Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré

Foreword

The present paper is dedicated to Antonio Giorgilli, in occasion of his 70$^{\mathrm{th}}$ birthday.

There is no need to stress here his scientific merits: his publications are worth more than a thousand words. And for sure he would deserve a much better paper than the present contribution, to celebrate his career.

We thus prefer to praise the human qualities of Antonio, which are perceived immediately even by those who know him since only a short period of time; and they get absolutely clear for those who had the luck and the honor to know him since a long time. Like one of us, who met Antonio almost 30 years ago as a professor at the second year of the physics degree.

For all of us, to various extent, he has been a master; we could even say a scientific father. And we are proud to claim he is more than a good friend.

1.   Introduction

The present paper deals with an extension of a classical result on periodic orbits in nearly integrable Hamiltonian Systems, due to Poincaré at the end of XIX century ^[28,29]. The problem considered is the continuation of periodic orbits foliating a completely resonant torus $I = I^*$ of maximal dimension $m$ of an integrable Hamiltonian $H_0(I)$ of $m$ degrees of freedom, once a small perturbation ${\epsilon} H_1(\theta, I)$ is added to the system. Given $I = I^*$ and $\theta^{(0)}(t) = \mathit{\boldsymbol{\omega}} t+\theta(0)$ the corresponding unperturbed periodic flow (where $\mathit{\boldsymbol{\omega}}: = D_I H_0(I^*)$), and assuming the invertibility of $D_I^2H_0(I^*)$, Poincaré's Theorem ensures the existence of the continuation at small $|{\epsilon}|$ for those choices $\theta^*(0)\in{\mathbb{T}}^m$ which are non-degenerate critical points of the time averaged perturbation ${\left\langle{{H_1\Big|_0}}\right\rangle}_T$ defined as

where ${\left\langle{{\cdot}}\right\rangle}_T$ and $\Big|_0$ in the rest of the paper will denote respectively the time-average over one period and the evaluation at the unperturbed dynamics on the torus $I^*$. Since any unperturbed periodic orbit on $I = I^*$ is uniquely identified by the quotient of the resonant torus with respect to the periodic flow, it turns out that the functional ${\left\langle{{H_1\Big|_0}}\right\rangle}_T$ is defined on the quotient manifold ${\mathbb{T}}^{m-1}$ and can be expressed in terms of $m-1$ suitable resonant angles ${\varphi}$ (see for example formula (10) in ^[19]), namely

Hence, in terms of these "phase-shift" variables ${\varphi}$, the values ${\varphi}^*$ fulfilling

identify unperturbed periodic solutions that can be continued via Implicit Function Theorem. Even more, Poincaré's result relates linear stability of these continued periodic orbits to the product of the two matrices ${\epsilon} D^2_\theta {\left\langle{{H_1\Big|_0}}\right\rangle}_T({\varphi}^*)$ and $D_I^2H_0(I^*)$, showing that the Floquet multipliers (different from one) are of order ${\mathcal{O}}(\sqrt{\epsilon})$; in particular, if $D_I^2H_0(I^*)$ is definite (positive or negative), linear stability is encoded in the nature of ${\varphi}^*$ as critical point on ${\mathbb{T}}^{m-1}$. An extension of this result to the completely degenerate case ${\left\langle{{H_1\Big|_0}}\right\rangle}_T\equiv \rm{constant}$ is included in ^[20]; with the same original approach of Poincaré, based on the ${\epsilon}$ Taylor-expansion of the solutions, the authors are able to replace the average ${\left\langle{{H_1\Big|_0}}\right\rangle}_T$ with a ${\mathcal{O}}({\epsilon}^2)$ higher order functional $F$ on the quotient torus ${\mathbb{T}}^{m-1}$, whose non-degenerate critical points ${\varphi}^*$ are the new candidate for the continuation. A further and complete generalization of Poincaré's result is instead given by ^[26], where a KAM-like normal form approach, combined with fixed point methods, has been used to explore any kind of Poincaré degeneracy. Among the various degenerate scenarios, the normal form approach allows to investigate the case of isolated critical points ${\varphi}^*$ whose hessian in (1.2) is not invertible (hence non-degeneracy does not hold). In this case the existence of the continuation depends on the higher order corrections, which are needed to be explicitly calculated.

One of the typical mechanism for the complete degeneracy to occur, is the lack of any terms, in the given leading order perturbation $H_1$, related to all the resonances of the torus; indeed in these cases ${\left\langle{{H_1\Big|_0}}\right\rangle}_T$ turns out to be a constant function on the torus. This kind of degeneracy is naturally met in the study of spatially localized solutions in Hamiltonian Lattices (see for example the Special Issue ^[21] for a recent collection of results on this topic), like Multibreathers in the Klein-Gordon model

or multi-pulse discrete solitons in standard discrete NLS model

where ${\mathcal{J}}$ is a finite^* set of indexes. Although quite trivial, let us remark that if ${\epsilon} = 0$ these models reduce to a collection of uncoupled identical nonlinear oscillators. One typically considers an unperturbed (${\epsilon} = 0$) periodic solution given by a subset $S = \{{j_1, \ldots, j_m}\}$ of $m$ of these uncoupled identical oscillators, having the same action $I_{j_l} = I^*$ and hence the same nonlinear frequency $\omega$; in other words, one is considering a low-dimensional completely resonant torus of the unperturbed system, corresponding to the lowest order resonance ${\left({1:1:\ldots:1}\right)}$. Once the perturbation $H_1$ is added (in the above models it is given by the weak linear interactions among the oscillators), the torus breaks down and only some, typically a finite number, of the unperturbed periodic orbits survive at small ${\epsilon}$: these solutions are still spatially localized (in terms of amplitude of the oscillations) around the unperturbed oscillators $j_l$. The mathematical investigation of these solutions naturally represents an extension of the original Poincaré's result to low-dimensional completely resonant tori. The literature on the topic provides two suitable methods to prove existence of such solutions, both based on variational arguments and leading to the study of critical points on the average perturbation ${\left\langle{H_1}\right\rangle}_T$ as in Poincaré. The first method is formulated and developed in ^{[2,3,6,11,14,15,17]} and applies to models like (1.3), while the second exploits the rotation symmetry in (1.4) and is developed and applied in ^[1,10,12,13]. All these results deal with the non-degenerate case, which occurs for example in (1.3) and (1.4) when, due to the nearest-neighbours linear interaction, consecutive oscillators $S = \{{1, 2, \ldots, m}\}$ are considered. If instead some of the unperturbed oscillators are not consecutive, degeneracy occurs: in particular total degeneracy happens in models like (1.3) and (1.4) when all the oscillators are not consecutive (the easiest case is $S = \{{j_1, j_2}\}$ with $|j_1- j_2|\neq 1$). Some results treating partial or total degeneracy in this context might be found in ^[24,25,27]: the first two works are based on the Lyapunov-Schmidt decomposition, while the last one represents the extension of ^[26] to the low-dimensional case, hence the perturbation scheme is performed at the level of Hamiltonian normal forms.

^*Part of the literature consider the lattice as infinite; we here prefer to take the finite case, in order to avoid technical difficulties related to the infinite dimensional phase space structure which are not essential with respect to the problem we are tackling.

The results we here present are the extensions to low dimensional tori of both the original non-degenerate Theorem, due to Poincaré, and of the totally degenerate one of Meletlidou-Stagika ^[20]; such extensions are performed in the special case when the unperturbed Hamiltonian $H_0$ is already decoupled into two non-interacting integrable subsystems

as in the Hamiltonian Lattice models (1.3) and (1.4). From a technical point of view, the continuation is obtained looking for fixed points of the period map, and the expansions in ${\epsilon}$ are performed at the level of the (analytic) solutions; the above restriction (1.5) allows to decompose the differential of the period map into a block triangular form, so that the Implicit Function Theorem can be easily applied. The same decomposition also allows to successfully investigate the linear stability of the periodic orbits thus obtained. For integrable Hamiltonian more general than (1.5) and for extensions to partial or higher order degeneracies, normal form techniques are needed, to put the system into a suitable form to investigate the continuation (see ^{[4,5,8,9,16,27,30,31]}).

Theorem 1.1. Consider a Hamiltonian system of the form

where $(\theta, I)\in{\mathbb{T}}^m\times{\mathcal{U}}(I^*)$ while $z = (\eta, \zeta)\in{\mathbb{C}}^{2n}$. Assume $z = 0$ to be an elliptic equilibrium for ${H}^{\flat}_0(z)$, with

and $I^*\in{\mathcal{U}}$ to be a value of the actions which identifies a completely resonant torus of maximal dimension for ${H}^{\sharp}_0(I)$, with frequency vector $\mathit{\boldsymbol{\omega}} = \omega \mathit{\boldsymbol{k}}$. Let ${{\varphi}^*}$ be a critical point of the average ${\left\langle{{H_1\Big|_0}}\right\rangle}_T({{\varphi}})$. If the following three assumptions hold true:

1).

2).

3).

then, there exists ${\epsilon}^*$ such that, for $|{\epsilon}| < {\epsilon}^*$, there exists a periodic orbit with initial datum $({{\varphi}}_{po}({\epsilon}), I_{po}({\epsilon}), z_{po}({\epsilon}))$ analytic in ${\epsilon}$ and ${\mathcal{O}}({\epsilon})$-close to $({{\varphi}^*}, I^*, 0)$.

Moreover, if the second Melnikov non-resonance condition holds true

then the spectrum $\Sigma$ of the monodromy matrix splits into two different subsets: $\Sigma_1$ given by multipliers which have characteristic exponents $\mu_j({\epsilon})$ becoming zero in the limit ${\epsilon}\to 0$, which have the asymptotic behaviour $\mu_j({\epsilon})\sim\sigma_j\sqrt{\epsilon}$, with $\sigma_j^2$ eigenvalues of $-T^2D^2_\theta {\left\langle{{H_1\Big|_0}}\right\rangle}_T({\varphi}^*) D^2_I{H}^{\sharp}_0(I^*)$ and $\Sigma_2$ given by multipliers having purely imaginary characteristic exponents $\pm2\pi{i}\frac{\Omega_j({\epsilon})}{\omega}$ with $\Omega_j({\epsilon}) = \Omega_j + {\mathcal{O}}({\epsilon})\in{\mathbb{R}}$.

In the sufficient assumptions needed for the existence statement above we recover the so called Kolmogorov-non-degeneracy (K-ND), which implies that the resonant torus $I^*$ is isolated in ${\mathcal{U}}$, and the so called Poincaré-non-degeneracy (P-ND), which similarly implies that the critical points ${{\varphi}^*}$ on the torus ${\mathbb{T}}^{m-1}$, selecting the periodic orbits, are isolated. Moreover, we need a third assumption in order to split the tori variables $(\theta_1, {{\varphi}}, I)$ from the transversal ones: this is easily obtained by assuming the so called First Melnikov condition (M1), which is a non-resonance assumption between the frequency of the unperturbed periodic orbit $\omega$ and the transversal (small) oscillations of frequencies $\Omega_j$. The linear stability statement is instead a consequence of the continuity of the spectrum with respect to ${\epsilon}$, which is enough in the easy case of distinct eigenvalues, and of the Krein signature theory, which here applies if the frequencies $\Omega_j$ all have the same sign. In particular, it is evident that if $D^2_I{H}^{\sharp}_0(I^*)$ is either positive or negative definite, the relevant role in the stability is played by the nature of ${{\varphi}^*}$ as critical point of ${\left\langle{{H_1\Big|_0}}\right\rangle}_T$. The second Melnikov condition (M2) is instead needed (see for example ^[18]) to preserve transversal ellipticity of the periodic orbits, namely that multipliers of $\Sigma_2$ do not leave the unitary circle, via Krein signature: indeed, combining (M2) with the assumption (1.7) on the (positive) sign of $\Omega_j$, one can derive the (positive) definite signature for all the unperturbed Floquet multipliers $e^{2\pi{i} \frac{\Omega_j}{\omega}}$ on the unitary circle. Linear stability then depends on the effect of the ${\mathcal{O}}({\epsilon})$ perturbation on the approximate internal characteristic exponents $\sigma_j\sqrt{\epsilon}$; for example, in the generic case of distinct $\sigma_j$, any sufficiently small perturbation would still give distinct $\mu_j({\epsilon})$ and linear stability is then encoded in the spectrum of $D^2_\theta {\left\langle{{H_1\Big|_0}}\right\rangle}_T({\varphi}^*)$.

Theorem 1.2. Consider a Hamiltonian system of the form (1.6) which fulfills assumptions (K-ND) and (M1). Suppose that ${\left\langle{{{H}_1\Big|_0}}\right\rangle}_T({{\varphi}})$ is identically constant on ${\mathbb{T}}^{m-1}$. Then there exists a function $F_2({{\varphi}}):{\mathbb{T}}^{m-1}\to{\mathbb{R}}$ such that if ${{\varphi}^*}$ is a nondegenrate critical point of $F_2$, namely

then there exists ${\epsilon}^*$ such that, for $|{\epsilon}| < {\epsilon}^*$, there exists a periodic orbit with initial datum $({{\varphi}}_{po}({\epsilon}), I_{po}({\epsilon}), z_{po}({\epsilon}))$ analytic in ${\epsilon}$ and ${\mathcal{O}}({\epsilon})$-close to $({{\varphi}^*}, I^*, 0)$. Moreover, under the same assumption (M2), the same splitting of $\Sigma$ as in Theorem 1.1 holds true, with the only difference that $\mu_j({\epsilon})\sim\sigma_j {\epsilon}$, with $\sigma_j^2$ eigenvalues of $-T^2 D^2_\theta F_2({{\varphi}^*}) D^2_I{H}^{\sharp}_0(I^*)$.

The precise definition of $F_2$ in the above statement requires the expansion of the solution up to order ${\mathcal{O}}({\epsilon})$, hence it is deferred to formula (2.19) in Section 2, where we prove the two Theorems.

The two statements claimed above are then applied to show existence of discrete solitons in dNLS models, by studying first a typical non-degenerate vortex-like configuration for the ZigZag model, and then showing the existence of only in/out-of-phase discrete solitons in the easiest totally degenerate case ($S = {-1, 1}$) for the standard model. This part, which reproduces results already existing in the literature (see for example ^[22,23]), is developed in Section 3.

2.   Proof

The present section includes the proofs of the two Theorems. A first part shows the formal perturbation scheme here used (which is the same of ^[20]); the second and third parts provide the continuation argument in the non-degenerate and totally degenerate case; the last part is dedicated to linear stability, hence giving the splitting of the spectrum and asymptotic behaviour of Floquet exponents in the limit ${\epsilon}\to 0$.

2.1.   Formal expansions and leading order approximations

Due to the near integrability of the model, it is known that any $T$-periodic solution of (1.6) $(\theta, I, z)(t; {\epsilon})$ is analytic in ${\epsilon}$; we can thus Taylor-expand all the variables in ${\epsilon}$ and write

According to the same splitting of the integrable part $H_0$ given in (1.5), we can decompose the perturbation $H_1$ as

where ${H}^{\sharp}_1$ depends only on the action-angle variables, ${H}^{\flat}_1$ depends only on the complex variables $z = (\eta, \zeta)$ while ${H}^{\natural}_1$ properly depends on all the variables, and provide the interaction among the two subsystems. Notice that the term ${H_1}^{\natural}$ has necessarily to be at least linear in $z$, so that it has to vanish when evaluated at the unperturbed flow, namely ${H}^{\natural}_1\Big|_0 = 0$. Hamilton equations read

If we expand w.r.t. ${\epsilon}$ both the Hamiltonian vector field and the time derivatives we get the two systems:

and

these necessarily have to coincide at any order in ${\epsilon}$. Equating terms of order $0$ we get

where $D: = D_z^2 {H}^{\flat}_0 \Big|_0$ is the diagonal matrix

We deduce immediately that actions keep their unperturbed initial values $I^{(0)}(0)$, the angles rotate on the torus ${\mathbb{T}}^m$ with frequencies $\nabla_I {H}^{\sharp}_0(I^{(0)})$, while the external variables $z$ evolve according to their unperturbed nonlinear dynamics

At order $\mathcal{O}({\epsilon})$ we have the time-dependent system

by inserting order $0$ approximation of the periodic solutions on the unperturbed torus $I^*$

we get the system

where equation for $\dot{I}^{(1)}$ and $\dot{\theta}^{(1)}$ are uncoupled from $\dot{z}^{(1)}$. Notice that equation for $z^{(1)}$ simplifies in those models with ${H}^{\flat}_1(z) = O(z^2)$; in fact in such cases one has $\nabla_z {H}^{\flat}_1(z) = O(z)$, which implies a vanishing contribute of $\nabla_z {H}^{\flat}_1$ in the third equation, if at ${\epsilon} = 0$ the variables $z^{(0)}$ stay at rest

Solving the equation for $\dot I$, we get the leading order corrections to $I^*$ of the actions

Since $I^{(1)}(t)$ have to be periodic of period $T$, namely $I^{(1)}(T) = I^{(1)}(0)$, we have to impose

Let us set for the sake of simplicity $k_1 = 1$ in the resonant frequency vector $\omega{k}$, with $\omega: = \frac{2\pi}{T}$. We introduce the $m-1$ resonant angles ${\varphi} = \{{{\varphi}_j}\}_{j = 1}^{m-1} = k_j\theta_1-\theta_j$, which are the natural coordinates of the quotient torus ${\mathbb{T}}^{m-1}$ of the resonant torus over the periodic flow. Hence we get (by setting $\theta_1^{(0)}(0) = 0$)

Indeed, in the previous average, we can exchange the time-variable with $\theta_1$, so that the first element of the gradient $\nabla_\theta{H}^{\sharp}_1$ has zero average and in the $m-1$ remaining components, only the dependence on the angles ${{\varphi}}$ is left in the average

Thus, the periodicity condition (2.7) easily reads

solutions ${{\varphi}^*}$ of (2.9) represent critical points of the functional (2.8) defined on the torus ${\mathbb{T}}^{m-1}$.

Le us define the Hessian matrix

thus the equation for $\dot\theta$ becomes

since also the angles $\theta^{(1)}$ have to be $T$-periodic (uniformly in ${\epsilon}$), we have to ask

where the first term is the average of $I^{(1)}(t)$ while the second represents the average of $\nabla_{I} {H}^{\sharp}_1$ restricted (as usual) to the unperturbed flow. Recalling (2.6) one has

which provides the correction at order ${\mathcal{O}}({\epsilon})$ of the initial values of the actions

It is clear from (2.11) that $I^{(1)}(0)$ depends on the unperturbed initial data $I^*$ and ${{\varphi}}_0$.

Let us now move to the equation for $\dot z$

In the Taylor expansions of ${H}^{\natural}_1$ and ${H}^{\flat}_1$ with respect to the external variable $z$, the important terms are linear in $z$; indeed all those terms which are ${\mathcal{O}}(z^2)$ produce contributions, in the equation, that vanish when evaluated at $z^{(0)} = 0$. Hence

represents a periodically forcing term in the inhomogeneous linear equation (2.12)

if we assume the non-resonance (M1) condition between $\Omega_j$ and the frequency $\omega$, then $h_1$ is non-resonant with respect to $JD$ and the solution of (2.12) is given by the convolution integral

The $T$-periodicity of $z^{(1)}$ provides, as for $I^{(1)}$, the correction at order ${\mathcal{O}}({\epsilon})$ of the initial value $z_0 = {\epsilon} z^{(1)}(0) + O({\epsilon}^2)$; also in this case, $z^{(1)}(0)$ has to depend on $I^*$ and ${\varphi}_0$. Indeed, from

one gets the correction

with ${\left({{\mathbb I}- e^{JD T}}\right)}$ invertible because of (M1).

2.2.   Continuation in the non-degenerate case

Given a generic initial datum $P_0$ of the unperturbed system (${\epsilon} = 0$), we look for a correction $P({\epsilon})$ at ${\epsilon}\neq 0$ such that the variation of the Hamiltonian flow after one period $T$ vanishes, namely $\Phi^T(P({\epsilon}), {\epsilon})-P({\epsilon}) = 0$. The condition we have to impose is then the $T$-periodicity of the flow, assuming its validity at ${\epsilon} = 0$, and exploiting the analyticity of the flow $\Phi^T(P({\epsilon}), {\epsilon})$, and of its initial datum $P({\epsilon})$, with respect to ${\epsilon}$.

In order to identify the periodic orbit, we can ignore its phase $\theta_1$ and consider as unknowns only the other initial $2n+2m-1$ "transversal" variables $\{{{{\varphi}}_0({\epsilon}), I_0({\epsilon}), z_0({\epsilon})}\}$. Moreover, since the Hamiltonian is kept constant along the orbit, the $2(n+m)$ equation $\Phi^T(P({\epsilon}), {\epsilon})-P({\epsilon}) = 0$ are not independent, and we can get rid of one of them; we choose to ignore the action $I_1$ associated to the angle $\theta_1$. In this way, the periodicity condition reduces to a set of $2n+2m-1$ equations in $2n+2m-1$ variables of the form

Since the velocity of variation of the actions $I_{l = 2, \ldots, m}$ (which are approximately constant in time) are of order ${\mathcal{O}}({\epsilon})$, we can rescale them by ${\epsilon}$; in this way the leading terms of (2.13), with respect to ${\epsilon}$, read

where now $\{{{{\varphi}}_0, I_0, z_0}\}$ are the variables which identify the generic unperturbed initial datum $P_0$, modulo a shift of $\theta_1$, and the components $F_j$ read

where $H_1(\tau)$ is the perturbation restricted to the unperturbed flow corresponding to the initial datum $({{\varphi}}_0, I_0, z_0)$. By redefining $X$ coherently with the previous scaling of the actions $J$, we get the compact form

Solving the above equation (2.14) at ${\epsilon} = 0$ is equivalent to solve the system $F\left({{\varphi}_0}, I_0, z_0\right) = 0$. Hence, if $I_0 = I^*$ corresponds to a completely resonant torus with frequency vector $\omega \mathit{\boldsymbol{k}}$, then $F_1(I^*) = 0$; given such $I^*$ and setting at rest the complex variables $z_0 = 0$, if ${{\varphi}_0} = {{\varphi}^*}$ is a critical point of ${\left\langle{{{H}^{\sharp}_1\Big|_0}}\right\rangle}_T$

then also $F_2({{\varphi}^*}, I^*) = 0$. Thus, the point $({{\varphi}^*}, I^*, 0)$ solves

In order to apply the Implicit Function Theorem, we need invertibility of $X'$, which is the differential of $X$ evaluated on the approximate solution $({{\varphi}^*}, I^*, 0)$ at ${\epsilon} = 0$; this coincides^† with the differential of $F$ at its zero and is block triangular. Indeed

^†Recall that to compute $X'$ we can first restrict to ${\epsilon} = 0$ and then differentiate w.r.t. the unperturbed initial datum $z_0$.

where

From the first assumption (K-ND) the matrix $C$ is invertible, and the same holds true for $D_{{\varphi}_0}^2{\left\langle{{{H}_1(\tau)}}\right\rangle}_T({{\varphi}^*}, I^*, 0) = D_{{\varphi}}^2{\left\langle{{{H}_1\Big|_0}}\right\rangle}_T({{\varphi}^*})$ due to the second one (P-ND); in particular notice that

because $D_{{\varphi}_0}^2{\left\langle{{{H}^{\natural}_1(\tau)}}\right\rangle}_T$ is at least linear in $z_0$. The invertibility of block $e^{JDT}-{\mathbb I}$ of $X'$ is due to the first Melnikov condition (M1), which ensures that $\frac{\Omega_j}{\omega}\notin \mathbb{Z}$. Due to the triangular structure of $X'$, the block $X'_{12}$ does not play any role, even if different from zero.

2.3.   Continuation in the totally degenerate case

The totally degenerate case corresponds to ${\left\langle{{{H}_1}}\right\rangle}_T\equiv \rm{const}$; what follows represents the extension of what has been already developed in ^[20] in the case of maximal tori.

In the degenerate scenario, the variation of the actions $I_l$ on the approximate periodic orbit is much slower than what happens in the non-degenerate case, being $\dot I_l = {\mathcal{O}}({\epsilon}^2)$; as a consequence, an expansion up to order ${\epsilon}^2$ of the actions $I_l$ is required. Such an expansion provides the following equation for $I^{(2)}$

Since the term ${H}^{\natural}_1(\theta, I, z)$ is at least linear in $z$, then two of the above terms will vanish when evaluated at $z = 0$

hence we can reduce (2.16) to the simplified form

where $D^2_{z\theta}{H}^{\natural}_1(t)z^{(1)}(t)$ is given only by linear terms in $z$, and the first order corrections $\theta^{(1)}(t), \, I^{(1)}(t)$ and $z^{(1)}(t)$ have been determined by the periodicity conditions at order ${\mathcal{O}}({\epsilon})$. By integrating on the period interval $[0, T]$, the periodicity condition $I^{(2)}(T)-I^{(2)}(0) = 0$ reads

where in the first addendum the initial datum $\theta^{(1)}(0)$ can be "added" without affecting the computation of the average; indeed one gets ${\left\langle{{D^2_\theta{H}^{\sharp}_1(\tau)\theta^{(1)}(0)}}\right\rangle}_T\equiv0$, due to the degeneracy assumptions. Moreover, thanks to the conservation of the energy along the periodic orbit, it is possible to prove that only $m-1$ of the above $m$ equations are independent. Hence, as in the non-degenerate case, we can ignore the evolution of the first action $I_1$ and consider the periodicity condition only for the remaining $m-1$ ones. Due to the complete degeneracy, the Taylor expansion of the $m-1$ actions $I_l$ starts with term of order ${\mathcal{O}}({\epsilon}^2)$; thus we rescale $I_l(T)-I_l(0)$ by ${\epsilon}^2$ and define the components of $F_2$ as

where the initial values $I^{(1)}_0$ and $z^{(1)}_0$ explicitly depend on the unperturbed initial values ${{\varphi}}_0$ and $I_0$. We want to apply the IFT to the periodicity condition (2.14), where now ${{\varphi}^*}$ has to solve

The above Eq (2.20) selects those ${{\varphi}}_0 = {{\varphi}^*}$ on the torus ${\mathbb{T}}^{m-1}$ which can be continued, and that provide the solution of (2.14) at ${\epsilon} = 0$

In order to apply again the IFT we need such ${{\varphi}^*}$ to be non-degenerate critical points for $F_2({{\varphi}}_0, I^*)$, namely

which is indeed hypothesis (1.8).

2.4.   Linear stability

In order to investigate the linear stability of the periodic orbits, we have to exploit the fact that the monodromy matrix $\Lambda({\epsilon})$ coincides with the differential of the Hamiltonian flow at time $T$ with respect to the initial datum of the periodic orbit. We develop the non-degenerate case, being the totally degenerate one a minor variation of the forthcoming arguments.

With standard calculations, which resemble those for the differential $X'$ of the period map, one obtains the following structure for $\Lambda({\epsilon})$

with:

● the $\Lambda_{11}$ block reads

where we have included in the ${\mathcal{O}}({\epsilon})$ also the term $-{\epsilon} T D^2_{\theta_0 I_0} {\left\langle{{{H_1}^{\sharp}\Big|_0}}\right\rangle}_T({{\varphi}^*}, I^*)$;

● the out-of-diagonal blocks are perturbations of the same order ${\mathcal{O}}({\epsilon})$

since the interaction between internal variables $(\theta, I)$ and the external ones $(\xi, \eta)$ is only due to the perturbation $H_1$;

● the block $\Lambda_{22}$ reads

By continuity with respect to ${\epsilon}$, the multipliers of $\Lambda({\epsilon})$ have to converge to those of $\Lambda(0)$ as ${\epsilon}\to 0$, the last ones being collected into two different sets (due to the block diagonal shape of $\Lambda(0)$): $2n$ multipliers of $\Lambda_{22}(0)$ are on the unitary circle, with purely imaginary exponents $\pm2\pi{i}\frac{\Omega_j}{\omega}$, and $2m$ multipliers of $\Lambda_{11}(0)$ equal to $1$ (due to the complete resonance of the $n$-dimensional torus). Moreover, due to the two Melnikov conditions^‡, the $2n$ multipliers on the unitary circle are different from $\pm 1$. By continuity in ${\epsilon}$, the two sets stay disjoint and provide the two different sets $\Sigma_1$ and $\Sigma_2$ of the statements: $\Sigma_1$ is made of multipliers $\lambda({\epsilon})$ which all bifurcate from $1$, hence having vanishing characteristic exponents, while $\Sigma_2$ is made of multipliers which are deformations of $e^{\pm2\pi{i}\frac{\Omega_j}{\omega}}\neq\pm 1$. If some (or all) of the unperturbed Floquet multiplier of $\Lambda_{22}(0)$ coincide, then having the same definite Krein signature (see ^[7,18,32]) is a sufficient condition for their perturbations not to leave the unitary circle. Due to the Melnikov conditions, and since the frequencies $\Omega_j$ share the same sign (rotation direction), the unperturbed Floquet multipliers $e^{\pm2\pi{i}\frac{\Omega_j}{\omega}}$ are all positive definite, so any sufficiently small perturbation $\lambda_j({\epsilon})$ cannot leave the unitary circle, still being different from $\pm 1$. If all $\Omega_j$ are distinct, then the same holds true for the corresponding unperturbed Floquet multipliers (which are all simple eigenvalues of $\Lambda({\epsilon})$) and for any sufficiently small perturbations, without invoking their Krein signatures.

^‡Notice that (M1) implies $\pm\frac{\Omega_j}{\omega}\neq l\in{\mathbb{Z}}$, hence the multipliers have to be different from $1$. The second condition (M2) implies in particular $\pm\frac{\Omega_j}{\omega}\neq \frac{l}2\in{\mathbb{Z}}$ hence the multipliers have to be different also from $-1$. We recommend reference ^[18].

It remains to show the prescribed asymptotic behaviour of the Floquet multipliers of $\Sigma_1$. These are solutions of

with vanishing characteristic exponents, hence $\lambda_j({\epsilon})\to 1$. We thus assume for these part of the spectrum the ansatz $\lambda_j({\epsilon}) = e^{\sqrt{{\epsilon}}\sigma_j+{\mathcal{O}}({\epsilon})}$, where the $\sqrt{\epsilon}$ scaling is expected (as in the original Poincaré result, but see also ^[30]) and will be clarified in a while. Since $M_{22}({\epsilon}) = e^{JDT}-\lambda{\mathbb I} + {\mathcal{O}}({\epsilon})$ is invertible for sufficiently small ${\epsilon}$, because of the first Melnikov assumption (M1), we can apply the Schur complement to rewrite the characteristic equation as

Since $M_{22}^{-1} = {\mathcal{O}}(1)$ and $M_{21} = {\mathcal{O}}({\epsilon}) = M_{12}$, it turns out that $M_{21} M_{22}^{-1} M_{12} = {\mathcal{O}}({\epsilon}^2)$ is a perturbation of $M_{11}$, thus it is enough to study $\mathsf{det}{\left({M_{11}(\sigma, {\epsilon})}\right)} = 0$, where

Here we can apply a second time the Schur complement, being the diagonal blocks invertible for ${\epsilon}\neq 0$ small enough, so that $\mathsf{det}(M_{11}(\sigma, {\epsilon})) = 0$ reads

Taking the limit ${\epsilon}\to 0$, we obtain the asymptotic expansion claimed in the statement of Theorem 1.1; this concludes the non-degenerate case, since by adding the smaller perturbation term $M_{12} M_{22}^{-1} M_{21}$ the same arguments hold true.

The totally degenerate case can be faced following the same steps above, with a minor but crucial variation which involves the perturbation $M_{12} M_{22}^{-1} M_{21}$. Indeed, since the magnitude of $\dot I$ is of order ${\mathcal{O}}({\epsilon}^2)$, it follows that $M_{12}$ has the special structure

which provides

hence it is component wise a perturbation of the leading term $M_{11}$

where $F_2$ is taken with all its $m$ components.

3.   Spatially localized solutions in discrete NLS models

The aim of this section is to exploit Theorem 1.1 to investigate the existence of a special class of spatially localized time-periodic solutions in a dNLS model of the form

where the (weak) linear term $L$ is beyond nearest-neighbours

the coupling parameter ${\epsilon}$ has to be considered small enough (so called anti-continuum limit) and the boundary conditions are of Dirichlet type in the case of ${\mathcal{J}}$ finite. The equations can be written in Hamiltonian form ${i}\dot\psi_j = {\frac{\partial {H}}{\partial {\overline\psi_j}}}$ with

where

Being interested in solutions which are spatially localized on many-sites, we introduce the set of excited sites

where we stress that the indexes do not have to be necessarily consecutive; in other words, we are including also configurations where the localization of the amplitude (hence of the energy), is clustered, with "holes" separating the different clusters along the chain.

The easiest and most popular spatially localized solutions in the considered dNLS model correspond to the so-called discrete solitons (or also multi-pulse solutions). These are periodic solutions $\psi(t)$ of (3.1) which at ${\epsilon} = 0$ take the form

where $\phi^{(0)}$ is independent of time. Since the unperturbed excited oscillators $\{{\psi_j^{(0)}}\}_{j\in{\mathcal{J}}}$ in (3.5) are required to keep the same frequency $\omega$, thus being in $1:1$ resonance, a common amplitude $R > 0$ is necessary, so that the unperturbed spatial profile $\phi^{(0)}$ reads

where

All these orbits are uniquely defined except for a phase shift, which corresponds to a change of the initial configuration in the ansatz (3.5). According to this geometrical interpretation, all the unperturbed periodic orbits foliate a completely resonant $m$-dimensional torus ${\mathbb{T}}^m$ of the phase space, and any orbit is uniquely identified by a point in the quotient space ${\mathbb{T}}^{m-1} = {\mathbb{T}}^m\setminus{\mathbb{T}}$; such a point can be well represented by introducing a set of $m-1$ new "phase shifts" variables

In order to put the dNLS model in the form (1.6) of Theorem 1.1, we introduce action-angle variables for the sites $j\in S$

and Birkhoff complex variables $z = (\eta, \zeta)$ (replacing the complex variable $\psi$) for the remaining sites

in this way the symplectic form reads $d\eta\wedge d\zeta + d\theta\wedge dI$ and the integrable part of the Hamiltonian takes the form

while the perturbation $H_1$ can be split in the three parts ${H}^{\sharp}_1, \, {H}^{\flat}_1, \, {H}^{\natural}_1$ depending on the shape of $L$ in (3.2) and of the choice of $S$ in the considered example. Moreover, once fixed at $I^*$ the common action of the excited nonlinear oscillators $\psi_j, \, j\in S$, the frequency (3.7) reads

We will benefit of rewriting the generic linear interaction term as

In this way, a term of ${H}^{\sharp}_1$ would read

one of ${H}^{\flat}_1$ would read

and one of ${H}^{\natural}_1$ (assuming $j\in S$ and $j+l\in{\mathcal{J}}\setminus S$)

3.1.   Non-degeneracy and consecutive excited sites

Let us consider, instead of the generic model (3.4), the dNLS Hamiltonian with nearest neighbor and next to nearest neighbor linear interactions, on a lattice ${\mathcal{J}} = \{{-N, \ldots, N}\}$

and take $m\ll N$ excited consecutive sites in the set $S = \{{1, \ldots, m}\}$. In order to find candidate solutions, we need to solve (2.15), i.e., to find critical points of ${\left\langle{{{H}^{\sharp}_1\Big|_0}}\right\rangle}_T$, which explicitly reads

where

while both ${H}^{\flat}_1$ and ${H}^{\natural}_1$ have to vanish, since $z^{(0)} \equiv 0$. Explicit calculations provide

by introducing the coordinates $\varphi_{i} = \theta_{i+1}^{(0)} - \theta_{i}^{(0)}$ one gets the ($\phi$-dependent part of the) functional on the torus ${\mathbb{T}}^{m-1}$

whose critical points provide the solutions ${\varphi}^*$ we were looking for.

For examples, let us consider three consecutive sites $m = 3$ with $S = \left\lbrace 1, 2, 3 \right\rbrace$, in the so-called ZigZag model $k_1 = k_2 = 1$. The average $\left\langle {H}^{\sharp}_1\Big|_0 \right\rangle_T$ defined on the torus ${\mathbb{T}}^2$ is

so that Eq (2.15) reads

which provides six different solutions: four of them correspond to in/out-of-phase configurations ${\varphi}^*\in\{{(0, 0), (0, \pi), (\pi, 0), (\pi, \pi)}\}$ and the other two represent the so-called vortex solutions (see ^[24]) with ${\varphi}^* = \pm(\frac23\pi, \frac23\pi)$. All the six solutions are isolated, hence we expect non-degeneracy to hold: this is confirmed by an explicit computation of the differential in (P-ND).

Since $C = \gamma{\mathbb I}$, approximate linear stability is completely encoded by the nature of the critical points ${\varphi}^*$. Indeed the nonzero eigenvalues of $D^2_\theta {\left\langle{{{H_1}^{\sharp}\Big|_0}}\right\rangle}_T$ are given by the eigenvalues of $-\gamma T^2 D^2_{\varphi}\left\langle {H}^{\sharp}_1\Big|_0 \right\rangle_T$, with

In particular we recover the property, already illustrated in the literature, that for consecutive excited sites the center and saddle directions are exchanged when changing the sign of the product ${\epsilon}\gamma$. If such a product is positive, we found one maximum $(0, 0)$, two minima $\pm(\frac23\pi, \frac23\pi)$ and the remaining are saddles of $\left\langle {H}^{\sharp}_1\Big|_0 \right\rangle_T$.

3.2.   Total degeneracy and non consecutive sites

Let us now consider the standard dNLS model on a lattice ${\mathcal{J}} = \{{-N, \ldots, N}\}$

and take two excited but not consecutive sites, namely $S = \{{-1, 1}\}$. The perturbation $H_1$ can be split in the three parts

The complete degeneracy of the continuation appears in the independence of ${H}^{\sharp}_1$ with respect to the angles

In this case, condition (2.18) further simplifies, being

hence the persistence condition reduces to

with $l$ chosen as either $l = -1$ or $l = 1$, since the two conditions are dependent. Due to ${i}\eta_j = \zeta_j$, it is evident that ${H}^{\natural}_1$ is a real function, and the same holds also for its gradient. We choose $l = -1$, so that

which is a real function since the second addendum is the complex conjugate of the first; in the above and in what follows, we have to keep in mind that

We can also write explicitly

which tells us that only evolutions of the complex configurations $\zeta^{(1)}_{0}$ and $\zeta^{(1)}_{-2}$ are needed. Their equations are (remember that $\zeta$ are the momenta of the canonical variables)

where the different forcing terms depend on the fact that the $0$-site is excited by two neighbours, while the $-2$-site only by one. The solutions, because of the non-resonance condition between the linear frequency $\Omega = 1$ and $\omega$, are then given by

where the Cauchy problems are given by the periodicity conditions $\zeta^{(1)}_{l}(T) = \zeta^{(1)}_{l}(0)$, thus

By inserting the above initial data in the general solutions, one obtains

which provides

Hence, being the above term constant in time, the average provides exactly

where we have used the explicit dependence of $\omega$ with respect to $I^*$; critical points of $F_2$ are only ${\varphi}^*\in\{{0, \pi}\}$, which are non-degenerate, and the IFT applies.

Linear stability immediately follows from $-{\epsilon} F_2'({\varphi}^*)$: we stress that here $\gamma$ does not play any role, and stability is exchanged only by switching from positive to negative values of ${\epsilon}$, hence from attractive to repulsive linear interaction.

Acknowledgments

T. P. thanks his former Master student Sarah Dal Lago and the group of Studitorium of Mechanics in the Department of Physics of the Aristotle University of Thessaloniki, for useful and friendly discussions which at the end of 2018 have motivated the present contribution. T. P. and V. D. have been supported by the GNFM - Progetto Giovani funding "Low-dimensional Invariant Tori in FPU-like Lattices via Normal Forms" and by the MIUR-PRIN 20178CJA2B "New Frontiers of Celestial Mechanics: theory and Applications".

Conflict of interest

The authors declare no conflict of interest.