Periodic and homoclinic travelling waves in infinite lattices

Nonlinear Analysis 74 (2011) 2071–2086

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Periodic and homoclinic travelling waves in infinite lattices Percy D. Makita ∗ Mathematical Institute, University of Giessen, Arndtstr. 2, D–35392 Giessen, Germany

article

abstract

info

Article history: Received 24 November 2009 Accepted 3 November 2010

Consider an infinite chain of particles subjected to a potential f , where nearest neighbours are connected by nonlinear oscillators. The nonlinear coupling between particles is given by a potential V . The dynamics of the system is described by the infinite system of second order differential equations

MSC: 37K60 34C25 34C37

q¨ j + f ′ (qj ) = V ′ (qj+1 − qj ) − V ′ (qj − qj−1 ),

Keywords: Infinite dimensional Hamiltonian systems Travelling waves Periodic and homoclinic motions

j ∈ Z.

We investigate the existence of travelling wave solutions. Two kinds of such solutions are studied: periodic and homoclinic ones. On one hand, we prove under some growth conditions on f and V , the existence of non-constant periodic solutions of any given period T > 0, and speed c > c0 , where the constant c0 depends on f ′′ (0) and V ′′ (0). On the other hand, under very similar conditions, we establish the existence of non-trivial homoclinic solutions, of any given speed c > c0 , emanating from the origin. Moreover, we prove that these homoclinics decay exponentially at infinity. Each homoclinic is obtained as a limit of periodic solutions when the period goes to infinity. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction We consider a Z lattice system in one dimension subjected to a potential f . In addition, each particle interacts with its nearest neighbours under a (nonlinear) interaction potential V . The dynamics of the system is described by the infinite system of second order ordinary differential equations: q¨ j (t ) + f ′ (qj (t )) = V ′ (qj+1 (t ) − qj (t )) − V ′ (qj (t ) − qj−1 (t )),

t ∈ R, j ∈ Z,

(1.1)

where f , V ∈ C (R). For f ≡ 0, (1.1) is the usual lattice equation, sometimes called the Fermi–Pasta–Ulam (FPU) lattice. For f (x) = K (1 − cos x), with K > 0, (1.1) is sometimes called the Frenkel–Kontorova model, even if V is not harmonic, i.e. V (x) ̸= α2 x2 . As is well known, (1.1) is an infinite-dimensional Hamiltonian system whose Hamiltonian is ‘formally’ given by 1

H =

− [1 j∈Z

2

]

p2j + f (qj ) + V (qj+1 − qj ) ,

where the first term represents the kinetic energy of the j-th particle and the remaining terms (containing qj ) represent its ‘potential energy’. The aim of this paper is to investigate the existence of periodic and homoclinic travelling waves. A travelling wave, say with speed c > 0 and profile u, is a solution of (1.1) of the form qj (t ) = u(j − ct ),

∗

j ∈ Z.

Tel.: +49 641 9932 173; fax: +49 641 9932 179. E-mail address: [email protected].

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.11.011

(1.2)

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P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

Plugging the Ansatz (1.2) into (1.1) yields the second order backward–forward differential equation for the wave profile:

˜ ), c 2 u′′ + f ′ (u) = V ′ (Au) − V ′ (Au

(1.3)

where the difference operators A and A˜ are defined by

˜ (s + 1), Au(s) = u(s + 1) − u(s) = Au

s ∈ R.

The study of systems of the type (1.1) goes back to Fermi et al. [1] who studied finite lattices with nearest-neighbour interaction by means of numerical methods. The use of variational methods is quite recent. One of the first results in that direction is due to Friesecke and Wattis [2]. They gave a global existence result for travelling waves for which u′ ∈ L2 in the case of a FPU lattice. Their approach was to minimize the ‘kinetic energy’ over the set of states with a given ‘potential energy’. They solved the variational problem, under some superquadratic growth condition on the quadratic part of V , using Lions’ concentration compactness principle. The speed of travelling waves was then given as some unknown Lagrange multiplier. Also, they showed the optimality of the growth condition satisfied by V by proving the non-existence of travelling waves for quadratic potentials. Smets and Willem [3] also proved the existence of travelling waves, with a prescribed speed instead. They used a completely different approach from the one in [2]. More precisely, using a variant of the mountain pass theorem, a travelling wave such that u ∈ L2 was obtained as a critical point of ‘the action functional’ defined on some Hilbert space on which the Palais–Smale compactness condition is not satisfied. Many authors have also studied the Frenkel–Kontorova model. For the sake of brevity, let us just mention the recent paper by Kreiner and Zimmer [4]. There, they proved the existence of non-constant periodic travelling waves with period bigger than some unknown constant. For this, they considered an interaction potential V of type (A.0) with α > 0 and a nonquadratic part W satisfying the global growth condition (A.3). Furthermore, they also proved the existence of homoclinic travelling waves with prescribed speed, but the non-quadratic part of the coupling has a special form, namely W (x) = a0 |x|β with a0 > 0 and β ≥ 3. Another class of lattice systems consists of chains of oscillators. In [5] travelling waves in chains of oscillators are studied using bifurcation theory. For an overview on the topic we refer to [6] and the references therein. In [7] chains of linearly coupled oscillators are considered. The existence of periodic travelling waves is then proven via the linking theorem under a global growth condition. However, the result does not say whether these periodic solutions are non-constants. In the present paper we shall prove two results (Theorems 1.1 and 1.2). Theorem 1.1 concerns the existence of nonconstant periodic travelling waves under growth conditions V and f . In Theorem 1.2 we prove the existence of travelling waves which are homoclinic to 0. Specifically, each of these travelling waves is obtained as a limit of periodic waves by letting the period go to infinity. This approach is borrowed from [8] where a homoclinic solution of (HS) is constructed as a limit of a sequence of subharmonics. In the process we also prove the existence of non-constants periodic travelling waves with high periods. Note that if V ′ (0) = 0 and x0 is a critical point of f , then the constant function s → x0 is a solution of (1.3). Note also that 1-periodic solutions of (1.3) satisfy the equation

(HS) u′′ + c −2 f ′ (u) = 0. It is known that under the assumptions of Theorem 1.1 this equation possesses at least a non-constant 1-periodic solution (see, e.g. [9]). As is well known in the Calculus of Variations, solutions of (1.3) can be obtained as critical points of the action functional

Φ ( u) =

∫ [ T

c2 2

] (u′ )2 − f (u) − V (Au) ds,

(1.4)

defined on some appropriate Hilbert space E, where T ⊆ R. When the Palais–Smale compactness condition is satisfied, the critical points of Φ can be detected with the aid of the mountain pass theorem, or some of its variants. It will be shown that under some conditions on f and V , Φ satisfies the PS condition when T is a segment. However, when T = R, this condition is never fulfilled. This is due to the invariance of the functional under the action R × E → E , (τ , u) → u(· + τ ). Organization of the paper The main results are stated in the end of the present section. Preliminary results are collected in the next section. The third section is devoted to the proof of the first result (Theorem 1.1), while the second result (Theorem 1.2) is proven in the last section. Main results Throughout this paper, we shall consider potentials V and f of the type (A.0) V (x) = α2 x2 + W (x), f (x) = − ω2 x2 + g (x), where α and ω are real constants.

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The non-quadratic part h ∈ {g , W } shall satisfy either (A.1) h(x) = o(x2 ) as x → 0, (A.2) h ≥ 0 and there are constants β > 2, r0 > 0 such that

β h(x) ≤ xh′ (x) for |x| ≥ r0 , or (A.3) there is a constant β > 0 such that 0 < β h(x) ≤ xh′ (x) for x ̸= 0. It is clear that (A.3) implies (A.1) and (A.2), respectively, and if h satisfies (A.2) then there are constants a0 , a1 > 0 such that the following holds true: (A.4) h(x) ≥ a0 |x|β − a1 for all x. Consider the set

C (α, ω) = {c > 0 : min(c 2 x2 − 4α sin2 (x/2) + ω) ≥ 0}. x∈R

Clearly if C (α, ω) is non-empty then ω ≥ 0. We define c0 : R × [0, ∞) → [0, ∞) by c0 (α, ω) := inf C (α, ω).

(1.5)

The main results are the following: Theorem 1.1. Let V and f be C 1 functions given by (A.0), with ω = 0, where g, and W satisfy (A.1), (A.2). Then for every T > 0, and c > c0 (α, 0), (1.3) possesses a non-constant T -periodic solution. Theorem 1.2. Let V and f be C 1 functions given by (A.0), with ω > 0, and the nonlinearities W and g satisfy (A.3). Then, for every c > c0 (α, ω), (1.3) possesses a non-trivial homoclinic solution emanating from the origin. We shall prove that these travelling waves decay exponentially (Lemma 4.6). Note that we require ω to be zero in Theorem 1.1. Indeed, the proof of the latter uses the linking theorem which only guarantees the existence of a non-trivial solution, which may still be constant. The assumption ω = 0 ensures that the solutions obtained are non-constants. One can also show that Theorem 1.2 remains true in the particular case of a linear coupling, i.e. V (x) = α2 x2 . 2. Preliminaries In this section we give some preliminaries that are needed in the sequel. Let us first fix some notations. Given T > 0, we denote by HT1 the space of T -periodic functions whose restriction to [0, T ] belongs to H 1 ([0, T ]). Similarly, we introduce the p notations LT and CTk for 1 ≤ p ≤ ∞ and k ∈ N ∪ {0}. 2 Lemma 2.1. The difference operator A : HT1 → L∞ T ∩ LT is bounded and for every u ∈ H1T we have

‖Au‖L2 ≤ ‖u′ ‖L2 ,

‖Au‖∞ ≤ l(T )‖u′ ‖L2 ,

T

T

(2.1)

T

with l(T ) =



[1/T ] + 1

if 0 < T < 1 if T ≥ 1,

1

(2.2)

where [η] denotes the integer part of η. Proof. We shall only establish the second estimate in the case when 0 < T < 1. For the other cases we refer to [6]. Let u ∈ HT1 and s ∈ R. By the Cauchy–Schwarz inequality we have:

|Au(s)| ≤

s+1

∫

′

s+1

[∫

(u )

′ 2

|u | ≤

]1/2

.

s

s

Setting n := [1/T ], we have nT ≤ 1 < (n + 1)T , and it follows that

|Au(s)| ≤

s+(n+1)T

[∫

( u′ ) 2

s

and the proof is complete.

]1/2

[ ]1/2 ∫ T √ = (n + 1) ( u′ ) 2 ≤ n + 1‖u′ ‖L2 , 0



T

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P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

Similarly one can prove that Lemma 2.2. The difference operator A : E → L∞ ∩ L2 is bounded and max(‖Au‖L2 , ‖Au‖∞ ) ≤ ‖u′ ‖L2

(∀u ∈ E ).

(2.3)

For the proof we refer to [3]. Let I be a compact interval. Then by the Sobolev embedding theorem, there is a positive constant C 1 such

‖u‖L∞ (I ) ≤ C ‖u‖H 1 (I ) (∀u ∈ H 1 (I )). Moreover, the embeddings H 1 (I ) ↩→ C (I ) and H 1 (I ) ↩→ L2 (I ) are compact. In the sequel, we shall denote by Cs the constant C given by the Sobolev embedding theorem. To end this section, let us recall the following well-known results: Proposition 2.1. Let h ∈ C 1 (R) and I a compact interval. Then, the functional h¯ I : H 1 (I ) → R defined by h¯ I (u) =

∫

h(u) I

is C 1 and its derivative is given by h¯ ′I (u)ξ = ⟨h′ (u), ξ ⟩L2 (I )

(∀u, ξ ∈ H 1 (I )).

Proposition 2.2. Let h ∈ C 1 (R) satisfies (A.1). Then, the functional h¯ ∞ : E = H 1 (R) → R defined by h¯ ∞ (u) =

∫

h(u) R

is C 1 and its derivative is given by h¯ ′∞ (u)ξ = ⟨h′ (u), ξ ⟩L2

(∀u, ξ ∈ E ).

3. Existence of periodic travelling waves In this section we are going to give a detailed proof of Theorem 1.1. The main tool to achieve this goal is a version of the mountain pass theorem, which we shall state below. Before doing so, we fix some terminology. Let X be a real Banach space and J ∈ C 1 (X , R). We say that a sequence {um } ⊂ X is a Palais–Smale for J if the sequence {J (um )} ⊂ R is bounded and J ′ (um ) → 0 as m → ∞. The functional J is said to satisfy the Palais–Smale (compactness) condition, for short the PS condition, if every Palais–Smale sequence is precompact. Theorem 3.1 (Rabinowitz [10]). Let X = X0 ⊕ Xˆ with dim X0 < ∞ and J ∈ C 1 (X ) satisfies the PS condition. Suppose in addition the following conditions are satisfied (J.3) J |X0 ≤ 0, (J.4) there are constants µ, ρ > 0 such that J > 0 in Xˆ ∩ (Bρ \ {0}) and J ≥ µ on Xˆ ∩ Sρ , (J.5) for each finite-dimensional subspace Y ⊂ X , there is an R = R(Y ) such that J ≤ 0 on Y \ BR . Then, Φ possesses a positive critical value b characterized by b = inf

max

h∈Γ u∈B¯ R(X1 ) ∩X1

J (h(u)),

where

Γ = {h ∈ C (B¯ R(X1 ) ∩ X1 , X )|h(u) = u if J (u) ≤ 0} and X1 = X0 ⊕ span{v}, for any non-zero v ∈ Xˆ . The notations Br , B¯ r and Sr stand for the open ball, the closed ball and the sphere centered at 0 with radius r, respectively. Proposition 3.1. Let V , f ∈ C 1 (R) satisfy the assumptions of Theorem 1.1. Then ΦT ∈ C 1 (HT1 , R), with

ΦT (u)ξ = ′

T

∫



c 2 u′ ξ ′ − f ′ (u)ξ − V ′ (Au)Aξ ,



0

for all u, ξ ∈ HT1 . Furthermore, any critical point of ΦT is a classical solution of (1.3). 1 It also holds true for I = R, but it is not needed here.

P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

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Proof. Write

ΦT (u) =

1 2

˜ T (u), BT (u, u) − GT (u) − W

with BT (u, v) =

T

∫



c 2 u′ v ′ + ωuv − α AuAv ,



0

GT (u) =

T

∫

˜ T (u) = g (u), W

T

∫

W (Au). 0

0

One can easily check that BT is a bounded (symmetric) bilinear form on HT1 , therefore the functional u → BT (u, u)

˜ T = WT ◦ A where WT is defined similarly to is C 1 (it is actually C ∞ ). Note that A is a bounded linear operator on HT1 and W ˜ T ∈ C 1 (HT1 , R). GT . Hence, by Proposition 2.1 we have GT , W If u ∈ HT1 is critical point of ΦT . Then, for every test function ξ we have T

∫

[c 2 u′ ξ ′ − f ′ (u)ξ − V ′ (Au)Aξ ]

0 = 0 T

∫

˜ ))ξ ] [c 2 u′ ξ ′ − f ′ (u)ξ + (V ′ (Au) − V ′ (Au

= 0

i.e. u is a weak solution of (1.3), and u is continuous because HT1 continuously embeds into CT0 . Since V ′ , f ′ ∈ C (R), thanks to (1.3), we have u′′ ∈ C 0 , and therefore u ∈ C 2 , i.e. it is a classical solution.  Remark 3.1. Using some Fourier analysis, one shows that for every α ∈ R, ω > 0, and c > c0 there are positive constants Λ0 and Λ1 (depending on α, ω and c) such that

Λ0 ‖u‖2H 1 ≤ BT (u, u) ≤ Λ1 ‖u‖2H 1 . T

(3.1)

T

In fact Λ0 and Λ1 are respectively the infimum and the supremum of the function c 2 x2 − 4α sin2 (x/2) + ω

x →

1 + x2

.

Proposition 3.2. Let f , V ∈ C 1 (R) be given by (A.0), where g and W satisfy (A.1) and (A.2). Then ΦT satisfies the PS condition. Proof. We first prove that PS sequences are bounded and next, that they are pre-compact. Boundedness: Given p > 0 we set Np (u) =

T

∫

c (u ) − α(Au) + pu



2

′ 2

2

2

  1/2

for u ∈ HT1 .

0

Then, thanks to (3.1), Np defines a norm on HT1 which is equivalent to the usual one. Let (um ) ⊂ HT1 be a PS sequence, i.e. for some constant M ≥ 0 we have

|ΦT (um )| ≤ M (∀m),

lim ΦT′ (um ) = 0.

and

m→∞

Then, for some positive integer m0 we have ‖ΦT′ (um )‖ ≤ 1 whenever m ≥ m0 . Fixing m ≥ m0 , we have

|ΦT′ (um )um | ≤ Np (um ), which implies T

∫

g ′ (um )um + W ′ (Aum )Aum = −ΦT (um )um + Np2 (um ) − p‖um ‖2L2

 0



T

≤ Np (um ) +

Np2

(um ).

Set I1 = {s ∈ [0, T ] : |um (s)| ≤ r0 },

I2 = {s ∈ [0, T ] : |Aum (s)| ≤ r0 },

(3.2)

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P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

and ¯Ii = [0, T ] \ Ii for i = 1, 2. Then,

∫

∫

g ( um ) +

[

W (Aum ) ≤ K0 := T

|x|≤r0

I2

I1

]

max g (x) + max W (x) , |x|≤r0

and thanks to (A.2) and (3.2), we get T

∫

[g (um ) + W (Aum )] =

∫

0

g (um ) +

∫

W (Aum ) +

∫ ¯I1

I2

I1

≤ K0 + β

−1

[∫ ¯I1

≤ K0 + β − 1

g (um )um + ′

∫ ¯I2

g (um ) +

∫ ¯I2

W (Aum )

W (Aum )Aum ′

]

T

∫

g ′ (um )um + W ′ (Aum )Aum





0

≤ K0 + β −1 (Np (um ) + Np2 (um )), i.e. T

∫

[g (um ) + W (Aum )] ≤ K0 + β −1 (Np (um ) + Np2 (um )).

(3.3)

0

Also, thanks to (A.2), there is a constant r ≥ r0 such that the condition |x| ≥ r implies

(∗) x2 ≤ xg ′ (x). Setting I = {s ∈ [0, T ] : |um (s)| ≤ r },

¯I = [0, T ] \ I ,

we then deduce from (∗) and (3.2) that

‖um ‖2L2 =

∫

T

u2m + I

∫ ¯I

u2m ≤ r 2 T +

≤ r 2T +

∫ ¯I

g ′ (um )um T

∫

g ′ (um )um , 0

that is,

‖um ‖2L2 ≤ r 2 T + Np (um ) + Np2 (um ).

(3.4)

T

Combining (3.2)–(3.4), it results K1 := K0 +

p 2 p

r 2T + M

≥ K0 + r 2 T + ΦT (um ) 2

=

1 2

Np2

p

(um ) − ‖um ‖2L2 − 2

p

2

≥ K0 + r T + 2

T

1 2

Np2

T

∫

[g (um ) + W (Aum )] + K0 + 0

p 2

r 2T

s

(um ) − (r 2 T + Np (um ) + Np2 (um )) − K0 − β −1 (Np (um ) + Np2 (um )), 2

i.e.

(1/2 − 1/β − p/2)Np2 (um ) − (1/β + p/2)Np (um ) ≤ K1 . Choosing p such that p < 1 − 2/β , we deduce from the above inequality that (um ) is bounded in (HT1 , Np ) and therefore in (HT1 , ‖ · ‖H 1 ). T

Precompactness: The boundedness of (um ) in HT1 allows us to extract a weakly convergent subsequent, which for simplicity we still denote by (um ). Let u ∈ HT1 be its (weak) limit. Then um converges to u strongly in CT0 as well as in L2T . Note that ΦT′ (u)u = 0. Indeed,

    ΦT′ (u)u = ΦT′ (um )u + ΦT′ (u)um − ΦT′ (um )u + ΦT′ (u)u − ΦT′ (u)um .    Σm

P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

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The first term goes to zero because (um ) is a PS sequence and the last one goes to zero too because um ⇀ u weakly in HT1 . The second term can be written as

Σm =

T

∫



g ′ (um )u − g ′ (u)um +



0



T

W ′ (Aum )Au − W ′ (Au)Aum ,





0



1 Σm

∫





2 Σm

and

∫ T |g ′ (u)u − g ′ (u)um | |g ′ (um )u − g ′ (u)u| + 0 0  √  + ‖ g ′ (u)‖L2 ‖u − um ‖L∞ . ≤ T ‖u‖L2 ‖g ′ (u) − g ′ (um )‖L∞ T T

|Σm1 | ≤

T

∫

T

T

1 Since um → u in and g is uniformly continuous on compact sets, it follows from the above estimate that Σm → 0. 2 Replacing g , u, and um by W , Au and Aum , respectively, one shows that Σm → 0. Hence Σm → 0 and we conclude that ΦT′ (u)u = 0. Thanks to the continuity of V ′ and f ′ we get

CT0

T

∫

f ′ (um )um + V ′ (Aum )Aum =



lim m

′



T

∫

0



f ′ (u)u + V ′ (Au)Au



0

= −ΦT′ (u)u + c 2 ‖u′ ‖2L2 T

= c 2 ‖u′ ‖2L2 . T

On one hand, because (um ) is a bounded PS sequence, we have ΦT′ (um )um → 0 as m → ∞. Consequently,

[

lim ‖u′m ‖2L2 = c −2 lim ΦT′ (um )um + m

m

T

∫

T

(f ′ (um )um + V ′ (Aum )Aum )

]

0

= ‖u ‖L2 . ′ 2

(3.5)

T

On the other hand the boundedness of (um ) implies the one of (u′m ) in L2T . It follows from (3.5) that u′m → u′ in L2T . Hence um → u in HT1 .  Set E0 = {u ∈ HT1 |∃ζ ∈ R s.t. u(s) = ζ for a.e. s}. Then E0 can be identified with R and HT1 = E0 ⊕ E0⊥ , where ⊥



E0 =

u∈

HT1

T

∫ |

 u=0

0

denotes the orthogonal complement of E0 in HT1 . We have Lemma 3.1. Under the assumptions of Theorem 1.1, ΦT satisfies the conditions (J.3)–(J.5) of Theorem 3.1, with X = HT1 , X0 = E0 , and Xˆ = E0⊥ . Proof. Condition (J.3) follows from the facts that g ≥ 0 and V (0) = 0. Condition (J.4): Let ϵ be such that 0 < ϵ < Λ0 , where Λ0 is defined in Remark 3.1. By (A.1), there is a δ > 0 such that max(f (x), W (x)) ≤

ϵ 2

x2

whenever |x| ≤ δ.

Set

ρ=

δ max(l(T ), Cs )

where l(T ) is given by (2.2), and Cs by the Sobolev embedding HT1 ↩→ CT0 . Choose a u ∈ E0⊥ such that 0 < ‖u‖H 1 ≤ ρ. T

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P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

Then, by the Sobolev embedding, and (2.1), we have max(‖Au‖L∞ , ‖u‖L∞ ) ≤ δ. T

T

Thanks to (3.1), we have

ΦT (u) ≥

Λ0 2

‖ u‖

2 HT1

ϵ

−

T

∫

2



u2 + (Au)2 ≥



Λ0 − ϵ 2

0

‖u‖2H 1 > 0. T

In particular

ΦT (u) ≥ (Λ0 − ϵ)ρ 2 /2 > 0 for ‖u‖H 1 = ρ. T

Condition (J.5): Let Y be a finite-dimensional subspace of HT1 . Then, any two norms on Y are equivalent. Therefore, there is a positive constant µ depending only on Y such that

‖u‖Lβ ≥ µ‖u‖H 1 ∀u ∈ Y . T

T

If S (Y ) denotes the unit sphere of HT1 (with respect to the Sobolev norm), then we have

  β β inf ‖u‖ β + ‖Au‖ β ≥ µβ .

u∈S (Y )

LT

LT

Given a non-zero u ∈ HT1 , we set u = r u˜ ,

r = ‖u‖H 1 , T

so that u˜ ∈ S (Y ), and

ΦT (u) =

r2 2

BT (˜u, u˜ ) −

≤

Λ1

≤

Λ1

2 2

T

∫

g (r u˜ ) + W (rAu˜ )





0

r2 −

T

∫

a0 r β (|˜u|β + |Au˜ |β ) − 2a1





0

r − a0 µβ r β + 2a1 T . 2

Since β > 2, there is an R > 0 depending on µ, and therefore on Y , such that if u ∈ Y , with ‖u‖H 1 > R, then ΦT (u) ≤ 0. Hence ΦT satisfies (J.5).

T



Thanks to Theorem 3.1, ΦT possesses a critical point u in HT1 which is, of course, a T -periodic solution of (1.3). If u were constant, the corresponding critical value, ΦT (u), would be non-positive, contrary to Theorem 3.1. Therefore u is necessarily non-constant. Remark 3.2. Theorem 1.1 still holds true when f ≡ 0. Note that we only require a growth condition on W at infinity. However, to carry out the construction of a travelling wave solution u for which u′ ∈ L2 , as it is done in [6], one definitely needs a global growth condition on W . 4. Existence of homoclinic travelling waves In this section we are going to construct homoclinic travelling waves as limits of periodic ones. We need the standard version of the mountain pass theorem in order to prove the existence of periodic solutions for (1.3) under the assumptions of Theorem 1.2. We have the following Lemma 4.1. Under the assumptions of Theorem 1.2, (1.3) possesses non-trivial periodic solutions of any given period T > 0 and any given speed c > c0 . This shall follow from the following Theorem 4.1 (Ambrosetti–Rabinowitz [11]). Let J ∈ C 1 (X , R) satisfy the PS condition, and J (0) = 0. Suppose the following conditions are satisfied (J.1) there are constants ω, ρ > 0 such that J |Sρ ≥ ω, (J.2) there is an e ∈ X \ B¯ ρ such that J (e) ≤ 0.

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Then, J possesses a critical value b ≥ ω characterized by b = inf max J (γ (t )),

(4.1)

Γ = {γ ∈ C ([0, 1], E )|γ (0) = 0 and γ (1) = e}.

(4.2)

γ ∈Γ 0≤t ≤1

where

Proof. We only have to check that ΦT satisfies the PS condition, and the conditions (J.1) and (J.2) of Theorem 4.1. The PS condition. We will only prove the boundedness of PS sequences. The precompactness can be dealt with following the same line of arguments as in the proof of Proposition 2.1. Let (um ) ⊂ HT1 be a PS sequence, i.e. for some constant M ≥ 0 we have |ΦT (um )| ≤ M for all m and Φ ′ (um ) → 0 as m → ∞. Then, there is an integer m0 ≥ 1 such that ‖ΦT′ (um )‖ ≤ 1 for all m ≥ m0 . Fixing m ≥ m0 , thanks to (A.3) and (3.1) we have M+

1

β

1

‖um ‖H 1 ≥ Φ (um ) − Φ ′ (um )um T β   ] ∫ T[ ] ∫ T[ 1 1 1 ′ 1 ′ − BT (um , um ) + g (um )um − g (um ) + W (Aum )Aum − W (Aum ) = 2 β β β 0 0   1 1 − ‖um ‖2H 1 . ≥ Λ0 T 2 β

Thus

Λ0 (β − 2)‖um ‖2H 1 − 2‖um ‖H 1 ≤ 2β M . T

T

As β > 2, the latter shows that (um ) is bounded in HT1 . Condition (J.1). Thanks to (3.1), we have

˜ T (u) = ΦT (u) + GT (u) + W ≥

1 2 1 2

BT (u, u)

Λ0 ‖u‖2H 1 . T

Now we only have to show that

˜ T (u) = o(‖u‖2 1 ). GT (u) + W H T

Thanks to (A.1), given ϵ > 0, there is a δ > 0 such that, if |x| ≤ δ , then max(g (x), W (x)) ≤ ϵ x2 /2. Set

ρ=

δ , max(l(T ), Cs )

where the constant Cs is given by the Sobolev embedding, and l(T ) by (2.2). Let u be a member of HT1 with

‖u‖H 1 = ρ. T

Then for almost every s we have

|u(s)| ≤ ‖u‖L∞ ≤ Cs ‖u‖H 1 ≤ δ. T

Also, thanks to (2.1), we have, for almost every s

|Au(s)| ≤ ‖Au‖L∞ ≤ l(T )‖u′ ‖L2 ≤ δ. T T

Thus,

˜ T (u) ≤ GT (u) + W

ϵ 2

∫

T

(u2 + (Au)2 ) ≤

0

Since ϵ > 0 is arbitrary, this shows that

˜ T (u) = o(‖u‖2 1 ). GT (u) + W H T

ϵ 2

‖u‖2H 1 . T

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P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

In particular, if we choose ϵ such that 0 < ϵ < Λ0 , then, thanks to (3.1), we have

ΦT (u) ≥ (Λ0 − ϵ)ρ 2 /2 > 0. Condition (J.2). Thanks to (A.4), there are constants a0 , a1 > 0 such that min(g (x), W (x)) ≥ a0 |x|β − a1 Let u be a non-zero element of

ΦT (ru) ≤ 2a1 T +

r2 2

HT1

for all x.

and r > 0. Then we have

T

∫

c (u ) + ωu − α(Au)



2

′ 2

2

2



− a0 r

β

T

∫

(|u|β + |Au|β ).

0

0

But β > 2, therefore ΦT (ru) → −∞ as r → ∞ and there is an ru > 0 such that ΦT (ru) ≤ 0 for r ≥ ru .



Remark 4.1. In order to construct a homoclinic solution of (1.3) we need non-constant periodic solutions. At this point we only have non-trivial periodic solutions. However, we will see that for a suitable choice of e (see condition (J.2) of Theorem 4.1) the periodic solutions whose existence is guaranteed by Lemma 4.1 are actually non-constant. We denote by Φ∞ the functional Φ defined on E = H 1 (R) by (1.4), i.e. with T = R. It is worth mentioning that members u of E satisfy u(s) → 0 as |s| → ∞. We are going to construct a family U [0] such that each of its members is a periodic solution of (1.3). Next, we shall prove the existence of a convergent sequence (¯uk )k∈N in U [0] whose limit u¯ belongs to E, and is a critical point of Φ∞ . Proposition 4.1. Under the assumptions of Theorem 1.2, Φ∞ ∈ C 1 (E , R) and its derivative is given by ′ Φ∞ (u)ξ =

∫



c 2 u′ ξ ′ − f ′ (u)ξ − V ′ (Au)Aξ .



R

Proof. Write 1

Φ∞ (u) =

2

˜ ∞ ( u) B∞ (u, u) − G∞ (u) − W

with B∞ (u, v) =

∫



c 2 u′ v ′ + ωuv − α(Au)(Av)



R

G∞ (u) =

∫

˜ ∞ (u) = W

g (u)

∫R

W (Au).

R

˜ ∞ = W∞ ◦ A, B∞ is a bounded (symmetric) bilinear form therefore u → B∞ (u, u) is C ∞ . Since A : E → E is bounded and W 1 ˜ where W∞ is defined similarly to G∞ , thanks to Proposition 2.2 we have G∞ , W∞ ∈ C (E , R).  Lemma 4.2. Under the assumptions of Theorem 1.2 the critical points of Φ∞ are classical solutions of (1.3). Moreover, any critical point u of Φ∞ satisfies u(s) → 0 as |s| → ∞, i.e. it is a homoclinic solution of (1.3) emanating from the origin. Proof. The first part can be dealt with like in [3]. Therefore we will only focus on the second statement. We shall prove that if u ∈ E is a critical point of Φ∞ , then u′′ ∈ L2 , which, obviously implies that u′ ∈ E, and therefore u′ (s) → 0 as |s| → ∞. Since g ′ (x) = o(x) and W ′ (x) = o(x) as x → 0, there is a δ > 0 such that max(|g ′ (x)|, |W ′ (x)|) ≤ |x|

for |x| ≤ δ.

Because u ∈ E, we have u(s) → 0 as |s| → ∞, and therefore, there is an r1 > 0 such that

|u(s)| ≤ δ/2 for all |s| ≥ r1 . It then follows that

|Au(s)| ≤ δ for s ∈ (−∞, −r1 − 1] ∪ [r1 , ∞),

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and

˜ (s)| ≤ δ for s ∈ (−∞, −r1 ] ∪ [r1 + 1, ∞). |Au Thanks to (1.3), we have, for |s| ≥ r1 + 1

  ˜ |2 ) + |g ′ (u)|2 + |W ′ (Au)|2 + |W ′ (Au ˜ )|2 (u′′ )2 ≤ 6c −2 ω2 u2 + α 2 (|Au|2 + |Au   ˜ |2 ) + (u2 + |Au|2 + |Au ˜ |2 ) ≤ 6c −2 ω2 u2 + α 2 (|Au|2 + |Au ˜ |2 ). ≤ 6c −2 (1 + max(α 2 , ω2 ))(u2 + |Au|2 + |Au Thus,

∫

(u′′ )2 ≤ c −2 R

≤c

−2

≤ c −2 ≤ c −2

r1 +1

∫

−r 1 −1 ∫ r1 +1 −r 1 −1 ∫ r1 +1 −r 1 −1 ∫ r1 +1

(u′′ )2 + 6c −2 (1 + max(α 2 , ω2 ))

∫

˜ |2 ) (u2 + |Au|2 + |Au |s|≥r1 +1

(u ) + 6c ′′ 2

−2

(1 + max(α , ω )) 2

2

∫

˜ |2 ) (u2 + |Au|2 + |Au R

(u′′ )2 + 6c −2 (1 + max(α 2 , ω2 ))(‖u‖2L2 + 2‖u′ ‖2L2 ) (u′′ )2 + 12c −2 (1 + max(α 2 , ω2 ))‖u‖2E .

−r 1 −1

This shows that u′′ ∈ L2 , and the proof is complete.



4.1. Construction of the family U [0] For convenience, we shall now define ΦT as an integral over [−T /2, T /2]. Let e0 be a non-zero member of C 1 ([−1, 1]) whose support lies inside (−1, 0). Denote by e2 the 2-periodic extension of e0 on R. Given T > 2, we denote by e the continuous extension of e0 on [−T /2, T /2], i.e. e agrees with e0 on [−1.1] and vanishes elsewhere. Let us now extend e on the whole R in a T -periodic fashion and denote that extension by eT , i.e. eT is T -periodic and agrees with e on [−T /2, T /2]. Obviously, for every T ≥ 2 we have

‖eT ‖ET = ‖e0 ‖H 1 (−1,0) , and for s ∈ [−T , 0] AeT (s) =

Ae2 (s) 0

if s ∈ [−2, 0] otherwise.



Therefore, for any r > 0, we have

∫

T /2

−T / 2

V (rAeT ) =

0

∫

V (rAeT ) = −T

∫

0

−2

V (rAe2 ) =

∫

1

V (rAe2 ). −1

The first equality is due to the fact that eT is T -periodic, while the last one follows from the fact that e2 is 2-periodic. Since f (0) = 0, it follows that

ΦT (reT ) = Φ2 (re2 ). e2 being a non-zero element of H21 , for r > 0 sufficiently large we have Φ2 (re2 ) ≤ 0. Now, set e¯ T = reT , where r > 0 is large enough so that Φ2 (re2 ) ≤ 0. Denote by bT the critical value of ΦT given by (4.1), with

Γ = ΓT := {γ ∈ C ([0, 1], HT1 )|γ (0) = 0 and γ (1) = e¯ T }. Let uT be the corresponding critical point. Lemma 4.3. Under the assumptions of Theorem 1.2, there are positive constants b0 , δ0 and M0 independent of T such that bT ≤ b0 and δ0 ≤ ‖uT ‖∞ ≤ M0 for all T ≥ 2. Furthermore, there is a T0 > 0 such that for any T ≥ T0 , uT is a non-constant T -periodic solution of (1.3).

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P.D. Makita / Nonlinear Analysis 74 (2011) 2071–2086

Proof. Uniform upper bound for bT : Let γT ∈ ΓT be given by γT (t ) = t e¯ T . Then

ΦT (γT (t )) = Φ2 (γ2 (t )), and we deduce that bT ≤ max Φ2 (γ2 (t )) =: b0 .

(4.3)

0≤t ≤1

Uniform upper bound for ‖uT ‖ET : Note that bT = ΦT (uT ) −

1

β

ΦT′ (uT )uT ≥



1 2

−

1



β

Λ0 ‖uT ‖2ET ,

where the inequality follows from (A.3). Thus, by (4.3) we have

1/2

2β b0

 ‖uT ‖ET ≤

(β − 2)Λ0

.

(4.4)

Thanks to the Sobolev embedding and (4.4) we get

√ ‖ uT ‖ ∞ ≤

β b0 ≤ M0 := 2 (β − 2)Λ0 

2‖uT ‖ET

1/2

.

(4.5)

Let Y1 , Y2 : [0, ∞) → [0, ∞) be defined by Y1 (r ) = 0 = Y2 (r ) if s = 0 and for r > 0 Y1 (r ) = max x−1 g ′ (x),

Y2 (r ) = max x−1 W ′ (x). 0<|x|≤r

0<|x|≤r

Clearly the two maps Y1 and Y2 are continuous, non-decreasing, and non-negative, and so is the map r → Y (r ) = Y1 (r ) + Y2 (2r ). Because |uT (s)| ≤ ‖uT ‖∞ , it readily follows from the properties of Y1 that g ′ (uT (s)) uT (s)

≤ Y1 (‖uT ‖∞ ),

for every s for which the left-hand side is well-defined. Similarly, we have W ′ (AuT (s)) AuT (s)

≤ Y2 (2‖uT ‖∞ ),

whenever the left-hand side is well-defined. Using the fact that uT is a critical point for ΦT , we get

Λ0 ‖uT ‖2H 1 ≤ BT (uT , uT ) T ∫ T /2  ′  = g (uT )uT + W ′ (AuT )AuT

∫

−T / 2 T /2

≤ −T / 2



Y1 (‖uT ‖L∞ )u2T + Y2 (2‖uT ‖L∞ )(AuT )2 T



T

≤ Y1 (‖uT ‖L∞ )‖uT ‖2L2 + Y2 (2‖uT ‖L∞ )‖u′T ‖2L2 T T T

T

≤ Y (‖uT ‖L∞ )‖uT ‖2H 1 . T T

Since uT is non-trivial, we have Y (‖uT ‖∞ ) ≥ Λ0 . Hence, thanks to the aforementioned properties of Y , it results that

≥ δ0 , ‖uT ‖L∞ T

(4.6)

for some δ0 > 0 which is independent of T . Existence of T0 . Suppose all the uT ’s are constants. Then we have

δ0 ≤ ‖uT ‖L∞ = |uT (0)| = T

‖ uT ‖ H 1 M0 √ T ≤ √ , T

2T

for all T ≥ 2, contradicting the fact that δ0 ̸= 0.



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We set U [0] = {uT }T ≥T0 ,

U [2] = {u′′T }T ≥T0 .

U [1] = {u′T }T ≥T0 ,

Lemma 4.4. For each i = 0, 1, 2, the family U [i] is uniformly bounded in the space Cb (R) equipped with the sup-norm. Furthermore the families U [0] and U [1] are equicontinuous. Proof. The boundedness of U [0] readily follows from (4.5). Boundedness of U [2] . Let T ≥ T0 . Since V ′ and f ′ are continuous, and the image of uT is contained in [−M0 , M0 ], it follows from (1.3) that, for all s,

  ˜ T (s))| |u′′T (s)| ≤ c −2 max |f ′ (uT (s))| + |V ′ (AuT (s))| + |V ′ (Au s∈R [ ] ≤ c −2 max |f ′ (x)| + 2 max |V ′ (x)| |x|≤M0

|x|≤2M0

= M2 . Thus

‖u′′T ‖∞ ≤ M2 (∀T ≥ 2). Boundedness of U [1] . Let T ≥ T0 . By the mean value theorem, we have uT (sT ) = ′

∫

s

u′T = uT (s) − uT (s − 1) s−1

for some sT ∈ [s − 1, s]. It follows that

 ∫ s ∫ s    |u′′T | |u′T (s)| = u′T (sT ) + u′′T  ≤ |uT (s) − uT (s − 1)| + sT sT ∫ s ≤ 2M0 + M2 s−1

= 2M0 + M2 . Thus

‖u′T ‖∞ ≤ M1 := 2M0 + M2 . Equicontinuity: Given ui ∈ U [i] , i = 0, 1, and s, s′ ∈ R, we have

∫ ′   s    u′  ≤ Mi |s − s′ | (∀i = 0, 1).  |ui (s) − ui (s )| =   s  ′

4.2. Existence of homoclinics Lemma 4.5. There is a sequence (¯uk ) ⊂ U [0] which converges to a non-trivial critical point u¯ ∈ E of Φ∞ . Proof. In view of Lemma 4.4 and thanks to Arzelà–Ascoli’s Theorem, there exists a sequence (¯uk ), with u¯ k = uTk , and 1 Tk → ∞ as k → ∞, such that u¯ k → u¯ in Cloc (R). Since each member of U [0] satisfies (1.3), the convergence is actually in 2 Cloc (R). By (4.5), one infers hat

∫

u¯ 2 + (¯u′ )2 ≤ M02 /2,





R

i.e. u¯ ∈ E. Let ξ be a test function on R. Denote by I0 and I1 the supports of ξ (·) and ξ (· + 1), respectively. Let k ∈ N be sufficiently large, so that I := I0 ∪ I1 ⊂ (−Tk /2, Tk /2). Then, we have ′ ′ Φ∞ (¯u)ξ = Φ∞ (¯u)ξ − ΦT′ k (¯uk )ξ

˜ I′ (¯u)ξ − W ˜ I′ (¯uk )ξ ), = (BI (¯u − u¯ k , ξ )) + (G′I (¯u)ξ − G′I (¯uk )ξ ) + (W

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with BI (v, ξ ) =

∫

c 2 v ′ ξ ′ + ωvξ − α Av Aξ ,





I

GI (v)ξ = ′

∫

˜ I′ (v)ξ = W

g ′ (v)ξ ,

∫I

W ′ (Av)ξ .

I

Note that

∫     |BI (¯u − u¯ k , ξ )| =  c 2 (¯u′ − u¯ ′k )ξ ′ − ω(¯u − u¯ k )ξ + α A(¯u − u¯ k )Aξ  I   ≤ |I |1/2 ‖ξ ‖E c 2 ‖¯u′ − u¯ ′k ‖L∞ (I ) + (ω + 2|α|)‖¯u − u¯ k ‖L∞ (I ) , where |I | is the length of I. Since (¯uk , u¯ ′k ) → (¯u, u¯ ′ ) uniformly on compact subsets of R, it follows from the above estimate that BI (¯u − u¯ k , ξ ) → 0 as k → ∞. Similarly, we have

|G′I (¯u)ξ − G′I (¯uk )ξ | ≤ |I |1/2 ‖ξ ‖E ‖g ′ (¯u) − g ′ (¯uk )‖L∞ (I ) ˜ I′ (¯u)ξ − W ˜ I′ (¯uk )ξ | ≤ |I |1/2 ‖ξ ‖E ‖W ′ (Au¯ ) − W ′ (¯uk )‖L∞ (I ) . |W It then follows from the continuity of g ′ , and W ′ , and the uniform convergence of (¯uk , u¯ ′k ) on compact subsets of R that lim |G′I (¯u)ξ − G′I (¯uk )ξ | = 0,

k→∞

˜ I′ (¯u)ξ − W ˜ I′ (¯uk )ξ | = 0. lim |W

k→∞

Thus ′ ′ |Φ∞ (¯u)ξ | = lim |Φ∞ (¯u)ξ − ΦT′ k (¯uk )ξ | = 0. k→∞

′ Since ξ is arbitrary, we conclude that Φ∞ (¯u) = 0, i.e. u¯ is a critical point of Φ∞ . To prove that u¯ ̸≡ 0, take the limit in (4.6), with uT replaced by u¯ k . We get

sup |¯u(s)| ≥ δ0 > 0, s∈R

which shows that u¯ ̸≡ 0.



4.3. Exponential decay As mentioned at the beginning, we will now prove that these homoclinics decay exponentially. More precisely we have the following Lemma 4.6. Under the assumptions of Theorem 1.2, there exists a positive constant λ0 = λ0 (α, ω, c ) such that, if u is a solution of (1.3) given by Lemma 4.5 then for every λ ∈ (0, λ0 ) there exists a positive constant Kλ such that

|u(s)| ≤ Kλ exp(−λ|s|) ∀s ∈ R. Let us denote by L the operator defined by Lu = −c 2 u′′ + α(A − A˜ )u + ωu and by G its Green’s function. Using Fourier transforms one easily shows that G(s, τ ) =

1 2π

ei(s−τ )x

∫ R

σ (x )

dx,

where the symbol σ of L is given by

σ (x) = c 2 x2 − 4α sin2 (x/2) + ω,

x ∈ R.

Clearly σ possesses no real zero for c > c0 . Denote by Z C (σ ) the set of complex numbers z = x + iy such that σ (z ) = 0. One can show that all these zeros are simple. For z0 ∈ C and δ > 0 set B(z0 , δ) = {z ∈ C : |z0 − z | < δ}. Using the Argument Principle, and the special form of σ one shows that for N ∈ N large enough #Z C (σ ) ∩ B(0, N ) ≤ 2N, where #S

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stands for the number of elements of the set S. In the same way, one shows that for every δ > 0 and for x > 0 large enough, #Z C (σ ) ∩ B(x, δ) = 0. Thus, for every δ > 0, the infinite strip |y| < δ contains at most finitely many elements of Z C (σ ). Hence, it makes sense to define

λ0 = λ0 (α, ω, c ) = min{|y| : z ∈ Z C (σL )} > 0.

(4.7)

We have the following Proposition 4.2. There exists a positive constant K = K (α, ω, c ) such that

|G(s, τ )| ≤ K exp(−λ0 |s − τ |) (∀s, τ ∈ R). Proof. It is possible to choose δ > λ0 and R > 0 in such a way that the triangular region ∆(R, δ) bounded by the segments [−R, R], [R, iδ] and [iδ, −R] contains no elements of Z C (σ ) but those with imaginary part y = λ0 . Set ZλC0 (σ ) = Z C (σ )∩{z ∈

C : y = λ0 }. Note that #ZλC0 (σ ) < ∞. Since all the (complex) zeros of σ are simple, applying the Residue Theorem to the function C \ Z C (σ ) ∋ z → exp(isz )/σ (z ) on ∂ ∆(R, δ), one shows that G(s, 0) = ie−λ0 s

eisx

− z ∈ZλC (σ )

∀s ≥ 0.

σ ′ (z )

0

In a similar way, one shows that G(s, 0) = ieλ0 s

eisx

− z ∈ZλC (σ )

∀s ≤ 0.

σ ′ (¯z )

0

Thus

|G(s, 0)| ≤ K (α, ω, c ) exp(−λ0 |s|) ∀s ∈ R, where K (α, ω, c ) =

1

− z ∈ZλC (σ )

|σ ′ (z )|

. 

0

Let u be a homoclinic solution of (1.3) given by Lemma 4.5. Setting

 ˜ (s)) − W ′ (Au(s))|  |g ′ (u(s)) + W ′ (Au ν(s) = K (α, ω, c ) |u(s)|  0

if u(s) ̸= 0 if u(s) = 0,

it follows from Proposition 4.2 that

∫    ′ ′ ˜  |u(s)| =  G(s, τ )[g (u(τ )) + W (Au(τ )) − W (Au(τ ))]dτ  ∫R ≤ e−λ0 |s−τ | ν(τ )|u(τ )|dτ . R

Now Lemma 4.6 follows from Lemma 4.8 in [6], which, for the sake of completeness we state here. Lemma 4.7. Let η, ν : R → R be non-negative functions such that ν(s) → 0 as |s| → ∞ and

η(s) ≤

∫

e−µ0 |s−τ | η(τ )ν(τ )dτ

∀s ∈ R,

R

for some µ0 > 0. Then for every µ ∈ (0, µ0 ) there is a constant Cµ such that η(s) ≤ Cµ e−µ|s| for all s ∈ R. References [1] [2] [3] [4]

E. Fermi, J. Pasta, S. Ulam, Studies in nonlinear problems, Rept. LA-1940, Los Alamos, Reprinted in: E. Fermi, Collected Papers, vol. II, 1955, pp. 978–988. G. Friesecke, J. Wattis, Existence theorem for travelling waves on lattices, Comm. Math. Phys. 161 (2) (1994) 391–418. D. Smets, M. Willem, Travelling waves with prescribed speed on infinite lattices, J. Funct. Anal. 149 (1) (1997) 266–275. C.-F. Kreiner, J. Zimmer, Travelling waves solutions for the discrete sine-Gordon equation with nonlinear pair-interaction, Nonlinear Anal. 70 (2009) 3146–3158.

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