q-breathers in Discrete Nonlinear Schroedinger lattices

q -breathers in Discrete Nonlinear Sc hr¨ odinger lattices K. G . Mishagin 1 , S. Fla ch 2 , O. I. Kanak o v 1 and M. V. Iv anc henk o 1 , 3 1 Dep artment of R adiophysics, Ni zhny Novgor o d Uni ve rsity, Gagarin Av enue, 23, 603950 Ni zhny Novgor o d, Rus sia 2 Max Planck Institut e for the Ph ysics of Complex S ystems, N¨ othnitzer Str. 38, D-01187 Dr esden, Germany 3 Dep artment o f Applie d Mathematics, University of L e e ds, L e e ds LS2 6JT, Unite d Kingdom (Dated: September 12, 2021) q -breathers are ex act time-p eriodic solutions of exten ded nonlinear systems contin u ed from the normal mo des of the corresp on d ing linearized system. They are lo cali zed in the space of normal mod es. T he ex istence of these solutions in a weakly anharmonic atomic chain explained essential features of th e F ermi-P asta-U la m (FPU) para dox. W e stud y q -breathers in one- t wo- and three- dimensional discrete nonlinear S c h¨ odinger (DNLS) lattices — theoretical playgrounds for ligh t prop- agation in nonlinear optical wa veguide n et works, an d th e dy namics of cold atoms in optical lattices. W e prov e the ex istence of th ese solutions for w eak nonlinearity . W e fi nd that the lo caliza tion of q -breathers is con trolled by a si ngle parameter which d epends on th e n orm density , nonlinearit y strength and seed w a ve vector. At a critical v alue of that parameter q -breathers delocalize via reso- nances, signali ng a breakdown of the normal mode picture and a transition in to strong mo de-mod e intera ction regime. In particular this breakdown tak es place at one of the edges of the normal mod e sp ectrum, and in a singular wa y also in the center of th at sp ectrum. A stabilit y an alysis of q -breathers supplements these findin gs. F or th ree-dimensional lattices, we find q -breather vo rtices, whic h violate time revers al symmetry and generate a v ortex ri ng flo w of energy in n ormal mo de space. P ACS num bers : 63.20.Pw, 63.20.Ry , 05.45.-a I. INTRO DUCTION A. Bac kground In 1955, F ermi, Pasta and Ulam (FPU) published their seminal r ep ort on the absence of thermalization in arra ys of par ticle s connected by w eakly nonlinear springs [1]. In particu- lar they observ ed that energy , initially seeded in a lo w-f requ ency normal mo de of the linear problem with a frequency ω q and corresp onding normal mo de n umber q , sta yed almost completely lo c k ed in a f ew neigh b or mo des in frequency space, instead of b eing distributed quic kly among a ll mo des of the system. The latter expectatio n w as due to the fact, that nonlinearit y do es induce a long- range net work o f in teractions among the normal mo des. ¿F ro m the presen t persp ectiv e, the FPU problem app ears to consist of the fo llowing parts: (i) for certain parameter ranges (energy , system size, nonlinearit y strength, seed mo de n um- b er), excitations appear to sta y exp onen tially lo calized in q -space of the normal mo des for long but finite times [2]; (ii) this in termediate lo calized state is reac hed on a fast time scale τ 1 , a nd equipartition is reac hed on a second time scale τ 2 , which can b e man y o rders of magnitude larger than τ 1 [3]; (iii) tuning t he con trol para meters may lead to a dra stical shortening o f τ 2 , until b oth time scales me rge, and the metastable regime of lo calization is replaced b y a fast relaxation tow ards thermal equilibrium [3], whic h is related to the non- linear res onance o verlap estimate b y Izrailev and Chirik ov [4]. F PU happ ened to compute cases with v alues of τ 2 whic h w ere inaccessible within their computation time. Among man y other in tr ig uing details of the evolution of the in termediate lo calized state, w e men tion the p ossibilit y of ha ving an almost regular dynamics of the few strongly excited mo des, leading to the familiar phenomenon of b eat ing . That b eating manifests as a recurrence of almost all energy in to the orig ina lly excited mo de (similar t o the beating among t w o w eakly coupled harmonic oscillators), and was explained by Zabusky and Krusk a l with t he help o f real space solitons of the Ko rdew eg-de- V ries (KdV) equation, for a particular case of long w a v elength seed modes, and perio dic boundary conditions [5]. The regular dynamics ma y b e replaced b y a w eak c ha otic one up on crossing ye t anot her threshold in con trol parameters [6]. Re- 2 mark ably , t ha t w eak c haos is still confined to the core of the lo calized excitation, leav ing the expo nen tia l lo calization almosts unc hanged, a nd p erhaps influencing most strongly the parameter de p endence of τ 2 . The in terested reader may also cons ult Ref.[7]. B. Motiv ation Single-site excitations in translationally in v ariant lattices of interacting anharmonic os- cillators are know n to show a similar behaviour of trapping the excitation on a few lattice sites around the originally excited one [8]. As conjectured almost 40 years ago [9], ex- act time -p erio dic a nd spatially lo calized orbits - discrete breathers (also in trinsic lo calized mo des, discrete solitons ) p ersis t in suc h latt ice s, and the dynamics on suc h an orbit, as w ell as the dynamics of the nearb y phase space flow , account for man y observ at io ns, and ha v e found their applications in man y differen t areas of phys ics [10]. Recen tly , it was shown, that the regime of lo calizatio n in normal mo de space for the FPU problem can b e equally explained by obtaining q -breathers (QB) - time-p erio dic and mo dal-space-lo calized orbits - whic h p ersist in the FPU mo del for nonzero nonlinearit y [11]. The originally computed FPU tra jectory sta ys close to such q -breat her solutions f or long time s, a nd therefor e man y features of its short- and medium-time dynamics hav e b een shown to b e captured b y the q -breather solution and small phase-space fluctuations aro und it. Delo calization t hresh- olds for q -breathers are related to resonances, and corresp onding o v erlap criteria [11]. The metho d of constructing q -breather solutions w as generalized to t wo- and three-dimensional FPU lattices [12]. Most imp ortan tly , a scaling theory w as dev elop ed, whic h allow ed to con- struct q -breathers for arbitrarily large system sizes, and to obtain analytical estimates o n the degree of lo calization of these solutions for ma cro scopic system siz e [13]. Exp ectatio ns w ere formulated, that the existence of q -breathers should b e generic to man y nonlinear spatially extended systems. C. Aim The sp ectrum of normal mo de frequencies of the linear part o f the FPU mo del is acoustic, con tains zero, reflecting the fact, that the mo del conserv es tota l mec hanical momen tum. In a dditio n it is b ounded b y a latt ice -induced cutoff, t he analog ue of the Deb ye cutoff in solid state ph ysics. W e may think of tw o path w ays of extending the ab o ve discussed results to other sys tem classes. First, w e could consider a s patially contin uous system, instead of a lattice. Ho w ev er, the initial FPU studies show ed confinemen t to a few long wa v elength mo des, and the results of Zabusky and Krusk al [5], confirm, that indeed in the spatially con tinuous system (KdV), tra jectories sim ilar to the FPU tra j ec tory p ersis t. As discussed ab ov e, the p ersistence , lo calization, and delo calization of q -breathers is due to a pro per av oiding of, or harv esting on, resonances b et wee n no rmal mo des. It therefore app ears to b e of substan tial in terest to extend the previous results to systems with a qualitativ ely differen t no r ma l mo de sp ectrum. Suc h a sp ectrum is o ne whic h do es not con tain zero, and is called optical. It may correspond to the excitation of certain degrees of freedom in a complex system, or to an F PU ty p e mo del whic h is dep osited on some substrate, or o therwis e exp osed to some external fields. T ogether with the acoustic band, these tw o band structures, when combined, describ e almost an y ty p e of normal mo de sp ectrum in a linear sy stem with spatially p erio dic mo dula t ed c hara cteristics. Adding w eak nonlinearities to suc h a system, taking lo cal action angle represen tations, and p erforming v arious ty p es of m ultiple scale analysis, suc h mo dels are mapp ed on discrete nonlinear Sc hr¨ odinger mo dels (D NLS) [1 4 ]. These mo dels enjo y a gauge in v aria nce, and a conserv ation of the sum o f the lo cal actions (norms), i.e. some global norm. A pa r t icular 3 v ersion of these equations is kno wn a s the discrete G ross-Pitaevs ky equation, and is derive d on mean fie ld grounds for Bose-Einstein condensates of ultracold b osonic atoms in o ptical lattices [15, 16]. The norm is simply the conserv ed n um b er of a toms in this case, and the nonlinearit y derive s from the a tom-atom in teraction. The propagatio n of lig h t in (spatially mo dulated) optical media is another researc h area where D NLS mo dels are used [1 7 ]. In that case the norm conserv ation de riv es from the conserv ation of electromagnetic w av e en- ergy along the propaga t io n distance (in t he assumed absence of dissipativ e mec hanisms). But DNLS models serv ed equally w ell in man y other areas of ph ysics, where w eak interac- tions star t to play a role, e.g. in the theory of p olaro n formation due to electron-phonon in teraction in solids. Releasing the norm conserv ation (e.g. b y a llo wing the n umber of ato ms in a condensate to fluctuate) will reduce the symmetries of the corresp onding mo del, but most imp ortan tly , it will lead to p ossible new r esonances of higher harmo nics. W e will discuss these limitat ions, and possible effects of releasing these limitations on q -breathers, in the discussion section. A particular consequence of norm conserv ation is a corresp onding symmetry of the normal mo de sp ectrum, whic h relates b oth (upp er and low er) band edges, and mak es the band cen ter a symmetry p oin t as w ell. The pap er is structured as follow s. Sec tion I I is in tro ducing the mo del and the main equations of motion. Sec tion I II give s an existence pro of for q -breathers. In section IV w e deriv e analytical expressions for the QB profiles using p erturbation theory . These are compared with the numeric al results for d = 1 in section V, whic h also con ta ins a stabilit y analysis of t he obta ined p erio dic o r bits. Section VI giv es results on p erio dic b oundary conditions (as oppo se d to fixe d b oundaries). W e generalize to d = 2 , 3 in se ction VI I, and discuss all results in section VII I. II. MODEL W e consider a discrete nonlinear Sc hr¨ odinger (DNLS) equation on a d -dimensional hy - p ercubic lattice of linear size N : i ˙ ψ n = X m ∈ D ( n ) ψ m + µ | ψ n | 2 ψ n . (1) Here ψ is a complex scalar whic h ma y describ e e.g. the proba bility amplitude o f an atomic cloud on an optical lattice site [16], or relates to the amplitudes of a propagating electromag- netic w av e in an optical w av eguide [17]. The lattice v ectors n, m hav e in teger comp onen ts, and D ( n ) is the set of nearest neighbors for the lattice site n . If no t noted otherwise, w e consider fixed b oundary conditions: ψ n = 0 if n l = 0 o r n l = N + 1 for any of the comp onen ts of n . Equation (1 ) is deriv ed fr o m the Hamiltonian H = X n   X m ∈ D ( n ) ψ m ψ ∗ n + µ 2 | ψ n | 4   (2) using the equations of motion i ˙ ψ n = ∂ H /∂ ψ ∗ n . In addition to energy , equation (1) conserv es the norm B = P n | ψ n | 2 . Note that t he change of the nonlinearit y parameter µ in (1) is strictly equiv alen t to c hanging the norm B . Here we will k eep the norm B (alternative ly the nor m densit y) fixed, a nd v a ry µ . W e p erform a canonical tra nsfor ma t io n to t he recipro cal space of normal mo des ( q -space 4 of size N d ) with new v ariables Q q ( t ) ≡ Q q 1 ...q d ( t ) ψ n ( t ) =  2 N + 1  d/ 2 N X q 1 ,...,q d =1 Q q ( t ) d Y i =1 sin  π q i n i N + 1  . (3) T ogether with (1) w e obtain the follo wing equations of motion in normal mo de space: i ˙ Q q = − ω q Q q + 2 d − 2 µ ( N + 1) d X p ,r ,s C q, p ,r ,s Q p Q ∗ r Q s , (4) where ω q = − 2 d P i =1 cos π q i N +1 are the no rmal mo de frequencies for the linearized system (1) with µ = 0. Nonlinearit y intro duce s a net w ork of in tera ctio ns a mo ng the normal mo de oscillators with the follo wing coupling co efficien ts: C q, p ,r ,s = d Q i =1 C q i ,p i ,r i ,s i , C q i ,p i ,r i ,s i = 1 P k ,l,m =0 ( − 1) k + l + m ( δ ( − 1) k p i +( − 1) l r i +( − 1) m s i ,q i + + δ ( − 1) k p i +( − 1) l r i +( − 1) m s i ,q i ± (2 N +2) ) . (5) II I. PROOF OF E XISTENCE OF t -REVERSIBLE q -BREA THER SOLUTIONS FOR WEAK NONLINEARITY W e lo ok for exact time-p eriodic solutions, whic h are stationary solutions of the D NLS equation (1), and whic h a re lo calized in normal mo de space: ψ n ( t ) = φ n exp ( i Ω t ) with frequency Ω and t ime-inde p enden t amplitudes φ n . In the space of normal mo des these stationary solutions hav e the form Q q ( t ) = A q exp ( i Ω t ), where the amplitudes of mo des A q are time-indep enden t and related to the real-space amplitudes b y the transformation (3). A t a give n norm B they satisfy a system of algebraic equations:      − Ω A q + ω q A q − 2 d − 2 µ ( N +1) d P p , r , s C q , p , r , s A p A ∗ r A s = 0 , q 1 ,..,d = 1 , N , P q | A q | 2 − B = 0 , (6) W e are fo cusing here (and throughout almost all of the pap er) on t -rev ersible p erio dic orbits. Therefore w e ma y consider a ll A q to b e real n um b ers. In this case system (6) con tains N + 1 equations for N + 1 v ariables. This system can b e condensed in to an equation for a v ector function: F ( X ; µ, B ) = 0 (7) with X = { . . . , A q , . . . , Ω } . The comp onen ts o f F are the left hand sides of (6), while µ , B are par a mete rs. F o r µ = 0 the normal modes in (4) are decoupled and eac h oscillator conserv es its norm in time: B q ( t ) = A 2 q . L et us consider the excitation of only o ne of the oscillators with the seed mo de n umber q 0 : B q = B q 0 δ q , q 0 . The excited normal mo de is a time-p erio dic solution of (4), and is lo calized in q -space. According to the Implicit F unction Theorem [18], the corresp onding solution of (6) can b e con tin ued in to the nonlinear case ( µ 6 = 0), if the Jacoby matrix of the linear solution ( ∂ F /∂ X ) µ =0 ,B is in v ertible, i.e. || ∂ F /∂ X | | µ =0 ,B 6 = 0. The Jacobian     ∂ F ∂ X     µ =0 = ( − 1) N d +1 2 A 2 q 0 Y q 6 = q 0 ( ω q − ω q 0 ) . (8) 5 In ve rtibilit y of the Jacoby mat r ix requires a non- degene rated sp ectrum o f normal mo de frequencies. Therefore, the contin ua t io n of a linear mo de with the seed mo de num b er q 0 will b e p ossible if ω q 6 = ω q 0 , for all q 6 = q 0 . That condition holds for the case d = 1, and th us q -breathers exist at least for suitably small v alues of µ there. F or higher dimensions, degeneracies of normal mo de frequencies app ear. T hese degeneracies are not an obstacle for n umerical contin uation o f q -breathers, but a formal p ersistence pro of has to deal with them accordingly . Analogous results for tw o- and three-dimensional β -FPU la t t ices ha ve b een obtained in Ref. [12]. IV. PER TURBA TION THEOR Y FOR q -BREA T HER PR OFILES T o ana lyz e the lo calization prop erties of q -breathers in q -space w e use a p erturbation theory approac h similar to [11],[12 ]. W e consider the general case of a d - dime nsional D NLS lattice (1). T aking the solution for a linear normal mo de with num b er q 0 = ( q 1 , 0 , . . . , q d, 0 ) as a zero-order approximation, an asymptotic expansion of the solution to t he first N equations o f (6) in p o we rs of t he small parameter σ = µ 2 ( d − 2) / ( N + 1) d is implemen ted. Note, that in the same wa y as it w a s done in [11],[12], w e fix the amplitude of the seed mo de A q 0 . Later on w e will a pply the nor m conserv atio n la w (the last equation of (6)) to express the norm B q 0 = | A q 0 | 2 via the total norm B fo r the case o f d = 1. Analytical estimations presen ted b elo w describ e amplitudes o f mo des lo cated along the directions of the lattice axes starting from the mo de q 0 . Mo de amplitudes of q -breathers ha ve the slow est deca y along these directions. Studying q -breather lo calization along a c hosen dimension i , for the sak e of compactness w e will use a sc alar mo de n umber to denote the i -th comp onen t of q , assuming all other c omp onen ts b e the same as in q 0 : q j 6 = i = q j, 0 . A. Close to the band edges Let us start with q -breathers lo calized in the low-frequen cy mo de domain ( q i, 0 << N , i = 1 , d ). According to the selection rules, if the seed mo de num b er q i, 0 is ev en (o dd), then only ev en (o dd) mo des are excited along i -th dimension. The n -th order of the asymptotic expansion is the leading one for the mo de q i,n = (2 n + 1) q i, 0 . When q i reac hes the upp er band edge in the i -th direction, a reflection at the edge in q -space tak es place if N + 1 is not divisible b y q i, 0 (cf. F ig. 1d). If N + 1 is divisible by q i, 0 then only mo des q i,n = (2 n + 1) q i, 0 < N are excited. In the analytical estimates we assume a large enough lattice size. Then the effect of band edge reflections is app earing in higher or ders of the p erturbation, whic h will not b e considered. In this case the appro ximate solution is A (2 n +1) q i, 0 = ( − sign( µ )) n  q λ ( i ) d  n A q 0 , i = 1 , d , Ω = ω q 0 − σ A 2 q 0 + O ( σ 2 ) . (9) The corres p onding exp onen tial deca y of mo de norms is B (2 n +1) q i, 0 = ( λ ( i ) d ) n B q 0 , q ( λ ( i ) d ) = | µ | A 2 q 0 ( N + 1) 2 − d 2 5 − d π 2 ( q i, 0 ) 2 = | µ | b k 0 2 5 − d ( k i, 0 ) 2 , i = 1 , d . (10) λ ( i ) d (0 < λ ( i ) d < 1) c hara cteriz e the exp onen tia l deca y along t he i -t h dim ension. Here k i, 0 = π q i, 0 (2 n +1) N +1 < π is the i -t h comp onen t of the seed w av e v ector k 0 , and b k 0 = B k 0 / ( N + 1 ) d is the norm densit y of the seed mode. When using intensiv e quantities – w av e n umber k i, 0 and norm density b k 0 – only , λ do es not dep end on the sys tem siz e. Equations (4 ) are in v ariant under t he symmetry o peration µ → − µ , t → − t , q i, 0 → N + 1 − q i, 0 for all i = 1 , d , whic h changes the sign of nonlinearit y , and maps mo des from 6 one band edge to the other. The replacemen t q i, 0 → N + 1 − q i, 0 for all i = 1 , d in (6) is equiv alen t to substitutions µ → − µ and Ω → − Ω. Using this symmetry , we can easily apply the ab o v e results t o the case of q -breathers lo calized near the upp er band edge, by coun ting mo de indices from the upp er edge: e q i = N + 1 − q i . W e neglect reflections fr o m the low er band edge, suc h that only mo des with nu m b ers e q i,n = (2 n + 1) e q i, 0 ( n is in teger, e q i, 0 << N ) are assumed to b e excited. It follo ws A N +1 − e q i,n = (sign( µ )) n  q e λ ( i ) d  n A q 0 (11) where n is an in teger and q ( e λ ( i ) d ) = | µ | A 2 q 0 ( N +1) 2 − d 2 5 − d π 2 ( N +1 − q i, 0 ) 2 = | µ | b k 0 2 5 − d ( π − k i, 0 ) 2 , i = 1 , d . (12) The analytical expression for the frequency of the q -br eat her solution turns out to b e the same as for the case of small seed mode n um b ers. B. Close to the band center W e implemen t the p erturbation theory approac h for seed mo des close to the band cen ter. Let us first consider the case of o dd N . W e in tro duce the new index p i , whic h is the n umber of a mo de counted fro m the middle of the sp ectrum along the i -th dimension: p i = q i − ( N + 1) / 2. W e c ho ose the s eed mo de with | p i, 0 | < < ( N + 1) / 2 . In the n -th order of p erturbation theory the newly excited mo de along the i -th dimension has the n umber p i,n = ( − 1) n (2 n + 1) p i, 0 , and A p i, 2 n = ( − λ ( i ) d ) n A p i, 0 , A p i, 2 n +1 = sign ( p i, 0 )( − 1) n +1 ( λ ( i ) d ) n +1 / 2 A p i, 0 , (13) where i = 1 , d , λ ( i ) d > 0, n = 1 , 2 , 3 , . . . , and q λ ( i ) d = | µ | A 2 q 0 2 4 − d π | q i, 0 − ( N + 1) / 2 | = | µ | b k 0 2 4 − d | k i, 0 − π / 2 | . (14) When using in tensiv e quan tities, λ again do es not de p end o n the sy stem s ize. F o r the case of ev en N w e use p i = q i − N / 2 a nd assume t ha t the seed mo de index p i, 0 > 0, i = 1 , d . The s et o f consecutiv ely p erturb ed mo des b ecomes more complicated: p i, 0 → ( − 3 p i, 0 + 2) → (5 p i, 0 − 2) → ( − 7 p i, 0 + 4) → (9 p i, 0 − 4) → ( − 11 p i, 0 + 6) → . . . . (15) F o r an ev en num b er o f p erturbation steps the new excited mo de along the i -th dimension has index p i, 2 n = (4 n + 1) p i, 0 − 2 n , and for an o dd n um b er of steps it ha s index p i, 2 n +1 = − (4 n + 3) p i, 0 + 2 n + 2, n = 0 , 1 , 2 , . . . . The amplitudes satisfy (13), but with q λ ( i ) d = | µ | A 2 q 0 2 4 − d π | q i, 0 − N / 2 | = | µ | b k 0 2 4 − d | k i, 0 − π / 2 + π / (2 N + 2) | . (16) F o r large enough N equation (16) approac hes the expression (14), therefore w e will use (14) for lar ge N only . V. q -BREA THERS FOR d = 1 : RESUL TS A. Close to the band e dge The k ey prop ert y of q -breathers is that they are lo calized in the space of linear normal mo des. Note that some q -breat her solutions may be compact in q -space and con tain only 7 one seed mo de q 0 [19], or a f ew mo des additionally , due to symmetries o f the in teraction net work spanned b y the nonlinear terms [20]. 0 0.1 0.2 0.3 0.4 0.5 −30 −20 −10 0 k/ π log 10 B k N=32 N=64 N=128 (a) 1 0.9 0.8 0.7 0.6 0.5 −30 −20 −10 0 k/ π log 10 B k N=32 N=64 N=128 (b) 0 0.5 1 −30 −20 −10 0 k/ π log 10 B k N=33 N=65 N=129 (c) 0.2 0.4 0.6 0.8 −30 −20 −10 0 k/ π log 10 B k (d) FIG. 1: Distribu tions of mo de norms for QBs in the one-dimensional DNLS mo del with parameters: B = 1, µ = 0 . 1 for (a), (b), (c); (a) QBs with lo w-frequency seed mo de q 0 = 1, (b) QBs with high-frequency seed mod e q 0 = N , (c) QBs with seed mod e near the midd le of t he sp ectrum q 0 = 1 + ( N + 1) / 2 (c). Dashed lines in (a), (b), (c) corresp ond to analytical estimations. (d) Ob serv ation of m ultiple reflections at the b oundaries of th e q -space, N = 31, B = 1, µ = 2, q 0 = 3. Dashed lines in ( d) are guidelines for the eye. 8 0 0.01 0.02 0.03 −150 −100 −50 0 k 0 / π S(k 0 ) µ =0.5,1,2 (N=1024) µ =−0.5,−1,−2 (N=1024) theory FIG. 2: The slop e S as a function of the se ed w av e num b er k 0 for d = 1 with fixed norm density b = 1 / 1025, µ = ± 0 . 5 , ± 1 , ± 2 (from b ottom to top). Dashed lines correspond t o the analytical estimates. Differen t sy m b ols: slope from the numerical calculation of QB (sq uares an d circles represent the results for N = 512). Solid lines guide the eye. Let us consider the one-dimensional case. W e compute q -br eat hers as the stationary solutions of the nonlinear equations (6) using the single-mo de solution fo r µ = 0 as an initial appro ximation. Fig. 1a,b shows t he distribution of mo de norms for q -breathers in the space of w av e n um b ers k = π q / ( N + 1 ) for differen t c ha in sizes and different seed w av e n umbers, lo cated near the lo w er ( k 0 << π , F ig. 1a) and upp er ( π − k 0 << π , Fig. 1b) edges of the linear mo de sp ectrum. W e find exp o nen tia l lo calization of q -breathers, with a lo calization length whic h depends strongly on the chosen parameters. The obtained analytical estimations for q -breathers lo calized at the low er band edge (9),( 10) resp ectiv ely upp er band edge (11),(12), are in quan titat ive agr eemen t with num erical results (see dashed lines in Fig. 1 a,b). The exp onen tial deca y in (10) is dep endin g on the seed mo de norm densit y . Many appli- cations (e.g. cold atoms in a condens ate) rather fix the total norm, or total norm den sit y . F o r the obtained q -breather solutions with q 0 << N , the relation b et w een these quantities can b e estimated by the sum of infinite geometric s eries: B ≈ ∞ X n =0 λ n B q 0 = B q 0 1 − λ , (17) where λ ≡ λ (1) 1 . F rom ( 17) it follow s that b k 0 = (1 − λ ) b , where b = B / ( N + 1) is the total norm densit y . Substituting the expression for b k 0 in to (10) and solving the equation for √ λ w e obtain: √ λ = √ 1+4 ν 4 /k 4 0 − 1 2 ν 2 /k 2 0 , ν 2 = | µ | b 16 . (18) 9 The same dep endence of √ λ on k 0 w as obta ined for lo w frequency q -breathers in the β -FPU mo del [1 3]. The exp onen tial deca y of mo de norms in the space of w a ve n um b ers, can b e no w written for q -breathers localized in low -frequency mo des: ln b k =  k k 0 − 1  ln √ λ + ln b k 0 . (19) T o c haracterize the degree of lo calization in k -space, we use the slop e of the profile of mo de norms in log-no r ma l plots (19) – S [13], where the absolute v alue of S is equiv alent to the in ve rse lo calization length ξ : S = 1 k 0 ln √ λ , | S | ≡ ξ − 1 . (20) Substituting the expression for √ λ (18) in to (20) we obtain: S = 1 ν z ln  p 1 + z 4 / 4 − z 2 / 2  , z = k 0 /ν . (21) Therefore the slop e (in v erse lo calization length) is a function of the rescaled wa v enum b er z . It therefore parametrically dep ends o n just one effectiv e nonlinearity pa rameter ν , whic h is giv en by the pro duct of the total nor m density and the a bsolute v a lue of nonlinearity strength. S v anishes for z → 0 , and it has an extrem um min( S ) ≈ − 0 . 7432 /ν at z min ≈ 2 . 577. The w av e n um b er k 0 = k min ≡ z min ν corresp onds to the strongest lo calization of a q -breather with fixed effectiv e nonlinear parameter ν . With increasing ν the lo calization length increases. Most imp ortan tly , the lo calization length dive rges fo r small k 0 ≪ k min since | S | ≈ z / (2 ν ) in t ha t case. This delo calization is due to resonances with nearb y nor ma l mo des close to the band edge. No t e, that our analytical estimations do not dep end on the sign o f the nonlinearity parameter µ . In F ig. 1a,b, due to the small sizes of the c hain, all v alues of k 0 are greater than k min , so a monotonous dep endence of the slop e on k 0 is observ ed. In Fig. 2 w e plo t theoretical and n umerically o btained dep endenc ies S ( k 0 ) for differen t v alues of the nonlinear param- eter µ and differen t system sizes N (we use large enough N to resolv e the theoretically predicted extrem um of S ). In all cases the nu merical r esults show, that the slop es indeed are c har a cteriz ed b y in t ens iv e quan tities only , and the ab o v e deriv ed scaling la ws hold. F o r p ositiv e v alues of µ and seed w av e n umbers close to the low er band edge, the extrem um in S is repro duced in the num erical data, tho ugh the nume rical curv es deviate from theo- retical curv es for small k 0 . W e also find that k min increases and | S ( k min ) | decreases with increasing µ as it is predicted b y o ur analytical results (increase of norm densit y b g ives the same effect). F o r negative v alues of µ and seed w a ve num b ers close to the low er band edge, w e do not obse rv e an ex trem um for S . In the region o f small k 0 the n umerically obtained curv es for positive and ne gativ e v alues of µ differ, while the t heoretically predicted slop es do not dep end on the sign of µ . Fig. 3 s ho ws the mo de norm pro files of q -breather solutions with small k 0 obtained for p ositiv e and negativ e v alues of µ . The reason for t he disc repancy b e- t wee n theoretical prediction and n umerical results for negativ e v alues of µ mus t b e strong con tributions from higher order terms in the p erturbation expansion. The standard argu- men t is, that the p erturbation theory is v a lid if the lo calization length is small, i.e. | λ | ≪ 1 as w ell. When | λ | becomes of the order of one, higher or der terms in perturbatio n theory ha ve to b e tak en in to accoun t, and it w ould b e tempting to conclude that delo calization will tak e place. That should b e true esp ecially when all higher order terms in the series carry the same s ign. That is the c ase for positive nonlinearity here, but for ne gativ e µ w e obtain alternating signs o f higher order terms. These alt ernat ing signs therefore e ffectiv ely 10 0 0.05 0.1 0.15 0.2 −8 −6 −4 −2 0 k/ π log 10 B k µ =2 µ =−2 FIG. 3: Distributions of mo de norms for QBs in the one-dimensional DNLS mo del for p ositiv e and negativ e v alues of µ ; q 0 = 2, N = 1024. 0 5 10 15 −1 −0.8 −0.6 −0.4 −0.2 0 z S m (z) FIG. 4: The master slop e function S m ( z ) (d as hed line). Different symbols and eye-guiding solid lines correspond to the scaled numerical estimates of t he slope presented in Fig. 2 (bottom line — µ < 0, top line — µ > 0). cancel most of the terms in the series, and are responsible for an increasing lo calization of q -breathers with negativ e µ in the limit of small w a v en um b ers. The obta ined results fo r q - breat hers with seed w a ve num b ers close t o the lo wer band edge ( k 0 << π ) are v alid for the case of k 0 close to the upp er band edge ( π − k 0 << π ) if w e c hang e µ → − µ . Th us, there is a strong asy mmetry in the lo calization prop erties o f q -breather solutions with k 0 < π / 2 and k 0 > π / 2 for a fixed sign of nonlinearit y . The num erical results in Fig. 2 sho w that slop e v alues calculated f o r differen t system sizes lie on the same curv es with correspo nding µ ev en in the r egio n of small k 0 , where higher order corrections to our analytical estimates ha ve to b e tak en in to a ccount. This result is in agreemen t with the exact scaling of q -breather solutions describ ed in [13 ]. W e plot in F ig. 4 t he master slop e function S m ( z ) = ν S , which dep ends on a single v ariable 11 z [13]. It implies that kno wing this single master slop e function is suffic ien t to predict the lo calization prop erty of a q -breather at a ny se ed w a ve n umber k 0 ≪ π , a t a ny ene rgy etc. Numerically obta ined slop es presen ted in Fig. 2 are rescaled and plotted in Fig. 4. W e see, that all results corresp onding to the same sign of µ condense on a single curve ev en (and esp ecially) for small k 0 , though these n umerical results differ from the analytical estimation. B. Close to the center of the band Fig. 1c shows the distribution of mo de norms for q -breathers in the space of w av e n um b ers k = π q / ( N + 1) for differen t c hain sizes and differen t seed w a ve n um b ers, lo cated close to the cen ter of the sp ectrum ( | k 0 − π / 2 | << π ). The analytical estimation for the amplitudes of these q -breathers are in go o d quan titative agreemen t with the n umerical results, for small enough pa rameters µ and b , cf. Fig. 1c. W e express again the norm densit y o f the seed mo de via the total norm densit y: b k 0 = (1 − λ ) b , ( λ ≡ λ (1) 1 ). Substituting this e xpression in to (14) w e find: √ λ = √ 1+4 ν 2 / ( k 0 − π / 2) 2 − 1 2 ν / | k 0 − π / 2 | , ν = | µ | b 8 . (22) The s lop e of the mo de no r m pro file in k -space is given b y S = 1 | k 0 − π / 2 | ln √ λ = 1 2 ν z ln  √ 1 + z 2 − z  , (23) where z = | k 0 − π / 2 | / (2 ν ). F or z << 1 : S = ( − 1 + z 2 / 6 + O ( z 4 )) / (2 ν ), for z >> 1: S = ln(1 / ( 2 z ) + O (1 /z 3 )) / (2 ν z ). In the limit z → 0, the slop e S → − 1 / (2 ν ). Th us, the strongest lo calization of a q -breather with fixed effectiv e nonlinear parameter ν should b e obtained for wa v e n umbers k 0 ≈ π / 2. The increase o f the effectiv e nonlinearity parameter ν ∼ | µ | b leads to a w eak er lo calization of q -breathers in k -space, since the absolute v alue of the slop e S decreases. Not e, that there is a special p oint for o dd N : k 0 = π / 2 , q 0 = ( N + 1) / 2. F or this see d mo de, the q - breather is compact in k -space [19]. In Fig. 5 the t heoretical and nume rically obtained dep endencies S ( k 0 ) for different v alues of µ and different system sizes N are plotted. F or small v alues of µ w e observ e go o d agreemen t b et w een analytical a nd n umerical results. But the increase of the nonlinearity leads to a deviation betw een the theoretical a nd n umerical curv es in the region of k 0 close to π / 2. These corrections, as it was for the case of q -breathers lo calized near the band edges, dep end on the lo cation of k 0 ( k 0 > π / 2 or k 0 < π / 2) and on the sign of nonlinearity . Therefore the curv es of S ( k 0 ) for strong nonlinearity ( µ = 30) in Fig . 5 a r e non-symmetric around the p oin t k 0 = π / 2: in con trast to k 0 < π / 2, for k 0 > π / 2 a lo cal minimum of S is observ ed. Still, the predicted scaling prop erties of q -breathers remain correct ev en for strong nonlinearity : the v alues o f S , computed for differen t system sizes N , lie on the same curv es for fix ed µ . C. Stability of q -breathers W e analyze the linear stabilit y of q - breathers as statio nary solutions of DNLS mo del con- sidering the ev olution of small p erturbations ε n in the rotating frame of the p erio dic solution [22]: ψ n ( t ) = ( φ 0 n + ε n ( t )) exp( i Ω t ), where φ 0 n are the non-p erturb ed time-indep enden t am- plitudes, ε n = α n + iβ n . A p erio dic or bit is stable when all p erturbations do not grow in time. Solving the linearized equations for the p erturbation, that condition t r a nslates in to the request, that all eigen v alues s m ( m = 1 , 2 N d ) of the linearized equations m ust b e purely imaginary . Otherwise the orbit is unstable. In Fig. 6 we plot the n umerical outcome of the 12 0.48 0.49 0.5 0.51 0.52 −10 3 −10 2 k 0 / π S(k 0 ) N=512 N=1024 theory FIG. 5: The slop e S as a function of the se ed w av e num b er k 0 for d = 1 with fixed norm density b = 1 / 1025, µ = 0 . 5 , 5 , 30 (from b ottom to top). Dashed lines correspond to the analytical estimates. Different symbols correspond to the estimates of the slop es from numerica l calculations of QBs. stabilit y analysis. W e smo othly con tinue q -breather solutions for each seed mo de nu m b er q 0 b y increasing the no nlinearity parameter µ and c hec k the stabilit y . If the maxim um absolute v alue o f the real parts of all eigen v alues is smaller than 10 − 6 , the q -breather is considered as stable (solid line), otherwise it is unstable (dashed line), crosses mark the c hange o f stability . F or small v alues of no nlinearit y all q -breathers are stable. Qualitativ ely differen t threshold v alues and dep enden cies on q 0 for q -breathers with seed mo des from differen t parts of the sp ectrum q 0 < N / 2 and q 0 > N / 2 are observ ed. This is in a greeme n t with the results presen ted in [21], where stability prop erties of nonlinear standing wa v es are studied. Note, that due to the symmetry of the equations, the stability prop erties o f a q -breather with seed mode q 0 for some ne gativ e v alue of the nonlinearit y parameter µ is the same as the stability prop ert y of the q -breather w ith seed mo de e q 0 = N + 1 − q for the nonlinearit y parameter e µ = − µ . Let us disc uss the p ossible link betw een linear stabilit y and lo calization. If a q -breather b ecomes delocalized, t hat ha pp ens b ecause of resonances b et w een different mo de freque n- cies. Therefore w e can exp ect, that the same resonances will also drive the state uns table. Indeed, these cor r elations can b e clearly observ ed from the nume rical data. How eve r, if a q -breather is we ll lo calized, it do es not follow that it will b e stable as w ell, sinc e instability can a rise due to resonan t in teraction of mo des in the breather core alone. 13 10 −2 10 0 10 20 30 40 50 60 µ q 0 FIG. 6: Domain of stabilit y of q -breathers for d = 1. Solid lines - stable, dashed - u nstable, crosses - switch from stable to unstable or vice versa. N = 64, B = 2. VI. PERIODIC B OUND AR Y CONDITIONS In the case of perio dic b oundary conditions, we hav e used the following t r ansformations b et w een real space and the recipro cal space of norma l mo des (for eve n N ): ψ n ( t ) = 1 N d/ 2 N/ 2 P q 1 ,...,q d = − N/ 2+1 Q q ( t ) d Q i =1 exp  2 π q i ( n i − 1) N  . (24) The DNLS mo del ( 1 ) with p erio dic b oundary conditions has exac t solutions for nonlinear tra v eling w a v es, whic h can b e written, for instance in case d = 1, a s: ψ n ( t ) = φ 0 exp( i Ω t − ik 0 n ), where Ω = − 2 cos k 0 − µφ 2 0 , k 0 = 2 π q 0 / N , q 0 ∈ [ − N/ 2 , N / 2]. These types of solutions can b e considered as compact q -breathers whic h contain o nly o ne mo de q 0 . T r a v eling mo des are also not inv arian t under time rev ersal. The con tinuation of a linear standing w av e, consisting of tw o trav eling w av es with the same norms and w a v e n umbers: k 0 and − k 0 , in to the nonlinear regime leads to a time-rev ersible q -breather solution (see Fig . 7), whic h is not compact, and its lo calization prop erties are similar to the prop erties of q -breathers in the case of fixed b oundary conditions. Here w e pres en t the result for deca y of mo de norms λ ( i ) d for the case | k i, 0 | << π , whic h differs from (10) by a prefa cto r : q ( λ ( i ) d ) = | µ | A 2 q 0 N 2 − d 32 π 2 ( q i, 0 ) 2 = | µ | b k 0 8( k i, 0 ) 2 , i = 1 , d , (25) where b k 0 = B k 0 / N d , k i, 0 = 2 π q i, 0 / N . Fig. 7 illustrates go o d quantitativ e agreemen t b et w een analytical and n umerical results obtained for small enough v alues of norm and nonlinearit y . 14 −0.5 0 0.5 −30 −20 −10 0 k/ π log 10 B k FIG. 7: Time-reversi ble q -breather in th e one-dimensional DNLS mo del with p eriodic b oundary conditions, con- tinued from the linear standing wa ve consisting of t w o tra veling wa ves with q 0 = 2 and q 0 = − 2, N = 64, B = 1, µ = 0 . 1. Dashed lines represent analytical estimations. VII. q -BREA THERS IN TWO- AND THREE-DIMENSIONAL LA TTICES F o r tw o - and three-dimensional s ymmetric DNLS lattices (1) with fixed b oundary condi- tions, only linear mo des with mo de n umbers q on the main diagonal ha v e non-degenerate frequencies. Using the Implicit F unction Theorem [18], it fo llo ws that these mo des are con tinued in to the nonlinear regime. How ev er, w e also success fully p erformed num erical con tinuations of q -breathers with seed mo de num b ers off the main dia gonal as it w a s done in [12] for the F PU mo del. F or the tw o- dimens ional D NLS mo del the q - breather, con tinued from the single linear mo de q 0 = (2 , 3), is presen ted in Fig. 8 in real space ( a ) and in q -space (b). The q - breather with the seed mo de q 0 = (3 , 2) exists a s w ell and has the same frequency . The slow est decay of the mo de norms happens to b e along the direction of the main axes. The deca y is exp onen tia l and in go o d agreemen t with the analytical estimation (10) for d = 2. In addition to suc h asymmetric single mo de q -breathers it is p ossible to construct multi-mo de q -breathers contin ued from a pair of degenerate linear no rmal mo des ( q 0 , p 0 ) and ( p 0 , q 0 ) ( q 0 6 = p 0 ) with the same norms in b oth mo des and in-phase (Fig . 8c) or an tiphase (Fig. 8d) osc illations. It is imp ortant to note that the problem o f degenerate frequencies is av o ided for t hese solutions. Indeed, system (4) has t w o in v a r ia n t manifolds Q q 1 ,q 2 = ± Q q 2 ,q 1 . Lo oking for a solution on a manifold, the n um b er of indep enden t v ariables of stat e is reduced from N 2 to t he dimensionalit y of the manifold, which equals ( N 2 + N ) / 2 for the symmetric manifold and ( N 2 − N ) / 2 for the antisy mmetric o ne. The reduced system of equations con tains only modes with non-degenerate frequencies . F o r d = 3 w e ha ve also ve rified, that the analytical estimations of mo de norm deca y (10), (14) a gree w ell with the results of n umerical calculations for single-mo de q -br eat hers. In addition to v arious time-reve rsible q -breather solutions, whic h are constructed in the same wa y as for d = 2, the three-dimensional DNLS mo del allows also f or non-time- rev ersible (“v ort ex”) multi-mo de solutio ns . Let us consider q -breather solutions on an in v ariant manifold of the system (4 ): Q q 1 ,q 2 ,q 3 = exp( 2 π i/ 3) Q q 3 ,q 1 ,q 2 = exp( 4 π i/ 3) Q q 2 ,q 3 ,q 1 (note, that this manifold has a counterpart with the opposite s ign of the phase shifts). On the manifold the n umber of v ariables of state is reduced to ( N 3 − N ) / 3. W e hav e constructed n umerically a v ortex q -breather solution con tinued fro m a degenerate triplet of seed mo des whic h hav e the same nor m and a relativ e pha se shift 2 π / 3: Q q 0 ,q 0 ,p 0 = exp (2 π i/ 3) Q p 0 ,q 0 ,q 0 = 15 10 20 30 10 20 30 −0.05 0 0.05 n 1 n 2 A n 1 n 2 (a) 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 k 1 k 2 −40 −35 −30 −25 −20 −15 −10 −5 (b) 0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k 1 k 2 −35 −30 −25 −20 −15 −10 −5 (c) 0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k 1 k 2 −40 −35 −30 −25 −20 −15 −10 −5 (d) FIG. 8: Different q -breather modes for d = 2: N = 32, B = 1, µ = 0 . 5. Mod e norm magnitude is plotted in color co de on logarithmic scale ( except for (a), where a linear scale is used). (a), (b) — asymmetric single-mo de q -breather with seed mode ( q 0 = (3 , 2)) in real space (a) and in q -space (b); (c), (d) — symmetric multi-mode q -breather with in-phase (c) and antiphase (d) seed mod e pair q = ( 3 , 1) , (1 , 3). Dashed line in (b) guides th e eye along the main diagonal. exp(4 π i/ 3) Q q 0 ,p 0 ,q 0 , q 0 6 = p 0 . The frequency o f suc h a triplet b ecomes non-degenerate in the reduced system o n the manifold. The energy flo ws in a v ortex-lik e manner in q -space fo r suc h excitations (Fig.9). VII I. D ISCUSSI ON Comparing the aims w e form ula t ed in the in tro duction with the ab o v e results, we can conclude, that indeed the q - breat her concept turns out to b e generic, and time-p erio dic or- bits, whic h are lo calized in normal mo de space, are probably as generic a s discrete breathers (although p erhaps in a differen t w ay). Along with the study of the details of q - breathers prop erties in DNLS mo dels, w e found some particularities, whic h w ere not (y et) obtained in a coustic (FPU) s ystems. Let us discuss some of these and other findings from ab o ve. 16 0 5 0 5 0 2 4 6 8 q 1 t=0 q 2 q 3 0 5 0 5 0 2 4 6 8 q 1 t=T/3 q 2 q 3 0 5 0 5 0 2 4 6 8 q 1 t=2T/3 q 2 q 3 −3 −2 −1 0 1 2 3 FIG. 9: A vor tex q - breather state for d = 3: N = 8, B = 1, µ = 0 . 1, at some fixed time. Mode norm magnitude is encod ed in sphere sizes on logarithmic scale, while the sphere color denotes the phase of the mode. Seed mo de triplet: q = (3 , 1 , 1) , (1 , 3 , 1) , (1 , 1 , 3). A. Scales, delo calization thresholds, and healing length of a BEC Let us consider a finite D NLS c hain with N sites. L et us fix the tot a l norm density b and the no nlinearit y µ . Therefore we fix the effectiv e nonlinearity pa r ameter ν 2 = | µ | b/ 16 (18). If the system size was large enough, w e will resolv e the extrem um in S , and therefore the QB with seed w a v e num b er k min ≈ 2 . 577 ν is the strongest lo calized one. It eviden tly sets an inv erse length scale ν . This length scale is know n in the GP equation for a BE C in a trap with (similar) fixed b oundary conditions. One has a condensate, whose a mplitude v anishes at some b oundary , y et getting bac k to s ome me an v alue a w a y from the b oundary - exactly at the healing length ξ h = (4 π an ) − 1 / 2 , where n is the condensate densit y , and a is the scattering length whic h is proport io nal to the atom- atom in tera ctio n, a nd therefore to the nonlinearity strength µ within the mean field GP equation [16]. Therefore, the in verse healing length corresp onds t o the w av e num b er scale, on whic h the most strongly localized QB is observ ed. W e now increase the system size further, and compute the lo calization length (o r resp ec- tiv ely its negative in v erse - the slop e S ) of the longest w av elength mo de. With increasing N the grid o f allo w ed k -v alues becomes denser, and at some critical N c ( ν ) the localization length will reac h the finite size of the normal mo de space, and the q -breather delo calizes. With a little algebra it follows N c ≈ π 2 2 ν 2 , k ( c ) 0 ≈ 2 ν 2 π . (26) F o r ev en lar ger (and finally infinitely large) lattices the critical v alue k ( c ) 0 is marking a 17 b order in k -space: at the giv en norm densit y ν , all QBs with seed w av e n umbers k ≫ k ( c ) 0 are lo calized, while w e obtain delo calization for k ≤ k ( c ) 0 . So there is a lay er of delo calized QBs at the band edge, whose width grows according to (2 6) with growing ν . Mo des launc hed inside this la y er (with the giv en norm densit y) will quic kly spread their energy among many other modes - they w ill quic kly relax. Mo des launc hed outside this lay er will sta y lo calized in normal mo de space, at least for sufficien tly long times. k ( c ) 0 is therefore separating a la y er of strongly in teracting mo des from we akly interacting ones. F or large enough effectiv e nonlinearit y parameter ν ∼ 1 the whole w a v e n um b er space is filled with strongly interacting normal mo des, and the normal mo de picture breaks dow n c ompletely . As lo ng as ν is smaller, the v alue o f k ( c ) 0 sets a new length scale 2 π / k ( c ) 0 , whic h is prop ortional to the squared healing length ξ 2 h . On that new length scale, the normal mo de picture breaks do wn. B. Delo calization thresholds and mo dulational instability It is instructiv e to remem b er, that delo calization thresholds of QBs are related to reso- nances. Indeed, in the prese n t case, the densit y of states at the band edge div erges in the limit of large system size, and man y mo des hav e a lmost the same frequencies. It is these small differences, whic h tend to zero as 1 / N 2 , and which are resp onsible fo r the resonan t mo de-mo de in teraction. No t ably the analysis of stability of band edge mo des [23] (not e: for p erio dic b oundary conditions) sho ws some in teresting correlations. The analyzed band edge mo des are compact in normal mo de space, y et they undergo a tangen t bifurcation at amplitudes, whic h are the smaller, the larger the system size. These instabilities app ear ho wev er only for a certain sign of t he nonlinearit y , whic h ex actly corresp onds to the actual observ ed delo calization of QBs. Therefore w e expect, that the (so fa r not studied) case of a FPU c ha in with negat ive quartic nonlinearit y , whic h is known not to yield an instabilit y for the band edge mo de, will not sho w delo calization of QBs close to the (upp er) band edge. It is furthermore instructiv e, that if a tangen t bifurcat io n of a band edge mo de ta kes place, the sim ulation of a lattice sho ws the onset of mo dulational instabilit y , whic h leads to a collection of energy in smaller system volumes , and finally to the formation of discrete breathers - i.e. to lo calization in real space. Th us w e may ex p ect, that the resonan t lay er of strongly interacting mo des may lead to the formatio n of lo calized states in real space, while the res t of the no rmal mo de space is evolv ing in the regime of lo calization in normal mo de space. Th us, w e ma y expect to observ e in an actual sim ulatio n lo calization b oth in normal mode s pace and in real space. C. Comparing analyti cal and n umerical resul ts While the delo calization at one band edge, as predicted by p erturbation theory , is repro- duced b y n umerical data , that do es not ha pp en a t the second band edge (both ba nd edges c hange their places, when the nonlinearit y in verts sign). The breakdow n of p erturbation theory follows from comparing the leading and next-to-leading order terms in the expansion. When terms are o f the same order, we conclude, that the se ries will div erge, and therefore the QB s olutions will delocalize. This is tr ue for the case when all terms in the se ries hav e the same sign. How ev er, when the terms ha ve alternating sign, the ab ov e conclusion m ust not b e correct. And indeed the numerical dat a sho w, tha t these sign alternatio ns lead to an effectiv e cancelation, and a final strong lo calization of a QB solution. A similar breakdown of p erturbation theory happ ens for QB solutions lo calized at the band cen ter for large no n- linearit y (or nor m). The p erturbation theory tells, that QB states are w ell lo calized from 18 b oth sides of the cen ter. Numerical data how ev er sho w, that this is true only on o ne s ide from the cen ter, while the other side sho ws a tendency to w ards delocalizatio n. D. Op en questions Belo w w e list some p oten tia lly interesting and imp ortan t op en questions. First, t he influence of differen t b oundary conditions has not b een systematically studied. QBs are extended states in real space, therefore their spectrum, and the w ay they interact, ma y to some extend b e sensitiv e to the c ho ice of boundary conditions. Second, it remains completely op en, what kind of excitation w e catc h by considering a v ortex state in normal mo de space . Finally , it w ould b e in teresting to see the c hanges in the QB prop erties , when the norm conserv ation is lifted. In general w e exp ect, that this will lead to the generation of higher harmonics, and new ty p es of resonances. In fact it is exactly resonances due to higher harmonics, whic h give the leading order contribution to the FPU problem. How ev er, for sufficien tly narrow optical ba nds , higher harmonics will b e lo cated outside the optical band. These res onances will be therefore impor t an t, when the optical band is wide, or whe n one considers a complex and bro ader band structure. Ac kno wledgemen t s P art o f this w or k w as completed while SF was visiting the Institute for Mathematical Sciences, National Univers it y of Singap ore in 20 0 7. KM, OK and MI ackno wledge s upp ort from RF BR, gran ts No. 06-02- 1 6499, 07-02- 01404. [1] E. F ermi, J. Pasta, and S. Ulam, Los Alamos Rep ort LA-1940, (1955); also in: Collected P ap ers of Enrico F ermi, ed. E. Segre, V ol. I I (Universit y of Chicago Press , 1965 ) p.977 -978; Many-Body Pro blems, ed. D. C. Mattis (W orld Scientific, Singap ore , 1993) [2] L. Galgani and A. Scotti, Phys. Rev. Lett. 28 , 1173 (1972). [3] L. Berc hialla, A. Giorgil li and S. P aleari, Phys. Lett. A 321 167 (2004). [4] F. M. Izrailev and B. V. Chiriko v, Sov. Phys. Dokl. 11 , 30 (1966). [5] N. J. Zabusky and M. D. Krusk al, Phys. Rev. Lett. 15 , 240 (1965). [6] J. D e Luca, A . Lich tenb erg and M. A. Lieberman, Chaos 5 , 283 (1995). [7] J. F ord, Phys. Rep. 21 3 , 271 (19 92); F o cus issue, The F ermi-Pasta-Ulam P roblem - The First Fift y Y ears , edited by D. K. Campb ell, P . Rosenau and G. M. Zaslavsky , Chaos 15 N o. 1 (2005). [8] A. J. S iev ers and S. T akeno, Phys. Rev. Lett. 61 , 970 (1988); T. Dauxois, M. P eyrard and C. R. Willis, Physica D 57 , 267 (1992); S. Flach and C. R. Willis, Phys. Lett. A 181 , 232 (1993). [9] A. A . Ovchinnik o v, Sov. Ph ys. JETP 57 , 147 (1970). [10] R. S . MacKay and S. A ubry , Nonlinearity 7 , 1623 (1994); S. Aubry , Physica D 103 , 201 (1997); S. Flach and C. R. Willis, Phys. R ep. 295 , 192 (1998); T. Dauxois, A . Litv ak- Hinenzon, R. S. MacKa y and A. S panoudaki (Eds), Energy lo calization and transfer , W orld Scientific, Sin gap ore (2004); S. Flach and A. V . Gorbach, submitt ed to Phys. Rep. (2007). [11] S. Flach, M. V. Iva nchenk o and O. I. Kanako v, Phys. R ev. Lett. 95 , 064102 (2005); S. Flach, M. V. I v an chenko and O. I. Kanako v, Phys. Rev. E 73 036618 (2006); T. Penati an d S. Flac h, Chaos 17 , 023102 (2007); S. Flac h and A. Ponno, Physica D, in print, arXiv 0705.0804 [nlin.PS] (2007). [12] M.V. I v anchenko, O.I. Kanako v, K.G. Mishagin, and S. Flac h , Phys. Rev. Lett. 97 025505 (2006). [13] O.I. Kanako v, S .Fl ac h, M. V. I v an chenko, and K.G. Mishagin, Phys. Lett. A 365 , 416 (2007); S. Flach, O.I. Kanako v, K.G. Mishagin and M.V. Iv anchenko , Int. J. of Modern Ph ys. B 21 , 3925 (2007). [14] Y. S. Kivshar and M. P eyrard, Phys. Rev. A 46 , 3198 (1992); M. Johansson, Physica D 216 , 62 (2006). [15] L. P . Pitaevskii, Sov. Phys. JETP 13 , 451 (1961); E. P . Gross, Nuov o Cimento 20 , 454 (1961). [16] O. Morsch and M. Ob erthaler, R ev. Mo d. Phys. 78 , 179 (2006). [17] Y u. S. Kivshar and G. P . Agra w al, Op tical solitons: from fib ers to p h otonic cry stal s , Elsevier Science, Amster- dam (2003). [18] J. Dieud onne, F oun dations of mo dern analysis , Academic, New Y ork (1969). [19] q -b reathers with seed linear mod es q 0 i = ( N + 1) / 3,( N + 1) / 2, 2( N + 1) / 3 ( i = 1 , d ) are compact in q -space b ecause th ese seed mo des do not excite th e other mo des due to the selection rules (5). [20] R. L. Bivins, N. Metrop olis and J. R . Pasta, J. Comput. Phys. 12 , 65 (1973); G. M. Chechin, N. V. Novik o v a and A. A. Abramenko, Physica D 166 , 208 (2002); B. Rink , Physica D 175 , 31 (2002). 19 [21] A.A. Morgante, M. Johansson, G. Kopid akis, and S. A ubry , Ph y sica D, 162 , 53 (2002); M. Johansson, A .M. Morgan te, S. Aubry , and G. Kopidakis, Eur. Phys. J. B 29, 2792 83 (2002) [22] A.V. Gorbach and M. Johansson, Eur. Phys. J. D, 29 , 77 (2004). [23] N. Budin sk y and T. Bountis, Physica D 8 , 445 (1983); K. W . Sandusky and K. B. Pa ge, Phys. Rev. B 50 , 866 (1994); S. Flach, Physica D 91 , 223 (1996); J. Dorignac and S. Flac h , Physica D 204 , 83 (2005).