Quasi periodic breathers in Hamiltonian lattices with symmetries

Summary. We prove existence of quasiperiodic breathers in Hamiltonian lattices of weakly coupled oscillators having some integrals of motion independent of the Hamiltonian. The proof is obtained by constructing quasiperiodic breathers in the anticontinuoum limit and using a recent theorem by N.N. Nekhoroshev [8] as extended in [5] to continue them to the coupled case. Applications to several models are given.


Introduction.
Existence of breathers (i.e. time periodic space localized solutions) in infinite Hamiltonian lattices of weakly coupled oscillators has been proved by Mac Kay and Aubry [6] (see also [3,15]) starting from the anticontinuoum limit.
Existence of (time) quasiperiodic breathers has also been investigated, but such objects are expected not to exist in generic models due to the presence of linear combinations of the frequencies which fall in the continuous spectrum. As pointed out in [11,12] such "resonances" are expected to lead to a decay in time of "quasiperiodic breather like solution".
Nevertheless existence of quasiperiodic breathers has been proved in some special models: in particular (1) quasiperiodic breathers with two frequencies have been constructed in two models which have a non trivial symmetry, namely the discrete nonlinear Schrödinger equation [10] and the adiabatic Holstein model [1]; (2) KAM theory has been used to construct breathers with an arbitrary number of independent frequencies in the case where the oscillators are coupled via a first neighborhood potential which is at least cubic [17].
In the present paper we start from the anticontinuoum limit and construct quasi periodic breathers in Hamiltonian systems having some integrals of motion independent of the Hamiltonian. Our approach is based on a theorem by N.N. Nekhoroshev [8] that was recently reformulated in a form suitable for our applications in [5]. This theorem is an extension of the Poincaré-Lyapunov theorem of continuation of periodic orbits to the case where some integrals of motion independent of the Hamiltonian exist. It allows to continue families of invariant tori to perturbations of the original system, and to describe precisely the dynamics on them. Applying this theorem one can reproduce the known results on the DNLS and on the Holstein model, and more generally obtain quasiperiodic breathers in any model with symmetries. Here we will explicitly reconstruct the quasiperiodic breathers of [10] for the DNLS, and we will construct a new type of quasiperiodic breathers in the Holstein model. As an example possessing 3frequencies quasiperiodic breathers we also study the vector discrete nonlinear Schrödinger equation. We think that similar results could be obtained also by the approach of [7,16].
We point out that the present approach allows also to give more details on the breather's dynamics. In particular breathers will appear as invariant tori, which form smooth families with as many parameters and as many dimensions as the number of independent integrals of motion of the system. In many cases the frequencies can be used to parametrize the family (in particular this is true for the models studied here), so, in particular the set of the tori on which the motion is dense has full measure.
We close this section by mentioning the very recent works [2,9] in which the authors prove existence of breathers in FPU type systems by new approaches. We did not investigate the possibility of finding quasiperiodic breathers in systems with symmetry using such approaches. Acknowledgement. We thank Andrea Carati for suggesting the coordinate transformation 3.15, 3.16.

The abstract theorem 2.1 Statement
We state here Nekhoroshev's theorem [8] in the improved form of [5]. For the proof see [5].
Let (B, Ω) be a (weakly) symplectic Banach space (with symplectic form Ω). Let H := {H 1 , ..., H s } be s real functions on B, defined for in a neighbourhood E of zero.
Consider the Hamiltonian vector field X i of H i (defined as the unique vector field such that dH i X = Ω(X i , X) ∀X ∈ B).
We assume that there exists an s dimensional compact submanifold Λ ⊂ B, invariant under the flows of the fields X 0 i ≡ X i =0 for all i = 1, ..., s. We also assume that i the functions H i and the vector fields X i are of class C ∞ in U × E, where U is a neighborhood of Λ, ii the vector fields X i are independent on Λ iii the functions H are in involution in U, namely Remark that by our assumptions the manifold Λ is diffeomorphic to an s dimensional torus.
Then we have to add the nonresonance assumption which extends that by the Poincaré Lyapunov theorem to the quasiperiodic case. To this end it is useful to restrict to the case where there exists a system of canonical coordinates (J, ψ, p, q) with ψ ∈ T s and (J, p, q) in an open set of a suitable Banach space, in which the manifold Λ has equation p = q = J = 0, and the functions whereH 0 i denotes a function with the property that the coefficients of its Taylor expansion in (J, p, q) are at least quadratic in J, p, q if they depend on J, while coefficients independent of J are at least cubic in p, q. As proved by Kuksin [13], such coordinates exist in quite general situations.
Consider the matrix A ≡ ω We will denote by A (k;j) the matrix obtained from A by substituting its k-th column with the j-th column of B. In the forthcoming theorem we will denote by |A| the determinant of the matrix A.
Theorem 2.2 In the above situation, assume that there exists n ≡ (n 1 , n 2 , ..., n s ) ∈ Z s and γ > 0 such that Then there exists * > 0, such that, for all ∈ E 0 , E 0 := (− * , * ), the following holds true: (1) there exists a family of symplectic submanifolds N , of dimension 2s, which is the union of s-dimensional tori T β , with β ∈ s small. For each ∈ E 0 , the tori T β are invariant under the flow of X i ; the tori T β and the manifold N depend in a C ∞ way on ∈ E 0 ; one has Λ = T 0 0 .
(2) There exist symplectic action angle coordinates (I 1 , ..., I s ; ϕ 1 , ..., ϕ s ) in N , such that the functions H i N depend only on the actions I j , namely with i = 1, ..., s. The coordinates (I ; ϕ ) depend in a C ∞ way on ∈ E 0 , and so do the functions H i N (I 1 , ..., I s ).
We remark that while statement (1) ensures the existence of the invariant tori and describes their shape, statement (2) describes completely the motion on the invariant tori. Indeed, in the coordinates I, ϕ (where we drop ) the equations of motion of H i , in N , have the forṁ which shows that on each of the tori the motion is quasiperiodic. Moreover, the fact that the invariant tori, the coordinates, and the Hamiltonians depend in a smooth way on ensures that the frequencies of the perturbed system are close to those of the unperturbed one. Moreover, if for example one has (for a fixed i) then the same holds for the perturbed system. In particular in this case one has that the action to frequency map is one to one, and therefore the tori can be parametrized by the frequencies.

Idea of the proof
First we recall the scheme of the proof of the Poincaré-Lyapunov theorem (corresponding to s = 1). In this case the manifold Λ reduces to a periodic orbit. Consider a Poincaré section and the corresponding Poincaré map P ; then periodic orbits of X 1 are found as solutions of P (x) = x; such solutions are constructed by using the implicit function theorem. The standard condition that the Floquet multiplier 1 have multiplicity 2 ensures that the implicit function theorem applies.
In the case s ≥ 2 one tries to mimic the above proof. To define the Poincaré map remark that, by the first step of the proof of Arnold's theorem, there exists a function K (p, q) := j α j H j (p, q) with the property that all solution of the Hamilton equations of K 0 with initial data in Λ are periodic with a definite period (actually there exist s different functions with this property). Then define a Poincaré section of one of these periodic orbits and the corresponding Poincaré map P . Due to the symmetries of the problem there exists a local foliation which is invariant under the flow of the X i , and moreover it turns out that P defines a natural mapP from one leaf of the foliation to an other. It turns out that fixed points ofP give rise to invariant tori. To find such fixed points one uses again the implicit function theorem: the corresponding invertibility condition is ensured by assuming that 1 is an isolated eigenvalue of multiplicity 2s of the Floquet operator. Finally it can be proved that such a condition is equivalent to 2.3.

Applications
All the applications will deal with systems of the form where (P k , Q k ) ∈ 2j are canonically conjugated variables, and H os is the Hamiltonian of the on site system that we will assume to have j degrees of freedom. In our cases we will have j = 1 or 2. The phase space is defined formally as follows: Fix β > 0 and define the Banach space β of the sequences P = {P k }, P k ∈ j , such that where P k denotes the euclidean norm. The phase space is (P, Q) ∈ β × β . The system 3.4 will also have s independent integral of motion in involution given by H 1 and by s − 1 more functions H 2 , ..., H s , that will turn out to be independent of . In our cases we will have s = 2 or 3.
Then we will proceed by first defining the manifold Λ for the different models and then introducing the coordinates such that the system takes the form 2.1; we will denote by ω 1 , ..., ω s the frequencies of motion of the unperturbed breather. We point out that in our cases the unperturbed quasiperiodic breather will be concentrated at the sites 1, ..., s of the lattice. Then we will write down explicitly the nonresonance condition 2.3. Actually in all the cases that we will consider it takes a quite simple form very similar to the nonresonance condition of the Poincaré-Lyapunov theorem. To obtain such a simple form we will compute explicitly the determinants involved in condition 2.3 and simplify as much as possible the so obtained expression. Proposition 3.6 Assume that the system satisfy a suitable nonresonance condition that depends on the model (see 3.11, 3.14 and 3.18 below); then there exists * and a function δ * = δ * ( ) defined for | | < * , such that for | | < * there exists a 2s-dimensional manifold N invariant under the flows of X 1 , ..., X s ; moreover one has N = |δi|<δ * ,i=1,...,s T δ1,...,δs with T δ1,...,δs an s-dimensional torus invariant under flow of X 1 ,..., X s . On T δ1,...,δs the dynamics of X 1 (i.e. of the model we are interested in) is quasiperiodic with the frequencies The tori T δ1,...,δs depend smoothly on δ i and on . In particular there exists a constant C such that, for all points in N one has P k + Q k < C e −β|k| , ∀k = 1, ..., s (3.7) We point out that equation 3.7 is a consequence of smooth dependence of the manifold N on in the topology of the phase space. The fact that it is possible to parameterize the tori by the frequencies is a consequence of the fact that in the case = 0 the application from the actions to the frequencies will turn out to be one to one in all the models we are interested in.

Discrete nonlinear Schrödinger equation
Consider the discrete nonlinear Schrödinger equation where n ≥ 2 is an arbitrary integer. The Hamiltonian is with p k + iq k = ψ k . The second integral of motion is given by In analogy with the standard Schrödinger such a quantity can be called electron probability. Fix positive ω 1 , ω 2 , then the manifold Λ is defined by Λ := p 2 1 + q 2 , p k = q k = 0 , ∀k = 1, 2 and introduce action variables at the sites 1,2 by I 1 = (p 2 1 + q 2 1 )/2, I 2 = (p 2 2 + q 2 2 )/2, and the corresponding angles. To introduce the coordinates we need in order to apply theorem 2.2 define J 1 := I 1 − ω 1/(n−1) 1 , and J 2 := I 2 − ω 1/(n−1) 2 so that the Hamiltonians take the form withH 1 having the same meaning as in 2.1. We remark that here the manifold N =0 is just the phase space of the first two oscillators, and the restriction of H 0 1 to N 0 is simply given by I n 1 n + I n 2 n .
Lemma 3.10 The nonresonance condition 2.3 takes here the form Proof. The matrices A and B are given by

from which one has
and thus the nonresonance condition 2.3 takes the form 'there exist (n 1 , n 2 ) ∈ Z 2 such that −n 1 ω 2 + n 2 ω 1 = N (ω 1 − ω 2 ) , for all N ∈ Z'. This can be rewritten as from which it is evident that the second term at left hand side (l.h.s.) does not affect the condition and can be dropped. So the condition is equivalent to 'there exist n ∈ Z such that n ω 2 ω 1 − ω 2 = N , (3.12) for all N ∈ Z', and in turn this is equivalent to 3.11, since, in case the fraction at l.h.s. of 3.12 is an integer, the l.h.s. of 3.12 is an integer for any choice of n, while in case the fraction is not an integer, just choose n = 1. So equation 3.11 is the wanted nonresonance condition under which the theorem 3.6 applies.
So we have that proposition 3.6 holds for DNLS. In this case the quasiperiodic breather is a solution in which the electron probability is essentially concentrated at two lattice cites.

Adiabatic Holstein model
The equations of motion of the adiabatic Holstein model [1] are given by which are Hamiltonian with Hamiltonian function (p k , q k ), (x i , y i ) are canonically conjugated variables, and one has ψ j = x j + iy j . The additional integral of motion is To construct the manifold Λ we proceed as follows: Following [4] we introduce action angle variables for H os at the sites 1,2 by first defining the variables (I i , ϕ i ) by x i = I i cos ϕ i , y i = I i sin ϕ i ; i = 1, 2 and then performing the canonical transformation which gives H os the form where we omitted the prime from I. Fix now two positive frequencies ω 1 and ω 2 , and define Λ := I 1 = ω 1 ω 2 0 , I 2 = ω 2 ω 2 0 , ξ i = η i = p k = q k = 0 , ∀i = 1, 2, k = 1, 2 Defining J 1 := I 1 − ω 1 ω 2 0 and J 2 := I 2 − ω 2 ω 2 0 one has that the Hamiltonians of the system take the form we need, namely Lemma 3.13 The nonresonance condition 2.3 takes here the form Proof. One has where the first two columns of the matrix B refer to the variables ξ, η at the lattice sites 1,2. The 2i−3 column refers to the variables (p i , q i ) (i ≥ 3), and the 2i−2 columns refer to the variables (x i , y i ) (i ≥ 3). We thus obtain the matrices A (k,j) , and from them the nonresonance conditions which have the form 'there exists (n 1 , n 2 ) ∈ Z 2 such that for all m ∈ Z'. In turn this is equivalent to the nonresonance conditions 3.14.
The kind of breathers we constructed here are new and consist of solutions in which the electron probability is essentially concentrated at two lattice sites, and the oscillators are at rest. Notice that the rest position of the oscillators at the sites where the breather is localized are translated with respect to the unperturbed ones.

Vector discrete nonlinear Schrödinger equation
Consider the vector DNLS equation whose equations of motion are with k ∈ Z. Introducing the real and imaginary parts of u k , v k as new variables, namely u k = x k,1 + iy k,1 , v k = x k,2 + iy k,2 one sees that the system is Hamiltonian with Hamiltonian function The additional integrals of motion are due to the phase shift symmetries, and to the rotation invariance in the plane of u and v. They are given by and by the angular momentum which are independent but not in involution. As functions independent and in so that the three independent integrals of motion take the form Then one can apply our theory and obtain a family of three dimensional invariant tori continuing to the coupled case the manifold Λ. In particular one has Lemma 3.17 The nonresonance condition 2.3 takes here the form Proof. A long but straightforward computation (just write down the determinants and compute them) shows that the nonresonance condition has here the form "there exist (n 1 , n 2 , n 3 ) ∈ Z 3 such that for all N ∈ Z'. Introducing the variable µ 1 := ω 2 − ω 1 in order to eliminate ω 2 , remarking that the last terms at l.h.s. of the first two equations are inessential and denoting m := n 1 − n 2 one sees that the above equations are equivalent to then it is clear that the terms containing µ 1 at l.h.s. do not affect the nonresonance conditions, and therefore this is equivalent to the stated conditions.

Discussion
We first discuss briefly the relation with the papers [11,12,17].
To fix ideas we consider a quasiperiodic breather of the DNLS, and consider the motion of one observable, for example q 1 , then one can consider the Fourier expansion of q 1 (t), namely write generically all coefficients c n1n2 are different from zero, i.e. the motion contains all the frequencies n 1 ω 1 + n 2 ω 2 . In particular some of these frequencies will fall in the continuous spectrum, and as shown by [11,12] in general this phenomenon creates a coupling between the quasiperiodic motion and the continuous spectrum, and as a consequence the breather begins to radiate energy and therefore to decay.
So at first sight it is quite surprising that quasiperiodic motions exist in the considered models. However, as it is clear from the perturbative construction of quasiperiodic solutions (see e.g. [14]), in order to destroy a quasiperiodic motion two ingredients are needed: the first one is a resonance, and the second one is a coupling term in the nonlinearity. Generically all possible coupling terms are present in the nonlinearity, but a system with symmetry is not generic! The symmetry actually prevents the existence of such coupling terms. For this reason the mechanism of [11,12] is not active in this case.
Concerning the relation with the work by Yuan [17] we point out that his situation is completely different from ours, indeed, while in our case radiation is not possible due to the non generiticity of the nonlinearity, in his case radiation is not possible due to the nongeneriticity of the linear part of the system, indeed in his case there is no continuous spectrum.
In conclusion we have shown that quasiperiodic breathers exist in Hamiltonian lattices having some integrals of motion independent of the Hamiltonian, and that Nekhoroshev's theorem is a powerful tool in order to actually construct such breathers in concrete cases.
We also emphasize that systems with symmetry are exceptional, but at the same time they are quite common and interesting: we think that the same is true for quasiperiodic breathers.