from (1.9) with $c_{\epsilon}^2 = c_{\star}^2+\epsilon^2$ and to think of the spaces $\mathcal{X}^r$ below as $\!\left\{\boldsymbol{\phi} \in H_{\operatorname{per}}^r(\mathbb{R}^2) \ \big| \ \langle\boldsymbol{\phi},\boldsymbol{\nu}_0\rangle = 0\right\}$ with the special case of ρ = 2. The conclusions of this theorem are a quantitative version of the results of Theorem 1.1 in a much more abstract context.

Theorem 6.2. Let $\{\mathcal{X}^r\}_{r \ge 0}$ be a family of Hilbert spaces such that $\mathcal{X}^{r+s}$ is continuously embedded in $\mathcal{X}^r$ for each $s \ge 0$. Denote the inner product on $\mathcal{X}^r$ by $\langle\cdot,\cdot\rangle_r$ and, for simplicity, let $\langle\cdot,\cdot\rangle := \langle\cdot,\cdot\rangle_0$; denote the norm on $\mathcal{X}^r$ by $\|\cdot\|_r$. Suppose that for $0 \lt \epsilon \lt \epsilon_0$ and some ρ > 0, there is a map

\begin{equation*} \boldsymbol{\Phi}_{\epsilon} \colon \mathcal{X}^{\rho} \times \mathbb{R} \to \mathcal{X}^0 \colon (\boldsymbol{\phi},\omega) \mapsto \boldsymbol{\Phi}_{\epsilon}(\boldsymbol{\phi},\omega) \end{equation*}

with the following properties.

1. (i) [Branch of trivial solutions] $\boldsymbol{\Phi}_{\epsilon}(0,\omega) = 0$ for all $\omega \in \mathbb{R}$.

2. (ii) [Regularity] The partial derivatives $D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_{\epsilon}$ and $D_{\boldsymbol{\phi}\boldsymbol{\phi}}^2\boldsymbol{\Phi}_{\epsilon}$ exist and are continuous from $\mathcal{X}^{\rho} \times \mathbb{R}$ to $\mathcal{X}^0$, and the partial derivative $D_{\boldsymbol{\phi}\omega}^2\boldsymbol{\Phi}_{\epsilon}(0,\cdot)$ exists and is continuous on $\mathbb{R}$.

3. (iii) [(Co)kernel dimensionality] There is $\omega_{\epsilon} \in \mathbb{R}$ such that

   \begin{equation*} \ker(D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_{\epsilon}(0,\omega_{\epsilon})) = \operatorname{span}(\boldsymbol{\nu}^{\epsilon}) \qquad\text{and }\qquad \ker(D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_{\epsilon}(0,\omega_{\epsilon})^*) = \operatorname{span}(\boldsymbol{\mu}_1^{\epsilon},\boldsymbol{\mu}_2^{\epsilon}) \end{equation*}

   for some vectors $\boldsymbol{\nu}^{\epsilon}$, $\boldsymbol{\mu}_1^{\epsilon}$, $\boldsymbol{\mu}_2^{\epsilon} \in \mathcal{X}^0$ with $\|\boldsymbol{\nu}^{\epsilon}\|_0 = \|\boldsymbol{\mu}_1^{\epsilon}\|_0 = 1$ (the case $\boldsymbol{\mu}_2^{\epsilon} = 0$ is allowed).

4. (iv) [Uniform transversality] $\inf_{0 \lt \epsilon \lt \epsilon_0} |\langle D_{\boldsymbol{\phi}\omega}^2\boldsymbol{\Phi}_{\epsilon}(0,\omega_{\epsilon})\boldsymbol{\nu}^{\epsilon},\boldsymbol{\mu}_1^{\epsilon}\rangle| \gt 0$.

5. (v) [Uniform coercivity] For each $r \ge 0$, there is $C_r \gt 0$ such that if $\boldsymbol{\psi} \in \mathcal{X}^{r+\rho}$ and $\boldsymbol{\eta} \in \mathcal{X}^r$ with

   \begin{align*} D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_{\epsilon}(0,\omega_{\epsilon})\boldsymbol{\psi} = \boldsymbol{\eta}, \quad \langle\boldsymbol{\psi},\boldsymbol{\nu}^{\epsilon}\rangle &= \langle\boldsymbol{\psi},\boldsymbol{\mu}_1^{\epsilon}\rangle = \langle\boldsymbol{\psi},\boldsymbol{\mu}_2^{\epsilon}\rangle = 0, \quad \text{and} \quad\\ \langle\boldsymbol{\eta},\boldsymbol{\nu}^{\epsilon}\rangle &= \langle\boldsymbol{\eta},\boldsymbol{\mu}_1^{\epsilon}\rangle = \langle\boldsymbol{\eta},\boldsymbol{\mu}_2^{\epsilon}\rangle = 0, \end{align*}

   then $\|\boldsymbol{\psi}\|_{r+\rho} \le C_r\|\boldsymbol{\eta}\|_r$.

6. (vi) [Bootstrapping] If $\boldsymbol{\phi} \in \mathcal{X}^{\rho}$ such that $D_{\boldsymbol{\phi}}\boldsymbol{\Phi}_{\epsilon}(0,\omega_{\epsilon})\boldsymbol{\phi} \in \mathcal{X}^r$ for some $r \ge 0$, then $\boldsymbol{\phi} \in \mathcal{X}^{r+\rho}$.

7. (vii) [Uniform mapping and Lipschitz estimates] There is $b_0 \gt 0$ such that the following estimates hold for each $r \ge 0$ (not necessarily uniformly in r):

   \begin{equation*} \sup_{0 \lt \epsilon \lt \epsilon_0} \|D_{\boldsymbol{\phi}\omega}^2\boldsymbol{\Phi}_{\epsilon}(0,\omega_{\epsilon})\|_{\mathcal{X}^{r+\rho} \to \mathcal{X}^r} \lt \infty, \end{equation*}

   \begin{equation*} \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ |\omega-\omega_{\epsilon}| \lt b_0, \ |\grave{\omega}-\omega_{\epsilon}| \lt b_0 \\ \omega \ne \grave{\omega}}} \frac{\|D_{\boldsymbol{\phi}\omega}^2\boldsymbol{\Phi}_{\epsilon}(0,\omega)-D_{\boldsymbol{\phi}\omega}^2\boldsymbol{\Phi}_{\epsilon}(0,\grave{\omega})\|_{\mathcal{X}^{r+\rho} \to \mathcal{X}^r}}{|\omega-\grave{\omega}|} \lt \infty, \end{equation*}

   \begin{equation*} \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ \|\boldsymbol{\phi}\|_{r+\rho} + |\omega-\omega_{\epsilon}| \lt b_0}} \|D_{\boldsymbol{\phi}\boldsymbol{\phi}}^2\boldsymbol{\Phi}_{\epsilon}(\boldsymbol{\phi},\omega)\|_{\mathcal{X}^{r+\rho} \times \mathcal{X}^{r+\rho} \to \mathcal{X}^r} \lt \infty \end{equation*}

   and

   \begin{equation*} \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ \|\boldsymbol{\phi}\|_{r+\rho} + |\omega-\omega_{\epsilon}| \lt b_0, \ \|\grave{\boldsymbol{\phi}}\|_{r+\rho} + |\grave{\omega}-\omega_{\epsilon}| \lt b_0 \\ (\boldsymbol{\phi},\omega) \ne (\grave{\boldsymbol{\phi}},\grave{\omega})}} \frac{\|D_{\boldsymbol{\phi}\boldsymbol{\phi}}^2\boldsymbol{\Phi}_{\epsilon}(\boldsymbol{\phi},\omega)-D_{\boldsymbol{\phi}\boldsymbol{\phi}}^2\boldsymbol{\Phi}_{\epsilon}(\grave{\boldsymbol{\phi}},\grave{\omega})\|_{\mathcal{X}^{r+\rho} \times \mathcal{X}^{r+\rho} \to \mathcal{X}^r}}{\|\boldsymbol{\phi}-\grave{\boldsymbol{\phi}}\|_{r+\rho} + |\omega-\grave{\omega}|} \lt \infty. \end{equation*}

8. (viii) If $\boldsymbol{\mu}_2^{\epsilon} \ne 0$, then there are a Banach space $\mathcal{W}_{\epsilon}$ with $\mathcal{X}^{\rho} \subseteq \mathcal{W}_{\epsilon} \subseteq \mathcal{X}^0$ and a nonzero linear operator $\mathcal{T}_{\epsilon} \colon \mathcal{W}_{\epsilon} \to \mathcal{X}^0$ with the following properties.

• • $\langle\boldsymbol{\Phi}_{\epsilon}(\boldsymbol{\phi},\omega),\mathcal{T}_{\epsilon}\boldsymbol{\phi}\rangle = 0$ for all $\boldsymbol{\phi} \in \mathcal{X}^{\rho}$ and $\omega \in \mathbb{R}$.

• • $\mathcal{T}_{\epsilon}\boldsymbol{\nu}^{\epsilon} = \boldsymbol{\mu}_1^{\epsilon}$.

• • There is $\tau_{\epsilon} \in \mathbb{R}\setminus\{0\}$ such that $\mathcal{T}_{\epsilon}\boldsymbol{\mu}_1^{\epsilon} = \pm\tau_{\epsilon}\boldsymbol{\mu}_2^{\epsilon}$ and $\mathcal{T}_{\epsilon}\boldsymbol{\mu}_2^{\epsilon} = \mp\tau_{\epsilon}\boldsymbol{\mu}_1^{\epsilon}$.

Then there is $a_{\star} \gt 0$ such that for $|a| \lt a_{\star}$ and $0 \lt \epsilon \lt \epsilon_0$, there exist $\boldsymbol{\phi}_{\epsilon}^a \in \cap_{r=0}^{\infty} \mathcal{X}^r$ and $\omega_{\epsilon}^a \in \mathbb{R}$ such that $\boldsymbol{\Phi}_{\epsilon}(\boldsymbol{\phi}_{\epsilon}^a,\omega_{\epsilon}^a) = 0$ with

\begin{equation*} \boldsymbol{\phi}_{\epsilon}^a = a(\boldsymbol{\nu}^{\epsilon}+\boldsymbol{\psi}_{\epsilon}^a), \qquad \langle\boldsymbol{\psi}_{\epsilon}^a,\boldsymbol{\nu}^{\epsilon}\rangle = 0, \qquad \boldsymbol{\psi}_{\epsilon}^0 = 0 \qquad\text{and }\qquad \omega_{\epsilon}^a = \omega_{\epsilon} + \xi_{\epsilon}^a, \qquad \xi_{\epsilon}^0 = 0. \end{equation*}

The following mapping and Lipschitz estimates also hold for each r (not necessarily uniformly in r):

(6.5)\begin{equation} \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ |a| \lt a_{\star}}} \|\boldsymbol{\psi}_{\epsilon}^a\|_r + |\xi_{\epsilon}^a| \lt \infty \qquad\text{and }\qquad \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ |a| \lt a_{\star}, \ |\grave{a}| \lt a_{\star} \\ a \ne \grave{a}}} \frac{\|\boldsymbol{\psi}_{\epsilon}^a-\boldsymbol{\psi}_{\epsilon}^{\grave{a}}\|_r}{|a-\grave{a}|} + \frac{|\xi_{\epsilon}^a-\xi_{\epsilon}^{\grave{a}}|}{|a-\grave{a}|} \lt \infty. \end{equation}

6.2. Application to periodic travelling waves in dimer FPUT

For long wave solutions, we are interested in rescaling the profiles as $\boldsymbol{\phi}(x) = \epsilon^2\boldsymbol{\varphi}(\epsilon{x})$, where ϵ > 0 measures the distance between the speed of sound $c_{\star}$ from (2.8) and the chosen wave speed c via $c^2 = c_{\star}^2+\epsilon^2$. In [Reference Faver16, Reference Faver and Wright20], the travelling wave problem was solved under this rescaling with the help of symmetry; here we obtain those long wave solutions as a consequence of Theorem 6.2 by introducing a rescaling of the amplitude parameter.

Specifically, let $\epsilon_0 = 1$ and let c_ϵ satisfy $c_{\epsilon}^2 = c_{\star}^2+\epsilon^2$. Let

\begin{equation*} \mathcal{X}^r := \!\left\{\boldsymbol{\phi} \in H_{\operatorname{per}}^r(\mathbb{R}^2) \ \big| \ \langle\boldsymbol{\phi},\boldsymbol{\nu}_0\rangle = 0\right\} \end{equation*}

and set

\begin{equation*} \boldsymbol{\nu}_{\epsilon} := \boldsymbol{\nu}_1^{c_{\epsilon}}, \qquad \qquad \boldsymbol{\mu}_1^{\epsilon} := \boldsymbol{\nu}_1^{c_{\epsilon}}, \qquad\text{and }\qquad \boldsymbol{\mu}_2^{\epsilon} := \boldsymbol{\nu}_2^{c_{\epsilon}}. \end{equation*}

Assume now that the spring potentials satisfy $\mathcal{V}_1$, $\mathcal{V}_2 \in \mathcal{C}^{\infty}(\mathbb{R})$. Then the map $\boldsymbol{\Phi}_{c_{\epsilon}}$ from (1.9) meets all of the hypotheses of Theorem 6.2. More precisely, Hypothesis (ii) follows from Lemma 3.6, Hypothesis (iii) from Corollary 2.2, Hypothesis (iv) from Corollary 2.3 and Hypothesis (v) from Corollary 2.4. The mapping and Lipschitz estimates in Hypothesis (vii) follow from the regularity properties of shift operators in Appendix A.4 and composition operators in Appendix A.5 and the uniform bounds on ω_c in c from (2.9).

We thus obtain solutions $\boldsymbol{\phi} = \boldsymbol{\phi}_{c_{\epsilon}}^a$ and $\omega = \omega_{c_{\epsilon}}^a$ to $\boldsymbol{\Phi}_{c_{\epsilon}}(\boldsymbol{\phi}_{c_{\epsilon}}^a,\omega_{c_{\epsilon}}^a) = 0$ for $0 \lt \epsilon \lt \epsilon_0$ and $|a| \le a_{\operatorname{per}}$ for some $a_{\operatorname{per}} \gt 0$. Returning to our original position coordinates, we see that

\begin{equation*} \mathbf{p}_{\epsilon}^a(x) := a\boldsymbol{\phi}_{c_{\epsilon}}^a(\omega_{c_{\epsilon}}^ax) \end{equation*}

solves the original travelling wave problem (1.5).

Now we expose the long wave scaling. Write a in the form $a = \alpha\epsilon^2$ for $|\alpha| \le a_{\operatorname{per}}$ and $0 \lt \epsilon \lt 1$; this ensures $|a| \lt a_{\operatorname{per}}$ and $0 \lt \epsilon \lt \epsilon_0$. Put

\begin{equation*} \boldsymbol{\varphi}_{\epsilon}^{\alpha} := \boldsymbol{\phi}_{\epsilon}^{\alpha\epsilon^2} \qquad\text{and }\qquad \Omega_{\epsilon}^{\alpha} := \frac{\omega_{c_{\epsilon}}^{\alpha\epsilon^2}}{\epsilon}. \end{equation*}

Then the solutions to (1.5) have the form

\begin{equation*} \mathbf{p}_{\epsilon}^a(x) = \alpha\epsilon^2\boldsymbol{\varphi}_{\epsilon}^{\alpha}(\epsilon\Omega_{\epsilon}^{\alpha}x), \end{equation*}

which reveals the long wave scaling. Moreover, these solutions have the same mapping and Lipschitz estimates previously established in [Reference Faver and Wright20, Thm. 4.1] and [Reference Faver16, Thm. 3.1]. For the frequency, the mapping and Lipschitz estimates from (6.5) and the bounds on $\omega_{c_{\epsilon}}$ from (2.9) give

\begin{equation*} \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ |\alpha| \lt a_{\operatorname{per}}}} |\epsilon\Omega_{\epsilon}^{\alpha}| = \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ |\alpha| \lt a_{\operatorname{per}}}} |\omega_{c_{\epsilon}}^{\alpha\epsilon^2}| \lt \infty \end{equation*}

and, for $0 \lt \epsilon \lt \epsilon_0$ and $|\alpha|$, $|\grave{\alpha}| \lt a_{\operatorname{per}}$

\begin{equation*} |\Omega_{\epsilon}^{\alpha} - \Omega_{\epsilon}^{\grave{\alpha}}| = \frac{|\omega_{c_{\epsilon}}^{\alpha\epsilon^2}-\omega_{c_{\epsilon}}^{\grave{\alpha}\epsilon^2}|}{\epsilon} \le \frac{C|\alpha-\grave{\alpha}|\epsilon^2}{\epsilon} = C\epsilon|\alpha-\grave{\alpha}|, \end{equation*}

where C > 0 is independent of ϵ, α, and $\grave{\alpha}$. This Lipschitz estimate is an improvement on the original $\mathcal{O}(1)$ Lipschitz estimates from [Reference Faver and Wright20, Thm. 4.1] and [Reference Faver16, Thm. 3.1].

For the profile, we first introduce the norm

\begin{equation*} \|\boldsymbol{\phi}\|_{\mathcal{C}_{\operatorname{per}}^r} := \|\boldsymbol{\phi}\|_{L^{\infty}} + \|\partial_x^r[\boldsymbol{\phi}]\|_{L^{\infty}} \end{equation*}

for r-times continuously differentiable, 2π-periodic functions ϕ. The mapping and Lipschitz estimates

\begin{equation*} \sup_{\substack{0 \lt \epsilon \lt \epsilon_0 \\ |\alpha| \lt a_{\operatorname{per}}}} \|\boldsymbol{\varphi}_{\epsilon}^{\alpha}\|_{\mathcal{C}_{\operatorname{per}}^r} \lt \infty \qquad\text{and }\qquad \sup_{0 \lt \epsilon \lt \epsilon_0} \|\boldsymbol{\varphi}_{\epsilon}^{\alpha} - \boldsymbol{\varphi}_{\epsilon}^{\grave{\alpha}}\|_{\mathcal{C}_{\operatorname{per}}^r} \lt C_r|\alpha-\grave{\alpha}| \end{equation*}

follow again from (6.5) and the Sobolev embedding.