Travelling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media

Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.

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Introduction
We consider the semi-linear wave equation where x, t, u(x, t) ∈ R, γ = ±1, and ρ, r are bounded, 2π-periodic, strictly positive, and even functions in the set The purpose of this paper is to prove the existence of travelling modulating pulse solutions.
Remark 1.1.The semi-linear wave equation (1) can be considered as a phenomenological model for the description of electromagnetic waves in photonic crystal fibres.Such fibres show a much larger (structural) dispersion than homogeneous glass fibres.As a consequence they are much better able to support nonlinear localised structures such as pulses than their homogeneous counterpart.Most modern technologies for the transport of information through glass fibres use these pulses, see [ISK+20].Sending a light pulse corresponds to sending the digital information 'one' over the zero background.Physically such a pulse consists of a localised envelope which modulates an underlying electromagnetic carrier wave.
The travelling modulating pulse solutions will be constructed as bifurcations from the trivial solution u = 0. Hence we first consider the linear wave equation Since ρ ∈ X 0 , the linear wave equation can be solved by the family of Bloch modes u (x, t) = e ±iωn(l)t e ilx f n (l, x) , n ∈ N, l ∈ B, where B := R\Z and where the pair (ω n (l), f n (l, •)) satisfies the eigenvalue problem subject to the boundary conditions f n (l, x) = f n (l, x + 2π) and f n (l, x) = f n (l + 1, x) e ix ∀l ∈ R, ∀x ∈ R.
The eigenfunctions f n are L 2 (0, 2π)-normalised according to ˆ2π The curves of eigenvalues l → ω n (l) are ordered such that where ω n (l) → ∞ for n → ∞, see [DLP+11].The positivity of ω 1 (l) follows from positivity of ρ ∈ X 0 .A prototypical pattern of the curves of eigenvalues is shown on figure 1.The modulating pulse solutions can be obtained via a weakly nonlinear multiple scaling ansatz which results in the nonlinear Schrödinger (NLS) equation for the description of slow temporal and spatial modulations of the envelope.In detail, for fixed n 0 ∈ N and l 0 ∈ B solutions of the semi-linear wave equation (1) can be approximated by the ansatz u app (x, t) = εA ε (x − c g t) , ε 2 t f n0 (l 0 , x) e il0x e −iωn 0 (l0)t + c.c., with complex amplitude A = A(X, T), group velocity c g := ω ′ n0 (l 0 ), and 0 < ε 1 being a small perturbation parameter.At the leading-order approximation, the envelope amplitude A satisfies the following NLS equation where The NLS equation (5) possesses travelling pulse solutions if ω ′ ′ n0 (l 0 )γ n0 (l 0 ) > 0 in the form: where c and ω are arbitrary parameters such that ωω ′ ′ n0 (l 0 ) < 0 and the positive constants γ 1 and γ 2 are uniquely given by Without loss of generality, we can set c = 0 and −ω = sgn(ω ′ ′ n0 (l 0 )) = sgn(γ n0 (l 0 )), due to the scaling properties of the NLS equation (5).
Remark 1.2.As an example consider the spatially homogeneous case with ρ(x) = 1 and r(x) = 1, i.e. the semi-linear wave equation with constant coefficients.Then, we can re-order the eigenvalues and define producing The travelling pulse solutions exist for γ = 1 with ω = −1 since ω ′ ′ n0 (l 0 ) > 0. Remark 1.3.In [BSTU06] an approximation result was established that guarantees that wavepacket solutions of the semi-linear wave equation (1) with periodic coefficients can be approximated by solutions of the NLS equation ( 5) on an O(ε −2 )-time scale via u app given by (4).In [DR20] this approximation was extended to the d-dimensional case.
Existence of standing and moving modulating pulse solutions in homogenous and periodic media has been considered beyond the O(ε −2 )-time scale.Depending on the problem, we have to distinguish between pulse solutions which decay to zero for |x| → ∞ and generalised pulse solutions which have some small tails for large values of |x|.
Remark 1.4.In the spatially homogeneous case, i.e. if ρ = r = 1, the modulating pulse solutions are time-periodic in a frame co-moving with the envelope.Time-periodic solutions with finite energy are called breather solutions.However, it cannot be expected that such solutions with finite energy do exist in general, according to the non-persistence of breathers result for nonlinear wave equations in homogeneous media [Den93,BMW94,Man21].Nevertheless, generalised breather solutions, i.e. modulating pulse solutions with small tails, do exist.Such solutions were constructed in [GS01] with the help of spatial dynamics, invariant manifold theory and normal form theory.In general, such solutions can only be constructed on large, but finite, intervals in R, see [GS05,GS08].
Remark 1.5.In the spatially periodic case standing generalised modulating pulse solutions of the semi-linear wave equation (1) have been constructed in [LBC+09].These solutions are time-periodic, i.e. again breather solutions, but in contrast to the homogeneous case true spatially localised solutions can be constructed by properly tailoring the periodic coefficients.In [BCLS11] breather solutions were constructed by spatial dynamics in the phase space of time-periodic solutions, invariant manifold theory and normal form theory.With the same approach in [Mai20] such solutions were constructed for a cubic Klein-Gordon equation on an infinite periodic necklace graph.The existence of large amplitude breather solutions of the semi-linear wave equation (1) was shown in [HR19,MS21] via a variational approach.23) below, are dense on the imaginary axis.However, due to the convolution structure w.r.t. the z-variable, see theorem 1.7, for a certain power of ε only a part of the linear operator has to be taken into account.For controlling the order O(ε) of the solution only the part A 1 (ω, c) has to be considered.The central spectrum of A 1 (ω, c) is sketched in the left panel.In the middle panel the central spectrum of A 1 (ω, c) and A 3 (ω, c) is sketched.It plays a role for controlling the order O(ε 3 ).The right panel shows a sketch of the central spectrum of A 1 (ω, c), A 3 (ω, c) and A 5 (ω, c) which plays a role for controlling the order O(ε 5 ).In all cases there is a spectral gap between zero and the rest of the spectrum.
Breather solutions were recently considered in [KR22] for quasi-linear wave equations with periodic coefficients.
Remark 1.6.To our knowledge travelling modulating pulse solutions have not been constructed before for the semi-linear wave equation (1) with spatially periodic coefficients.For the Gross-Pitaevski equation with a periodic potential such solutions were constructed in [PS08] by using the coupled-mode approximation and in [Pel11, chapter 5.6] by using the NLS approximation.The Gross-Pitaevski equation has a phase-rotational symmetry which is not present in the semi-linear wave equation (1).Another new aspect is the fact that in the present paper the normal form transformations are infinite-dimensional in contrast to the existing literature.
In the spatially periodic case travelling modulating solutions of the semi-linear wave equation (1) in general are quasi-periodic in the frame co-moving with the envelope.Hence their construction requires the use of three spatial variables rather than two spatial variables used in the previous works [LBC+09,PS08].However, although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided by considering the problem on a finite spatial domain and by allowing for small tails, as illustrated in figure 2.
The following result will be proven in this work.Figure 3 illustrates the construction of a generalised modulating pulse solution as described in the following theorem.
Theorem 1.7.Let ρ, r ∈ X 0 and γ = 0. Choose n 0 ∈ N and l 0 > 0 such that the following conditions are satisfied: ) with an envelope advancing with group velocity cg = ω ′ n0 (l 0 ), modulating a carrier wave advancing with phase velocity cp = ωn 0 (l 0 )/l 0 , and leaves behind the standing periodic Bloch wave.The wavelength of the carrier wave and the period of the coefficients ρ, r are of a comparable order. and for some fixed N ∈ N. If assumption 4.11 below is satisfied, then there are ε 0 > 0 and C > 0 such that for all ε ∈ (0, ε 0 ) there exist travelling modulating pulse solutions of the semi-linear wave equation (1) in the form where The constants γ 1 , γ 2 are defined in (7), and f n0 (l 0 , •) ∈ H 2 per (T) is a solution of (3).
Remark 1.8.Assumption 4.11 is of technical nature and guarantees the existence of infinitedimensional invariant manifolds in the construction of the modulating pulse solutions.It is satisfied, for instance, if eigenvalues of the linearised operators on and near iR \ {0} are semisimple.An extended result can be obtained in the case of double eigenvalues, see remark 4.15 below.
Remark 1.9.The function h solves a second-order differential equation that is an O(ε)perturbation of the stationary NLS equation.We select h to be a homoclinic orbit with exponential decay to 0 at infinity that is O(ε 2 )-close to the NLS approximation (16), see (15), computed at the pulse solution (6) for c = 0 and ω = −sgn(ω ′ ′ n0 (l 0 )) = −sgn(γ n0 (l 0 )).Remark 1.10.If the non-resonance condition (12) is satisfied for all odd m ⩾ 3, then N can be chosen arbitrarily large, but has to be fixed.The result of [GS05] was improved in [GS08] to exponentially small tails and exponentially long time intervals w.r.t.ε.It is not obvious that the exponential smallness result can be transferred to the spatially periodic case.We also do not use the Hamiltonian setup from [GS01] because it is not clear how the Hamiltonian structure of the semi-linear wave equation (1) can be developed in the spatial dynamics formulation.
The solution v of theorem 1.7 is only defined on a large but finite spatial interval for the semi-linear wave equation (1).However, due to the finite speed of propagation in the semilinear wave equation (1), it is also an approximate solution of the initial-value problem for a very long time, on a very large, but shrinking, spatial domain.The corresponding result is given by the following theorem.
Theorem 1.11.Let v be the solution of theorem 1.7 and take an arbitrary The corresponding solution of the semi-linear wave equation (1) with u(•, 0) = u 0 and Remark 1.12.By theorem 1.11, the modulated pulse solutions are approximated by h app much longer than on the O(ε −2 )-time scale guaranteed by the approximation theorem given in [BSTU06].For instance, on the spatial interval [− 1 2 ε −(2N+1) , 1 2 ε −(2N+1) ] the approximation holds up to time t = 1 2 ε −(2N+1) .We shall describe the strategy of the proof of theorems 1.7 and 1.11.As in [GS01, GS05,GS08] the construction of the modulating pulse solutions is based on a combination of spatial dynamics, normal form transformations, and invariant manifold theory.Plugging the ansatz (13) into (1), we obtain an evolutionary system w.r.t. the unbounded space variable ξ, the spatial dynamics formulation, i.e. we obtain a system of the form with M u linear and N nonlinear in u which is a vector containing v and derivatives of v.For all values of the bifurcation parameter 0 < ε 1 there are infinitely many eigenvalues of M(∂ z , ∂ x , x) on the imaginary axis, see figure 2, and hence the centre manifold reduction is of no use.However, the system is of the form where u 0 is a vector in C 2 corresponding to the eigenvalues of M which are close to zero and where u r corresponds to the infinite-dimensional remainder, i.e. to all the eigenvalues of M which are bounded away from zero for small |ε|.For ε = 0 all eigenvalues of M 0 are zero.The nonlinearity in the u r -equation is split into two parts such that N r ( u 0 , 0) = 0.By finitely many normal form transformations in the u r -equation we can achieve that the remainder term where N is an arbitrary, but fixed number, if certain non-resonance conditions are satisfied, see remark 3.6.Concerning orders of ε, we have ).Hence, the finite-dimensional subspace { u r = 0} is approximately invariant, and setting the highest order-in-ε term H r (ũ 0 ) = O(ε 2N+2 ) to 0, we obtain the reduced system For the reduced system, a homoclinic solution inside the subspace { u r = 0} can be found, which bifurcates with respect to ε from the trivial solution.The persistence of this solution for the system (18) and ( 19) cannot be expected, since the finite-dimensional subspace { u r = 0} is not truly invariant for (18) and ( 19), and therefore the necessary intersection of the stable and unstable manifolds is unlikely to happen in an infinite-dimensional phase space.However, the approximate homoclinic orbit can be used to prove that the centre-stable manifold intersects the fixed space of reversibility transversally which in the end allows us to construct a modulating pulse solution with the properties stated in theorem 1.7.The proof of theorem 1.11 is based on the energy method in the backward light cone associated to the point (x 0 , t 0 ).The point is arbitrarily chosen in the upper half-plane inside applicability of the solution v obtained in theorem 1.7, that is, for ξ ∈ [−ε −(2N+1) , ε −(2N+1) ].
The paper is organized as follows.In section 2 we introduce the spatial dynamics formulation by using Fourier series and Bloch modes.We develop near-identity transformations in section 3 for reducing the size of the tails and increasing the size of the spatial domain.A local centre-stable manifold in the spatial dynamics problem is constructed in section 4. The proof of theorem 1.7 is completed in section 5 by establishing an intersection of the centre-stable manifold with the fixed space of reversibility.Theorem 1.11 is proven in section 6.

Spatial dynamics formulation
Here we introduce the spatial dynamics formulation by using Fourier series and Bloch modes.We fix l 0 ∈ B and define where ω and c are to be determined and v(ξ, •, •) satisfies Inserting (20) into the semi-linear wave equation (1) and using the chain rule, we obtain a new equation for v: In order to consider this equation as an evolutionary system with respect to ξ ∈ R, we use Fourier series in Equation ( 21) is converted through the Fourier expansion (22) into the spatial dynamics system for every c = ±1: ) and the double convolution sum is given by The spatial dynamics system (23) can also be written in the scalar form as Remark 2.1.If ρ ∈ L ∞ per (T), then the domain D and the range R of the linear operator A m (ω, c) : Solutions of the dynamical system (23) are then sought such that at each ξ ∈ R they lie in the phase space with the range in where ℓ 2,k (Z, H s ), with k ∈ N, is a weighted ℓ 2 -space equipped with the norm The solution map for the initial-value problem associated to the dynamical system (23) is then defined as [0, ξ 0 ] ξ → (ṽ m , wm ) m∈Z ∈ C 1 ([0, ξ 0 ], D).The phase space D in (26) is equivalent to the function space X in theorem 1.7 under the Fourier series (22).
Remark 2.2.Real solutions v = v(ξ, z, x) after the Fourier expansion (22) enjoy the symmetry: The cubic nonlinearity maps the space of Fourier series where only the odd Fourier modes are non-zero to the same space.Hence, we can look for solutions of the spatial dynamics system (23) in the subspace Hence the components (ṽ m , wm ) for −m ∈ N odd can be obtained from the components (ṽ m , wm ) for m ∈ N odd by using the symmetry (28).

Linearised problem
Truly localised modulating pulse solutions satisfy i.e. such solutions are homoclinic to the origin with respect to the evolutionary variable ξ.If these solutions exist, they lie in the intersection of the stable and unstable manifold of the origin.However, the modulating pulse solutions are not truly localised because of the existence of the infinite-dimensional centre manifold for the spatial dynamics system (23).
) can be reformulated in the scalar form: Eigenvalues λ are obtained by setting V(x) = f n (ml 0 − iλ, x) and using the spectral problem (3) for l ∈ C, where both f n (l, x) and ω n (l) are analytically continued in l ∈ C. The eigenvalues are the roots of the nonlinear equations Zero eigenvalues λ = 0 exist if and only if there exist solutions of the nonlinear equations ω 2 n (ml 0 ) = m 2 ω 2 0 .Since ω 0 = ω n0 (l 0 ), ω 2 n (ml 0 ) = m 2 ω 2 0 is satisfied for m = 1 and n = n 0 .Due to the non-degeneracy assumption (10), ω 2 n (l 0 ) = ω 2 0 does not hold for any other n.This shows the geometric simplicity of λ = 0 for m = 1.It follows from ( 10) and ( 12) that no other solutions of ω 2 n (ml 0 ) = m 2 ω 2 0 exist for m ∈ {1, 3, . . ., 2N + 1}.It remains to prove that the zero eigenvalue for m = 1 is algebraically double.To do so, we again employ the equivalence of the eigenvalue problem (3) and ( 29) for λ = 0, l = l 0 , n = n 0 , and m = 1.For n = n 0 , and l = l 0 , this equation and its two derivatives with respect to l generate the following relations: The non-degeneracy condition (11) implies that c 2 g = 1.Computing the Jordan chain for A 1 (ω 0 , c g ) at the zero eigenvalue with the help of ( 31) and (32) yields and We use f n0 and ∂ l f n0 to denote f n0 (l 0 , •) and ∂ l f n0 (l 0 , •) respectively.It follows from (32) and ( 33) that where f ′ n0 denotes ∂ x f n (l 0 , •) and where we have used the normalisation where ϕ, ψ = ´2π 0 φψ dx is the standard inner product in L 2 (0, 2π).With some abuse of notation in the following we write f, g for f, g .
Using complex conjugation, transposition, and integration by parts, the adjoint operator to A 1 (ω, c) in L 2 (0, 2π) is computed as follows: for which we obtain where the normalisation has been chosen such that G 0 , F 1 = 1 due to the relation (37).Note also that G 0 , F 0 = 0 due to the relation (36).
For the generalised eigenvector of A * 1 (ω 0 , c g ) we have with As A 1 (ω 0 , c 0 ) has a compact resolvent, a standard argument using Fredholm's alternative guarantees that there exists a 2π-periodic solution of the inhomogeneous equation if and only if F 1 is orthogonal to Ker(A * 1 ), i.e. to G 0 .However, since G 0 , F 1 = 1, the Jordan chain for the zero eigenvalue terminates at the first generalised eigenvector (35), i.e. λ = 0 is algebraically double.
Remark 2.5.By using the same argument, we verify that also the adjoint operator A * 1 (ω 0 , c g ) has a double zero eigenvalue.This follows from the existence of solutions in (39) and (40) and non-orthogonality of the generalised eigenvector In the low-contrast case, i.e. when the periodic coefficient ρ is near ρ ≡ 1, the nondegeneracy conditions (10), (11) and the non-resonance condition (12) are easy to verify.If ρ(x) = 1, the eigenvalues ω n (l) are known explicitly, see (8).The following lemma specifies the sufficient conditions under which the non-resonance assumption (12) is satisfied.
Remark 2.7.The non-resonance condition (41) is satisfied for all m ∈ N if l 0 ∈ R \ Q.

Formal reduction
Let us now consider a formal restriction of system (24) to the subspace leading to the NLS approximation (16).As S is not an invariant subspace of system (24), this reduction is only formal and a justification analysis has to be performed, which we do in the remainder of this paper.
The scalar equation (24) on S reduces to for m = 1 and to the complex conjugate equation for m = −1.Using the Jordan block for the double zero eigenvalue in lemma 2.3, we write the two-mode decomposition: where , where the O(ε 3 ) terms are neglected.Using ω = ω 0 + ωε 2 and c = c g with ω 0 = ω n0 (l 0 ) and c g = ω ′ n0 (l 0 ), we obtain the following equation at order O(ε 3 ): where equations ( 31) and (32) have been used and f n0 again denotes f n0 (l 0 , •).Projecting (43) onto span{f n0 } and using (37) yields the stationary NLS equation which recovers the stationary version of the NLS equation (5) for A(X, T) replaced by A(X)e −i ωT with real A(X).The modulated pulse solution corresponds to the soliton solution of the stationary NLS equation (44), where γ 1 and γ 2 are given by (7).Note that among the positive and decaying at infinity solutions of the stationary NLS equation (44) the pulse solution (45) is unique up to a constant shift in X.

Reversibility
Because ρ and r are even functions in X 0 given by (2), the semi-linear wave equation (1), being second order in space, is invariant under the parity transformation: u(x, t) → u(−x, t).
Similarly, since it is also second-order in time, it is invariant under the reversibility transformation: u(x, t) → u(x, −t).
The two symmetries are inherited by the scalar equations ( 21) and ( 24) Since the symmetry is nonlocal in x, one can use the Fourier series in x given by and similarly for wm to rewrite the symmetry in the form: The implication of the symmetry (28) and ( 47) is that if a solution {v m,k (ξ), ŵm,k (ξ)} (m,k)∈N odd ×Z constructed for ξ ⩾ 0 satisfies the reversibility constraint: then the solution {v m,k (ξ), ŵm,k (ξ)} (m,k)∈N odd ×Z can be uniquely continued for ξ ⩽ 0 using the extension This yields a symmetric solution of the spatial dynamics system (23) for every ξ ∈ R after being reformulated with the Fourier expansion (46).

Near-identity transformations
By lemma 2.3, the Fredholm operator A 1 (ω 0 , c g ) has the double zero eigenvalue, whereas A m (ω 0 , c g ) for 3 ⩽ m ⩽ 2N + 1 admit no zero eigenvalues.In what follows, we decompose the solution in X into a two-dimensional part corresponding to the double zero eigenvalue and the infinite-dimensional remainder term.

Separation of a two-dimensional problem
Like in the proof of lemma 2.3, we denote the eigenvector and the generalised eigenvector of A 1 (ω 0 , c g ) for the double zero eigenvalue by F 0 and F 1 , see (34) and (35), and the eigenvector and the generalised eigenvector of A * 1 (ω 0 , c g ) for the double zero eigenvalue by G 0 and G 1 , see (39) and (40).
We define Π as the orthogonal projection onto the orthogonal complement of the generalised eigenspace span(G 0 , G 1 ), i.e.
The orthogonality follows from our normalisation, which was chosen in the proof of lemma 2.3 so that G 0 , . Moreover, we have Compared to the two-mode decomposition (42), we write where q 0 , q 1 : R → C are unknown coefficients and where S 1 (ξ, Similarly, we write and define w m = εW m , and use the notation V := (V m ) m∈N odd and V ⩾3 := (V m ) m∈N odd \{1} .
For ω = ω 0 + ε 2 ω and c = c g , we write the spatial dynamics system (23) with ω = ω 0 + ωε 2 and c = c g is now rewritten in the separated form: and for m ∈ N odd \{1}, where the correction terms H 0 and (H m ) m∈N odd are given by , and Remark 3.1.System (50) does not have an invariant reduction at S 1 = 0 and V ⩾3 = 0 because which contributes to Π H 1 and H 3 (as well as to H 0 ).

Resolvent operators for the linear system
In order to derive bounds ( 14) and (15), we need to perform near-identity transformations, which transform systems (50b) and (50c) to equivalent versions but with residual terms of the order O(ε 2(N+1) ).To be able to do so, we will ensure that the operators Π A 1 (ω 0 , c g )Π and A m (ω 0 , c g ), 3 ⩽ m ⩽ 2N + 1 are invertible with a bounded inverse.By lemma 2.3, these operators do not have zero eigenvalues but this is generally not sufficient since eigenvalues of these infinite-dimensional operators may accumulate near zero.However, the operators A m (ω 0 , c g ) have the special structure which we explore to prove invertibility of these operators under the non-degeneracy and nonresonance conditions.The following lemma gives the result.
Lemma 3.2.If the conditions ( 10)-( 12) are satisfied, then there exists a C 0 > 0 such that where D and R are defined in (25).
Proof.Under the non-degeneracy condition (11) which yields c g = ±1, the entries of A m (ω, c g ) are all non-singular.To ensure the invertibility of A m (ω 0 , c g ) with m ∈ N odd \{1}, we consider the resolvent equation for a given (f, g) ∈ R. The solution (v, w) ∈ D is given by w = f and v obtained from the scalar Schrödinger equation Under the non-resonance conditions (12), 0 is in the spectral gap of L m making the linear operator L m : H 2 per → L 2 is invertible with a bounded inverse from L 2 to H 2 per .Hence v = L −1 m (g − M m f) and A m (ω 0 , c g ) is invertible with a bounded inverse from R to D. The operator A 1 (ω 0 , c g ) is not invertible due to the double zero eigenvalue in lemma 2.3.However, it is a Fredholm operator of index zero and hence by the closed range theorem there exists a solution of the inhomogeneous equation for every (f, g) ∈ R. The solution is not uniquely determined since span(F 0 , F 1 ) can be added to the solution (v, w) ∈ D, however, the restriction to the subset defined by the condition removes projections to span(F 0 , F 1 ).Consequently, the operator (Π A 1 (ω 0 , c g )Π) is invertible with a bounded inverse from R to D.
In the limit δ → 0 of lemma 2.6 we can calculate the eigenvalues λ of A m (ω 0 , c g ) graphically, see figure 4, and explicitly.The following lemma summarises the key properties of eigenvalues which are needed for assumption 4.11 in theorem 1.7.
Lemma 3.3.Let ρ ≡ 1.For every fixed m ∈ N odd , the operator A m (ω 0 , c g ) has purely imaginary eigenvalues, Jordan blocks of which have length at most two, and complex semi-simple eigenvalues with nonzero real parts bounded away from zero.Moreover, if l 0 ∈ R\Q, then all nonzero, purely imaginary eigenvalues are semi-simple.
Proof.Eigenvalues of A m (ω 0 , c g ) are found as solutions of the nonlinear equation (30).For ρ ≡ 1 we use Fourier series (46) and write ω 2 n (l) = 1 + (n + l) 2 with n = k ∈ Z.After simple manipulations, eigenvalues λ are found from where m ∈ N odd , n 0 ∈ N, and l 0 ∈ B are fixed and where ω 0 = 1 + (n 0 + l 0 ) 2 .Setting κ := k − mn 0 , we get Eigenvalues λ are found explicitly as For each k ∈ Z, the value of κ ∈ Z is fixed.Eigenvalues are double if for some m ∈ N odd and κ ∈ Z, in which case the Jordan blocks have length two.If l 0 ∈ R\Q and κ = 0, then (m − κ(n 0 + l 0 )) 2 − 1 = 0 and the eigenvalues are semi-simple, in which case there are no Jordan blocks.If κ = 0, then λ = ±iω 0 √ m 2 − 1 which includes a double zero eigenvalue for m = 1 and pairs of semi-simple purely imaginary eigenvalues.If κ = 0, then complex eigenvalues off the imaginary axis arise for each (m, κ) ∈ N odd × Z with |m − κ(n 0 + l 0 )| < 1; the two complex eigenvalues appear in pairs symmetrically about iR.Therefore, complex eigenvalues with nonzero real parts are semi-simple.Moreover, Im(λ) = −κω 2 0 and hence |Im(λ)| ⩾ ω 2 0 for each complex eigenvalue.
Remark 3.4.Conditions (41) ensure that As a result, the purely imaginary non-zero eigenvalues of lemma 3.3 are bounded away from zero by The inequality D m > 0 follows from the fact that as κ → ∞, where ω 0 = 1 + (n 0 + l 0 ) 2 has been used.We do not need invertibility of A m (ω 0 , c g ) as m → ∞ since we only use the near-identity transformations for 1 ⩽ m ⩽ 2N + 1.Therefore, we do not need to investigate whether D m → 0 as m → ∞.
In the next two subsections we proceed with near identity transformations by using the bounds (52) and prove the following theorem.

S (N)
where Φ (2j) depends polynomially on q 0 , q0 , q 1 , and q1 , and analogously for V Moreover, the transformations preserve the reversibility of the system, see section 2.3.Remark 3.6.It is well known that in the equation for u j with eigenvalue i λ j a term of the form u nj 1 j1 . . .u nj r jr can be eliminated by a near identity transformation if the non-resonance condition λ j − r k=1 n j k λ nj k = 0 is satisfied, see section 3.3 in [GH83].Since the eigenvalues for the (q 0 , q 1 )-part vanish and since the eigenvalues for the S (N)

-part and the Y (N)
m -part do not vanish, all polynomial terms in (q 0 , q 1 ) can be eliminated in the equations for the S (N) m .This elimination is done by theorem 3.5 up to order O(ε 2N ).Some detailed calculations can be found in the subsequent sections 3.3 and 3.4.

Removing polynomial terms in q 0 and q 1 from (50c)
In order to show how the near-identity transformations reduce (50c) to (53c), we consider a general inhomogeneous term ε 2j q k1 0 qk2 0 q k3 1 qk4 1 in the right-hand side of (50c) with some 1 ⩽ j ⩽ N and positive integers k 1 , k 2 , k 3 , k 4 .The transformations are produced sequentially, from terms of order O(ε 2 ) to terms of order O(ε 2N ) and for each polynomial order in (q 0 , q 1 ).
At the lowest order j = 1, there exists only one inhomogeneous term in (50c) for m = 3, see remark 3.1, which is given by The choice of the second components in each term of Y 3 is dictated by the fact that ∂ ξ q 0 − q 1 and ∂ ξ q 1 are of the order of O(ε 2 ) due to equation (50a).We are looking for scalar functions h j ∈ H 2 per from the sequence of linear inhomogeneous equations obtained with the help of (51): Since L 3 is invertible with a bounded inverse by lemma 3.2, there exist unique functions h j ∈ H 2 per which are obtained recursively from h 0 to h 3 .After the inhomogeneous terms are removed by the choice of h j , the transformed right-hand side H3 becomes where H rest 3 is also modified due to the transformation.Substituting for q0 − q 1 and q1 from (50a) shows that H3 (q 0 , q 1 , 0, 0) = O(ε 2 ), hence the first step of the procedure transforms (50c) into (53c) with N = 1.One can then define H3 = H(q) 3 + Hrest with Hrest 3 = 0 if (S 1 , V ⩾3 ) = (0, 0) and proceed with next steps of the procedure.
A general step of this procedure is performed similarly.Without loss of generality, since the principal part of system (50a) is independent of q0 and q1 , we consider a general polynomial of degree M at fixed m ∈ {3, 5, . . ., 2N + 1}: where (a j , b j ) T depend on x only.Substituting q M−j 0 q j 1 h j g j into (50c) yields (53c) with N being incremented by one if h m , g m ∈ H 2 per are found from two chains of recurrence equations for j ∈ {0, 1, . . ., M}: which are truncated at h −1 = g −1 = 0. Since L m for 3 ⩽ m ⩽ 2N + 1 are invertible with a bounded inverse by lemma 3.2, the recurrence equations are uniquely solvable from g 0 to h 0 , then to g 1 and h 1 and so on to g M and h M .The aforementioned first step is obtained from here with M = 3 and a 0 = a 1 = a 2 = a 3 = 0.
3.4.Removing polynomial terms in q 0 and q 1 from (50b) Similarly, we perform near-identity transformations which reduce (50b) to (53b).The only complication is the presence of the projection operator Π in system (50b).
Since Π A 1 (ω 0 , c g )Π is invertible with a bounded inverse by lemma 3.2, the two chains of equations are uniquely solvable: from S (0) to S (1) and from S (2) to S (7) .
It is now straightforward that a general step of the recursive procedure can be performed to reduce (50b) to (53b).

Construction of the local centre-stable manifold
By theorem 3.5, the spatial dynamical system can be transformed to the form (53), where the coupling of (q 0 , q 1 ) ∈ C 2 with S (N) ).We are now looking for solutions of system (53) for ξ on [0, ξ 0 ] for some εdependent value ξ 0 > 0. In order to produce the bound ( 14), we will need to extend the result to ξ 0 = ε −(2N+1) .
It remains to prove the approximation bound (58).The symmetry (61) generates the twodimensional kernel of the linearised operator associated with the leading-order part of the truncated system (54): 5. Transversal intersection of the homoclinic orbit of system (54) with the fixed space of the reversibility operator.
The symmetry modes (63) do not satisfy the reversibility constraints (57) because A(0) = 0 and A ′ ′ (0) = 0, whereas the truncated system (54) inherits the reversibility symmetry (57) of the original dynamical system (23).Therefore, if we substitute the decomposition into (54), then the correction term (q 0 , q 1 ) satisfies the nonlinear system where the residual terms of the order of O(ε), see (62), are automatically orthogonal to the kernel of the linearised operator.By the implicit theorem in Sobolev space H 1 (R), one can uniquely solve the nonlinear system for the correction term (q 0 , q 1 ) under the reversibility constraints (57) such that for some ε-independent C > 0. This yields the approximation bound (58) in the original variables due to the Sobolev embedding of H 1 (R) into L ∞ (R).
Remark 4.3.Referring to system (18), we have now constructed the homoclinic solution for the approximate reduced system It remains to prove the persistence of the homoclinic solutions as generalised breather solutions under considering the higher order terms ε 2N+2 Z (m) ⩾3 ) in (53b) and (53c) for m ∈ N odd which lead to (S ⩾3 ) = (0, 0).We do so by constructing a centre-stable manifold nearby the approximate homoclinic solution, see the rest of section 4, and by proving that the centre-stable manifold intersects the fixed space of reversibility transversally, see section 5.
Let us denote the ε-dependent reversible homoclinic orbit of lemma 4.1 by (Q 0 , εQ 1 ) and introduce the decomposition (q 0 , q 1 ) = (Q 0 , εQ 1 ) + (q 0 , εq 1 ).We abbreviate c 0,hom := (Q 0 , Q 1 , Q 0 , Q 1 ) and c 0,r := (q 0 , q 1 , q 0 , q 1 ).Furthermore, we collect the components S for m ∈ N odd \{1} in c r .With these notations system (53) can now be rewritten in the abstract form: where the vector c 0,r (ξ) ∈ C 4 is controlled in the norm • C 4 , whereas the vector c r (ξ) is controlled in the phase space D defined by (26) with the norm • D .The operator εΛ 0 is the linearisation around the homoclinic orbit c 0,hom and hence G depends nonlinearly on c 0,r .Note that including the complex conjugated variables in c 0,r is needed in order for the linearised system ∂ ξ c 0,r = εΛ 0 (ξ)c 0,r to be linear with respect to the complex vector field.
Remark 4.4.At leading order in ε the matrix Λ 0 is derived from (62) in the form where using the fact that A is real.
Remark 4.5.Although not indicated by our notation, the operators Λ 0 , Λ r and the functions G, G R , F, and F R depend on ξ and ε continuously.
Remark 4.6.We lose one power of ε in front of G and G R by working with Q 1 instead of εQ 1 .

Residual terms of system (64)
Residual terms are controlled as follows.
Lemma 4.7.There exists ε 0 > 0 such that for every ε ∈ (0, ε 0 ) the residual terms of system (64) satisfy the bounds for every ξ ∈ R: where C > 0 is a generic ε-independent constant, which may change from line to line.
Proof.The residual terms are defined in theorem 3.5.Functions G, G R , F, and F R map D into D since D forms a Banach algebra with respect to pointwise multiplication.Using (q 0 , q 1 ) = c 0,hom + c 0,r and the fact that c 0,hom (ξ) C 4 is bounded independently of ε, the bounds on G, G R , F, and F R follow.

Linearised operator of system (64a)
The linear part of system (64a) is the linearisation around the approximate homoclinic orbit q 0,hom from lemma 4.1.Due to the translational and the SO(2)-symmetry of the family of homoclinic orbits generated by these symmetries applied to the reversible homoclinic orbit, the solution space of the linearised equation includes a two-dimensional subspace spanned by exponentially decaying functions.
Lemma 4.8.Consider the linear inhomogeneous equation with a given The homogeneous equation has a two-dimensional stable subspace spanned by the two fundamental solutions then there exists a two-parameter family of solutions c 0,r ∈ C 0 b (R) in the form c 0,r = α 1 s 1 + α 2 s 2 + c0,r , where (α 1 , α 2 ) ∈ C 2 and c0,r ∈ C 0 b (R) is a particular solution satisfying the constraints (67) and the bound for an ε-independent constant C.
Proof.As already said, the existence of the two-dimensional stable subspace spanned by (66) follows from the translational symmetries due to spatial translations and phase rotations of the truncated system (54).Since the truncated system is posed in C 4 , the solution space is four dimensional and the other two fundamental solutions of the homogeneous equation are exponentially growing as ξ → ∞.This can be seen from the limit of Λ 0 (ξ) as ξ → ∞.Indeed, we have lim ξ→∞ Λ 0 (ξ) = K 0 0 K , the eigenvalues of which are ± −2 ω(ω ′ ′ n0 (l 0 )) −1 , each being double, where ω(ω ′ ′ n0 (l 0 )) −1 < 0 by assumption of theorem 1.7.

Estimates for the local centre-stable manifold
We are now ready to construct a local centre-stable manifold for system (64).Let us split the components in c r in three sets denoted by c s , c u , and c c , where c s , c u , and c c correspond to components of Λ r with eigenvalues λ with Re(λ) < 0, Re(λ) > 0, and Re(λ) = 0 respectively.Remark 4.10.These coordinates correspond to the stable, unstable, and reduced centre manifold of the linearised system in lemma 2.3, where the reduced centre manifold is obtained after the double zero eigenvalue is removed since the eigenspace of the double zero eigenvalue is represented by the coordinate c 0,r .
We study the coordinates c s , c u , and c c in subsets of the phase space D denoted by D s , D u , and D c respectively.Similarly, the restrictions of Λ r to the three subsets of D are denoted by Λ s , Λ u , and Λ c respectively.Moreover, let P j for j = s, u, c be the projection operator from D to D j satisfying P s D→D + P u D→D + P c D→D ⩽ C for some C > 0.
We make the following assumption on the semi-groups generated by the linearised system, see [LBC+09].
Assumption 4.11.There exist K > 0 and ε 0 > 0 such that for all ε ∈ (0, ε 0 ) we have The following theorem gives the construction of the local centre-stable manifold near the reversible homoclinic orbit of lemma 4.1.It also provides a classification of all parameters of the local manifold which will be needed in section 5 to satisfy the reversibility conditions.The centre-stable manifold is constructed for ξ ∈ [0, ε −(2N+1) ] and not for all ξ ⩾ 0. The bound of O(ε 2N ) on the coordinates c 0,r and c r is consistent with the bound (14) in theorem 1.7.
We are looking for a global solution of system (72) and (73) for ξ ∈ [0, ∞) which may be unbounded as ξ → ∞.This global solution coincides with a local solution of system (64) on the interval [0, ξ 0 ] ⊂ R.
We write c 0,r = α 1 s 1 + α 2 s 2 + c0,r and rewrite (73) as an equation for c0,r .By the construction of the vector field in system (73), the vector field satisfies the reversibility constraints (67).By the bounds of lemma 4.7 and the invertibility of the linear operator in lemma 4.8, the implicit function theorem implies that there exists a unique map from Using the variation of constant formula the solution of system (72) projected to D c ⊕ D s ⊕ D u can be rewritten in the integral form and We denote due to the bound (59), it follows from the previous estimates that there exists C > 0 such that Using a bootstrapping argument, we show that S(ε −(2N+1) ) ⩽ Cε 2N if ε > 0 is small enough.
This bound still implies S(ξ 0 ) ⩽ Cε 2N but for ξ 0 = ε −N−1 and a Dc + b Ds ⩽ ε 3N+1 .Thus, the justification result of theorem 1.7 can be extended on the scale of ξ ∈ O(ε −N−1 ) to the generic case of eigenvalues with Jordan blocks of length two when assumption 4.11 cannot be used, see lemma 3.3.

End of the proof of theorem 1.7
In theorem 4.12 we constructed a family of local bounded solutions of system (64) on [0, ε −(2N+1) ].These solutions are close to the reversible homoclinic orbit of lemma 4.1 in the sense of the bound (14) for appropriately defined v and h but only on [0, ε −(2N+1) ].It remains to extract those solutions of this family which satisfy (14) not only on [0, ε −(2N+1) ], but also on [−ε −(2N+1) , ε −(2N+1) ].We do so by extending the local solutions on [0, ε −(2N+1) ] to the interval [−ε −(2N+1) , ε −(2N+1) ] with the help of the reversibility constraints.Obviously this is only possible for the solutions which intersect the fixed space of reversibilty.Hence, for the proof of theorem 1.7 it remains to prove that the local invariant centre-stable manifold of system (64) intersects the subspace given by the reversibility constraints (48).
• The initial data c s (0) and c u (0) are not arbitrary since the stable and unstable manifold theorems are used for construction of c s and c u in the proof of theorem 4.12.Combining c s/u := (c s , c u ) together for the complex eigenvalues outside iR, we can write Remark 5.2.Since the local centre-stable manifold intersects the plane given by the reversibility constraints (48), we have thus constructed a family of reversible solutions on [−ε −(2N+1) , ε −(2N+1) ] while preserving the bound (15).Tracing the coordinate transformations back to the original variables completes the proof of theorem 1.7.

Proof of theorem 1.11
For any given point (x 0 , t 0 ) ∈ R × (0, ∞), we define a triangular region in the backward light cone of the wave equation: To prove theorem 1.11, we show that solutions of the semi-linear wave equation (1) have a finite propagation speed (bounded in absolute value by 1).We start with the following lemma.Lemma 6.1 enables us to use the energy method in the proof of uniqueness of solutions of the semi-linear wave equation (1) inside the backward light cone C (x0,t0) for any fixed (x 0 , t 0 ) ∈ R × (0, ∞).The uniqueness is equivalent to the property of the finite propagation speed in (1).Then, u(x, t) = v(x, t) for every (x, t) ∈ C (x0,t0) .
Proof.Let w := u − v. Then w ∈ C 2 (R × R) satisfies ˆt 0 E (t) dt, for some C > 0 that depends on ρ, r ∈ X 0 .By Gronwall's inequality, this yields E(t) = 0 for every t ∈ [0, t 0 ] so that w(•, t) = 0 in H 1 (B (x0,t0,t) ) and ∂ t w(•, 0) = 0 in L 2 (B (x0,t0,t) ) for every t ∈ [0, t 0 ].By Sobolev's embedding of H 1 (R) into C 0 b (R), this implies that w(x, t) = 0 for every x ∈ B (x0,t0,t) and t ∈ [0, t 0 ], that is, for every (x, t) ∈ C (x0,t0) .Remark 6.3.Well-posedness of the initial-value problem for the semi-linear wave equation (1) with ρ, r ∈ X 0 and bootstrapping can be used to state that the first partial derivatives of u(x, t) and v(x, t) in (x, t) are also equal for every (x, t) ∈ C (x0,t0) .Remark 6.4.Theorem 1.11 is a restatement of lemma 6.2, where v (x, t) = v ext (x − c g t, l 0 x − ωt, x) is constructed based on theorem 1.7 and an arbitrary function ϕ.The function v ext (ξ, z, x) belongs to C 2 (R, X ) which is weaker than C 2 (R × R) for the function v(x, t) in variables (x, t).However, lemmas 6.1 and 6.2 can be extended by a simple density argument to functions for which the energy E(t) is bounded in C (x0,t0) and for which the linear wave equation ( 80) is satisfied almost everywhere C (x0,t0) .Functions u(x, t) and v(x, t) constructed from v ext (ξ, z, t) as in theorem 1.11 belong to the required function space.

Figure 1 .
Figure 1.The curves of eigenvalues {±ωn(l)} n∈N plotted as functions of the Bloch wave numbers l ∈ B in a typical situation.

Figure 2 .
Figure2.Eigenvalues of the spatial dynamics formulation, see (23) below, are dense on the imaginary axis.However, due to the convolution structure w.r.t. the z-variable, see theorem 1.7, for a certain power of ε only a part of the linear operator has to be taken into account.For controlling the order O(ε) of the solution only the part A 1 (ω, c) has to be considered.The central spectrum of A 1 (ω, c) is sketched in the left panel.In the middle panel the central spectrum of A 1 (ω, c) and A 3 (ω, c) is sketched.It plays a role for controlling the order O(ε 3 ).The right panel shows a sketch of the central spectrum of A 1 (ω, c), A 3 (ω, c) and A 5 (ω, c) which plays a role for controlling the order O(ε 5 ).In all cases there is a spectral gap between zero and the rest of the spectrum.

Figure 3 .
Figure 3.A generalised modulating pulse solution as constructed in theorem 1.7 with O(ε 2N ) tails existing for x in an interval of length O(ε −(2N+1)) with an envelope advancing with group velocity cg = ω ′ n0 (l 0 ), modulating a carrier wave advancing with phase velocity cp = ωn 0 (l 0 )/l 0 , and leaves behind the standing periodic Bloch wave.The wavelength of the carrier wave and the period of the coefficients ρ, r are of a comparable order.
c s/u (0) = b + cs/u (0), where cs/u (0) are uniquely defined of the order of O(ε 2N ) and depend on b ∈ D s in higher orders.By the implicit function theorem, there exists a unique solution b ∈ D s of b = c s/u (0) − cs/u (0) satisfying the reversibility constraints (48) and this unique b ∈ D s satisfies the bound (70).Remark 5.1.There still exist infinitely many parameters after a ∈ D c have been chosen to satisfy the constraint (48), namely, Re â(v) m,k and Im â(w) m,k for (m, k) ∈ N odd × Z.These parameters of the solution of theorem 1.7 must satisfy the bound (70) in theorem 4.12.
[Eas73]ion 4.11 can be satisfied for smooth small-contrast potentials, see lemmas 2.6 and 3.3.For ρ with a small non-zero contrast spectral gaps occur in figure4.Smoothness of ρ allows to control the size of the spectral gaps for large λ, see[Eas73].Assumption 4.11 can be weakened and Jordan-blocks can be allowed, see remark 4.15.The equivalent bound for the estimate of sup ξ ∈[0,ξ0] c c (ξ) Dc is given by sup Remark 4.13.The proximity bound (71) yields the estimate (14) in theorem 1.7.Remark 4.14.