$q$-Breathers in finite lattices: nonlinearity and weak disorder

q -Breathers in nite latties: nonlinearit y and w eak disorder M. V. Iv an henk o Dep artment of Applie d Mathematis, University of L e e ds, LS2 9JT, L e e ds, Unite d Kingdom Nonlinearit y and disorder are the reognized ingredien ts of the lattie vibrational dynamis, the fators that ould b e diminished, but nev er exluded. W e generalize the onept of q -breathers  p erio di orbits in nonlinear latties, exp onen tially lo alized in the reipro al linear mo de spae  to the ase of w eak disorder, taking the F ermi-P asta-Ulan  hain as an example. W e sho w, that these nonlinear vibrational mo des remain exp onen tially lo alized near the en tral mo de and stable, pro vided the disorder is suien tly small. The instabilit y threshold dep ends sensitiv ely on a par- tiular realization of disorder and an b e mo died b y sp eially designed impurities. Basing on it, an approa h to on trolling the energy o w b et w een the mo des is prop osed. The relev ane to other mo del latties and exp erimen tal miniature arra ys is disussed. P A CS n um b ers: 63.20.Pw, 63.20.Ry , 05.45.-a Nonlinearit y and disorder are ubiquitous and una v oid- able features of disrete extended systems, the k ey pla y- ers in a w ealth of fundamen tal dynamial and statistial ph ysial phenomena su h as thermalization, thermal on- dutivit y , w a v e propagation, eletron and phonon sat- tering. Studying lattie vibrational mo des is vital to gain full understanding of these problems. Nonlinear- it y indues in teration b et w een normal mo des and en- ergy sharing if strong enough (kno wn as the F ermi-P asta- Ulam (FPU) problem) [1 ℄, and time-p erio di exp onen tial lo alizations in diret spae (disrete breathers) [2 ℄. Lin- ear systems with disorder supp ort another generi lass of exp onen tially lo alized mo des (Anderson mo des) [3 ℄. A t the same time a satisfatory full understanding of the  onurr ent ee t of nonlinearit y and disorder is missing. This gap is b eing progressiv ely lled for str ongly disor- der e d and we akly nonline ar latties b y in tensiv e resear h on on tin uation of Anderson mo des in to nonlinear regime [4℄, w a v epa k et spreading [5 ℄, ligh t propagation in pho- toni latties [6 ℄, and Bose-Einstein ondensate (BEC) lo alization in random optial p oten tials [7℄. Little, ho w ev er, is kno wn on ho w the systems with pr o- noun e d nonline arity and we ak disor der b eha v e. Remark- ably , it is a demand in a n um b er of exp erimen tal and appliational on texts, b eside a  hallenge from theory . Miro- and nano-eletro-me hanial systems are rapidly dev eloping omp onen ts in miroinstrumen ts design (ul- trafast sensors, radio frequeny lters, added mass sen- sors) [8 ℄. Their arra y strutures oer broadband exita- tions, elasti w a v es, and eets of disp ersion to b e uti- lized [9, 10 ℄. Imp ortan tly , they are often suggested to op erate in the nonlinear regime, while maturing te hnol- ogy redues fabriation errors, hene diminishing spatial disorder. On the atomi sale, three-dimensional gold nano-luster strutures are found to b e ativ e and se- letiv e atalysts for a v ariet y of  hemial reations, the surfae vibrational mo des b eing p ossible initiators [11 ℄. One of the fundamen tal t yp es of nonlinear osillatory mo des is q-breathers (QBs) - exat time-p erio di solu- tions on tin ued from linear mo des and preserving ex- p onen tial lo alization in the linear normal mo de o or- dinates. Originally prop osed to explain the FPU para- do x (the energy lo  king in lo w-frequeny mo des of a w eakly nonlinear  hain, reurrenies, and size-dep enden t sto  hastiit y thresholds) [12 ℄, they ha v e b een diso v ered in t w o and three dimensional aousti latties, saled to innite systems, found in disrete nonlinear S hrö dinger (DNLS) arra ys, and quan tum QBs w ere observ ed in the Bose-Hubbard  hain [13 ℄. Reen tly , QBs ha v e b een sug- gested as ma jor ators in a BEC pulsating instabilit y and a four-w a v e mixing pro ess in a nonlinear rystal [ 14 ℄. In this pap er w e extend the onept of QBs to ran- dom nonlinear media, exemplifying in the FPU  hain. The ornerstones of our approa h are on tin uation of QBs in to non-zero 'frozen' disorder, taking a nonlinear lo alized solution as a seed, and statistial analysis of on tin ued solutions. W e sho w that QBs demonstrate the rosso v er from the exp onen tial lo alization near the en- tral mo de to plateaus at a distane. The a v erage sta- bilit y threshold in nonlinearit y remains the same in the rst order of appro ximation. In on trast, the standard deviation inreases linearly with disorder, manifesting a high sensitivit y on realizations. W e analyze the eet of the harmoni in spae inhomogeneities and disuss the energy o w on trol b y impurities design. The FPU- β  hain of N equal masses, oupled b y springs with disorder in linear o eien ts and quarti nonlinearit y in p oten tial, is desrib ed b y the Hamilto- nian H = 1 2 N X n =1 p 2 n + N +1 X n =1  1 2 (1 + D κ n )( x n − x n − 1 ) 2 + β 4 ( x n − x n − 1 ) 4  (1) where x n ( t ) is the displaemen t of the n -th partile from its original p osition, p n ( t ) its momen tum, x 0 = x N +1 = 0 , κ n ∈ [ − 1 / 2 , 1 / 2] are random, uniformly distributed, and unorrelated with h κ n κ m i = σ 2 κ δ n,m , σ 2 κ = 1 / 12 in our ase. A anonial transformation 2 x n ( t ) = q 2 N +1 N P q =1 Q q ( t ) sin  π qn N +1  denes the reipro- al w a v e n um b er spae with N normal mo de o ordinates Q q ( t ) , b eing solutions to the linear disorder-free ase. The normal mo de spae is spanned b y q and represen ts a  hain similar to the situation in real spae. The equations of motion read ¨ Q q + ω 2 q Q q = − β 2( N + 1) N X p,r,s C q,p,r,s ω q ω p ω r ω s × Q p Q r Q s − D √ N + 1 N X p ω q ω p K q,p Q p . (2) Here ω q = 2 sin π q 2( N +1) are the normal mo de frequen- ies. The oupling o eien ts C q,p,r,s [12 ℄ indue the se- letiv e nonlinear in teration b et w een distan t mo des and K q,p = 2 √ N +1 N +1 P n =1 κ n cos π q ( n − 1 / 2) N +1 cos π p ( n − 1 / 2) N +1 reet the all-to-all linear in teration due to disorder. Our metho dology onsists of t w o steps. Firstly , w e tak e a kno wn QB solution for non-zero nonlinearit y [12 ℄. A partiular realization of { κ n } is  hosen and d = D / √ N + 1 regarded as the disorder parameter. T o- gether with the nonlinearit y parameter ν = β / ( N + 1 ) , it is assumed to b e small: ν, d ≪ 1 . Then, an asymp- toti expansion in p o w ers of { ν, d } is dev elop ed. Linear stabilit y analysis is based on the onstruted solution. Seondly , w e address the statistial prop erties of the QB solution and instabilit y threshold alulating resp etiv e a v erages and v arianes. Con tin uation of QBs to β , D 6 = 0 from β 6 = 0 , D = 0 emplo ys the same te hnique as to β 6 = 0 , D = 0 from β = D = 0 [12 ℄. F or ν, d << 1 and small amplitude ex- itations the q -osillators get eetiv ely deoupled, their harmoni energy E q = 1 2  ˙ Q 2 q + ω 2 q Q 2 q  b eing almost on- serv ed in time. Single q -osillator exitations E q 6 = 0 for q ≡ q 0 only are trivial time-p erio di and q -lo alized so- lutions for β = D = 0 . F or the disorder-free  hain su h p erio di orbits an b e on tin ued in to the nonlinear ase at xed total energy [12 ℄ b eause the non-resonane ondition nω q 0 6 = ω q 6 = q 0 ( n b eing an in teger) holds for an y nite size [15 ℄ and the Ly apuno v theorem [16 ℄ applies. Same ideas are exp eted to w ork for d ≪ 0 , as the sp etrum remains non-resonan t with the probabilit y 1 [4 ℄. Su h on tin uation sueeded for all parameters w e to ok. W e on tin ue QBs from nonlinear disorder-free solu- tions inreasing D and k eeping a partiular random real- ization { κ n } xed. The total energy of the  hain is E = 1 in all examples, and 100 realizations of disorder are tak en and used for a v eraging. Dep endene of the a v erage QB energy distribution on the lev el of disorder is rep orted in Fig.1 . W e observ e the  harateristi QB exp onen tially lo alized prole on the almost at disorder-indued ba k- ground. The heigh t of the plateau gro ws with D , grad- 5 10 15 20 25 30 −30 −20 −10 q log 10 〈 E q 〉 D=0 D=5*10 −9 D=5*10 −7 D=5*10 −5 D=5*10 −3 D=0.1 FIG. 1: The a v erage mo de energy distribution in QBs with q 0 = 5 , β = 0 . 01 , N = 32 under inreasing disorder. Dashed lines are theoretial estimates (5). ually absorbing lo alized mo des. F or β = 0 . 0 1 , q 0 = 5 , and N = 32 the seond large mo de is o v erome near D = 0 . 01 , and in terp olation predits the en tral one re- mains w ell ab o v e the ba kground ev en for D > 2 , when some linear elastiit y o eien ts ma y b eome negativ e with a non-zero probabilit y . Let us reall, that in ase D = 0 the QB solution ˆ Q N L q ( t ) with a lo w-frequeny seed mo de n um b er q 0 an b e written as an asymptoti expansion in p o w ers of the small nonlinearit y parameter ν [12 ℄. The energies of the mo des q 0 , 3 q 0 ,. . . , (2 n + 1) q 0 ,. . . ≪ N read E N L (2 n +1) q 0 = λ 2 n E q 0 , λ = 3 β E q 0 ( N + 1) 8 π 2 q 2 0 , (3) and the frequeny ω N L = ω q 0 (1 + 9 / 4 ν E q 0 ) . No w w e de- v elop a p erturbation theory to (2) in terms of the small disorder parameter d : ˆ Q q ( t ) = Q (0) q ( t ) + dQ (1) q ( t ) + . . . , its frequeny b eing ˆ ω = ω (0) + dω (1) + . . . , substituting Q (0) q ( t ) = ˆ Q N L q ( t ) and ω (0) = ω N L . In the rst order appro ximation (2) b eomes the equation of a fored os- illator: ¨ Q (1) q + ω 2 q Q (1) q = − ω q ω q 0 K q,q 0 Q (0) q 0 . It follo ws that all mo des get exited b y disorder, their amplitude the bigger the loser its frequeny to ω q 0 : A (1) q = − ω q ω q 0 ω 2 q − ω 2 q 0 K q,q 0 A q 0 , q 6 = q 0 , (4) the frequeny b eing ˆ ω = ω q 0 (1 + 9 / 4 ν E q 0 + d/ 2 K q,q 0 ) . As h K q,q 0 i = 0 , the rst order orretions in d v anish for the a v erages D A (1) q E =  ω (1)  = 0 . Naturally , the v arianes are non-zero as the amplitude and frequeny orretions v ary dep ending on realization of { κ n } . The mo de energy (a v eraged o v er p erio d of a QB) appro ximately separates in to nonlinearit y and disorder-indued parts E q ≈ E N L q + E DO q , where  E DO q  = d 2 E q 0 ω 4 q 2( ω 2 q − ω 2 q 0 ) 2  K 2 q,q 0  = d 2 E q 0 σ 2 κ ω 4 q 2( ω 2 q − ω 2 q 0 ) 2 (5) T w o limit ases are of partiular in terest: (i) q ≫ q 0 , then  E DO q  ≈ d 2 E q 0 σ 2 κ / 2 that giv es a q -indep enden t plateau energy (dashed lines in Fig.1 ), and (ii) q = 3 0 1 2 3 4 5 x 10 −3 0 2 4 6 x 10 −3 D σ β * , 〈β * (D) 〉 − 〈β * (0) 〉 (b) 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.999 0.9995 1 1.0005 1.001 (a) β θ q 0 =3, N=16 q 0 =3, N=32 q 0 =6, N=64 D=0 D=0.002 D=0.0035 D=0.005 FIG. 2: (a) The maximal and minimal absolute v alues of the eigen v alues of QBs with q 0 = 6 , N = 64 for t w o real- izations of { κ n } . F or one realization the instabilit y thresh- old in nonlinearit y is inreasing with D , for another  de- reasing. (b) Empt y mark ers, dotted line: dep endene of the v ariane of the instabilit y threshold σ β ∗ on the disor- der strength. Solid lines are theoretial estimates (8). Filled mark ers: h β ∗ ( D ) − h β ∗ (0) i . q 0 + 1 , then  E DO q  ≈ d 2 σ 2 κ ω 2 q 0 ( N + 1) 2 E q 0 / 2 , that yields the QB lo alization riterion E q 0 ≫ E q 0 +1 if ω q 0 D σ κ ≪ 2 E q 0 / ( N + 1) . Expressions (i) and (3) pre- dit the rosso v er b et w een the exp onen tial lo alization and the plateau at q c ≈  ln Dσ 2 κ 2( N +1) / ln λ + 1  q 0 . Expres- sion (ii) suggests the 'small' D σ κ ≪ p 2 π 3 / ( N + 1) 3 and 'large' D σ κ ≫ p 8( N + 1) disorder riteria, the former implying that all QBs are lo alized and the latter that ev en QBs with q 0 = 1 are delo alized. It also reo v ers the b oundary q ∗ 0 ∝ √ N + 1 b et w een the lo alized (QBs) and delo alized in the q-spae (but lo alized in the diret spae) solutions (Anderson mo des), that agrees with the previous results [17 ℄. Note, that the parameters tak en in this pap er orresp ond to the 'small' disorder. The linear stabilit y of the on tin ued p erio di orbits is determined b y linearizing the phase spae o w around them and omputing the eigen v alues θ i , i = 1 , 2 N of the orresp onding sympleti Flo quet matrix [12 ℄. A QB is stable if | θ i | = 1 , ∀ i . The maximal and minimal abso- lute v alues of θ i of QBs with q 0 = 6 , N = 64 for sev eral inreasing v alues of D and t w o dieren t { κ n } are plot- ted vs. β in Fig.2(a). Remark ably , while the instabilit y threshold v aries monotonously with D , it ma y not only derease, but inrease as w ell, dep ending on a partiu- lar { κ n } . Moreo v er, stabilizing realizations are ommon, neatly balaning destabilizing ones. The observ ed devi- ation of the a v erage instabilit y threshold h β ∗ i from the disorder-free v alue β ∗ 0 w as m u h smaller that the v ariane (Fig.2(b)). The latter gro ws almost linearly in D , up to σ β ∗ ≈ 0 . 2 5 β ∗ 0 , as seen for q 0 = 6 , N = 64 (Fig.2(b); note 4 6 8 10 12 14 0.04 0.05 0.06 q β * (a) 5 10 15 20 25 30 0.04 0.045 0.05 0.055 0.06 p β * (b) ϕ=0 ϕ=π ϕ=π/2 ϕ=0 ϕ=π/2 FIG. 3: QB stabilit y in ase of spatially harmoni mo dula- tions: N = 32 , D = 0 . 0025 and (a) p = 10 , the en tral mo de q is  hanged, (b) the en tral mo de q = 8 , the mo dulation w a v e n um b er p is v aried. Dashed lines are theoretial estimates. that for larger D the linear t ma y b eome violated, due to the lo w er b ound β ∗ > 0 ). The monotonous dep endene of the instabilit y thresh- old on D suggests that it is aused b y the same resonane with the mo des q 0 ± 1 as in the disorder-free ase. Let us explore the impat of disorder on this bifuration. Lin- earizing equations of motion (2) around a QB solution Q q = ˆ Q q ( t ) + ξ q ( t ) , one gets ¨ ξ q + ω 2 q ξ q = − 3 ν ω q E q 0 cos 2 ( ˆ ω t ) X p C q,q 0 ,q 0 ,p ω p ξ p − dω q X p ω p K p,q ξ p + O ( ν 2 , ν d, d 2 ) (6) The strongest instabilit y is due to primary parametri resonane in (6) and in v olv es a pair of the resonan t mo des ˜ q , ˜ p = q 0 ± 1 . Omitting non-resonan t and O ( ν 2 , ν d, d 2 ) terms it is redued to ( ¨ ξ ˜ q + ω 2 ˜ q (1 + dK ˜ q, ˜ q ) ξ ˜ q = − 3 ν ω ˜ q ω ˜ p E q 0 cos 2 ( ˆ ω t ) ξ ˜ p ¨ ξ ˜ p + ω 2 ˜ p (1 + dK ˜ p, ˜ p ) ξ ˜ p = − 3 ν ω ˜ p ω ˜ q E q 0 cos 2 ( ˆ ω t ) ξ ˜ q (7) Th us, the disorder do es not reate new resonan t terms, its impat b eing onned to the QB and resonan t mo des frequeny shifts. The analysis analogous to [12 ℄ yields the instabilit y threshold β ∗ , its mean and v ariane: β ∗ = β ∗ 0  1 − 2 d ( N + 1) 2 π 2 ∆ K  , h β ∗ i = β ∗ 0 , σ β ∗ = 2 σ κ D √ N + 1 /E q 0 , (8) where the disorder-free v alue is β ∗ 0 = π 2 6 E q 0 ( N +1) and ∆ K = K ˜ q, ˜ q − 2 K q 0 ,q 0 + K ˜ p , ˜ p . It agrees w ell with the n umerial results (Fig.2(b)). 4 One ma y ask no w, whi h partiular realizations fa- v or or disfa v or stabilit y? F urthermore, if some re- eipts are dislosed, an they b e used in on trolling the energy o w in the mo de spae? The disorder- determined part of (8) an b e rewritten as ∆ K = − 4 √ N +1 N +1 P n =1 κ n cos π 2 q 0 ( n − 0 . 5) N +1 sin 2 π (2 n − 1) 2( N +1) . It is linear with resp et to κ n , th us w e an represen t it as a sum of spatial F ourier omp onen ts, their on tributions b eing additiv e. Th us, onsider κ n = 0 . 5 cos ( π p ( n − 0 . 5) N +1 + ϕ ) , where ϕ is the phase shift. It is natural to exp et the min- im um of ∆ K (and the maximal gain in stabilit y), when p = 2 q 0 , and it indeed yields ∆ K = 0 . 5 √ N + 1 cos ϕ , and the maxim um β ∗ = β ∗ 0  1 + D ( N + 1 ) 2 /π 2  for ϕ = 0 . Immediately , a high sensitivit y on ϕ is seen: the zero shift β ∗ = β ∗ 0 for ϕ = ± π / 2 ; the minim um β ∗ = β ∗ 0  1 − D ( N + 1) 2 /π 2  for ϕ = − π . The ef- fet of p = 2 q 0 on adjaen t QBs q ′ 0 = q 0 ± 1 is t wie as small and rev erse: for example, if ϕ = 0 then β ∗ = β ∗ 0  1 − D ( N + 1 ) 2 / (2 π 2 )  . Remark ably , while for p = 2 q 0 extremal shifts orresp ond to ϕ = 0 , π and zero ones to ϕ = ± π / 2 , for p = 2 q 0 ± 1 the zero shift ap- p ears for ϕ = 0 , π , and the extrema for ϕ = ± π / 2 : β ∗ = β ∗ 0  1 ∓ 8 D ( N + 1) 2 / (3 π 3 )  . These results are illustrated in Fig.3 , and sho w a go o d orresp ondene to the n umerially determined QB sta- bilit y . That is, dep ending on the phase ϕ , the spatially harmoni mo dulation of springs elastiities with the w a v e n um b er p = 2 q 0 , ma y signian tly augmen t, w eak en, or lea v e the stabilit y in tat (Fig.3(a)). Mo dulations with p = 2 q 0 ± 2  hange the stabilit y rev ersely and with t wie a smaller amplitude for the same ϕ , and those with p = 2 q 0 ± 1  just a bit w eak er than 2 q 0 , but with a π / 2 shift in ϕ (Fig.3(b)). Notably , mo dulations with other w a v e n um b ers ha v e only a minor eet. Therefore, the spatial F oirier omp onen ts with p ∈ [2 q 0 − 2 , 2 q 0 + 2] of { κ n } are deisiv e for the q 0 -QB stabilit y . These ndings suggest a p ossibilit y of on trolling the energy o w b et w een mo des. Indeed, b y imp osing a prop er p erio di mo dulation of the linear elastiit y one an destabilize ertain QB exitations and (i) promote equipartition or (ii) stabilize others, where the energy will b e radiated; new QBs ma y also b e sub jet to the same pro edure to arrange the further energy o w. Exp eri- men tally , elastiit y mo dulations ould b e a hiev ed, for example, b y laser heating, either as harmoni or sp ot im- purities, lik e it w as designed to on trol disrete breathers lo ation in an tilev er arra ys [9 ℄. In onlusion, w e ha v e demonstrated, that the onept of QBs an b e suessfully applied to analyzing nonlinear vibrational mo des in w eakly disordered latties. They es- sen tially retain exp onen tial lo alization and stabilit y in the mo de spae, if the disorder is suien tly small. W e sho w, that the stabilit y trend dep ends sensitiv ely on a partiular realization of disorder, and delib erately re- ated inhomogeneities oer a promising te hnique of on- trolling the energy o w b et w een nonlinear mo des. W e exp et that these ideas and metho ds to b e appliable to a v ariet y of nonlinear w eakly disordered latties  and w e ha v e already applied them to the DNLS  hain (to b e rep orted elsewhere)  inluding the on texts of a dif- feren t soure of disorder (masses, nonlinearities), higher dimensions, and quan tum arra ys. The results on the non- linear mo des sustainabilit y , stabilit y , and on trolling are strongly exp eted to b e in demand from exp erimen ts and appliations. W e thank S. Fla h for stim ulating and extremely v alu- able disussions. [1℄ E. F ermi, J. P asta, and S. Ulam, Los Alamos Rep ort LA- 1940, (1955); J. F ord, Ph ys. Rep. 213 , 271 (1992); F o us issue in Chaos 15 No.1 (2005). [2℄ R.S. MaKa y and S. Aubry , Nonlinearit y 7 , 1623 (1994); S. Fla h and A. Gorba h, Ph ys. Rep. 467 , 1 (2008). [3℄ P . W. Anderson, Ph ys. Rev., 109 , 1492 (1958). [4℄ C. Albanese and J. F ro ehli h, Comm. Math. Ph ys. 138 , 193 (1991); J. F. R. Ar hilla, R. S. MaKa y , J. L. Martin, Ph ysia D 134 , 406 (1999); G. K opidakis and S. Aubry , Ph ysia D 130 , 155 (1999); Id., 139 , 247 (2000). [5℄ A. S. Pik o vsky and D. L. Shep ely ansky , Ph ys. Rev. Lett. 100 , 094101 (2008); S. Fla h, D. Krimer, Ch. Sk ok os, (2008); Sh. Fishman, Y e. Kriv olap o v, A. Soer, J. Stat. Ph ys. 131 , 843 (2008). [6℄ T. Sh w artz et al., Nature 446 , 52 (2007). [7℄ J. Billy et. al, Nature 453 , 891 (2008); G. Roati et al., Nature 453 , 895 (2008). [8℄ K. L. Ekini, M. L. Rouk es, Rev. Si. Instr. 76 , 061101 (2005); M. Li, H.X. T ang, and M.L. Rouk es, Nature Nan- ote h., 2 114 (2007). [9℄ M. Sato, B. E. Habbard, and A. J. Siev ers, Rev. Mo d. Ph ys. 78 , 137 (2006); M. Sato, A. J. Siev ers, Lo w T emp. Ph ys. 34 , 543 (2008). [10℄ E. Buks and M. L. Rouk es, J. Mirome h. Sys. 11 , 802 (2002); M. Zalalutdino v et al., Appl. Ph ys. Lett. 88 , 143504 (2006). [11℄ Z. Y. Li et al., Nature 451 , 46 (2008). [12℄ S. Fla h, M. V. Iv an henk o and O. I. Kanak o v, Ph ys. Rev. Lett. 95 , 064102 (2005); S. Fla h, M. V. Iv an henk o and O. I. Kanak o v, Ph ys. Rev. E 73 , 036618 (2006). [13℄ M. V. Iv an henk o et al., Ph ys. Rev. Lett. 97 , 025505 (2006); O. I. Kanak o v et al., Ph ys. Lett. A 365 , 416 (2007); K. G. Mishagin et al., New J. Ph ys. 10 , 073034 (2008); J. P . Nguenang, R. A. Pin to, S. Fla h, Ph ys. Rev. B 75 , 214303 (2007). [14℄ U. Shrestha, M. K ostrun, and J. Ja v anainen, Ph ys. Rev. Lett. 101 , 070406 (2008); Sh. Jia, W. W an, and J. W. Fleis her, Opt. Lett. 32 , 1668 (2007). [15℄ J. H. Con w a y and A. J. Jones, A ta Arith. XXX , 229 (1976). [16℄ M. A. Ly apuno v, The General Problem of the Stabilit y of Motion, pp. 166-180 (T a ylor & F ranis, London, 1992). [17℄ K. Ishii, Suppl. Prog. Theor. Ph ys., No 53, 77 (1973).