q-Breathers in Discrete Nonlinear Schroedinger arrays with weak disorder
Nonlinearity and disorder are key players in vibrational lattice dynamics, responsible for localization and delocalization phenomena. $q$-Breathers -- periodic orbits in nonlinear lattices, exponentially localized in the reciprocal linear mode space …
Authors: M.V. Ivanchenko
q -Breathers in Disrete Nonlinear S hrö dinger arra ys with w eak disorder M. V. Iv an henk o Dep artment of Applie d Mathematis, University of L e e ds, LS2 9JT, L e e ds, Unite d Kingdom Nonlinearit y and disorder are k ey pla y ers in vibrational lattie dynamis, resp onsible for lo aliza- tion and delo alization phenomena. q -Breathers p erio di orbits in nonlinear latties, exp onen tially lo alized in the reipro al linear mo de spae is a fundamen tal lass of nonlinear osillatory mo des, urren tly found in disorder-free systems. In this pap er w e generalize the onept of q -breathers to the ase of w eak disorder, taking the Disrete Nonlinear S hrö dinger hain as an example. W e sho w that q -breathers retain exp onen tial lo alization near the en tral mo de, pro vided that disorder is suien tly small. W e analyze statistial prop erties of the instabilit y threshold and uno v er its sensitiv e dep endene on a partiular realization. Remark ably , the threshold an b e in ten tionally inreased or dereased b y sp eially arranged inhomogeneities. This eet allo ws us to form ulate an approa h to on trolling the energy o w b et w een the mo des. The relev ane to other mo del arra ys and exp erimen ts with miniature me hanial latties, ligh t and matter w a v es propagation in optial p oten tials is disussed. P A CS n um b ers: 63.20.Pw, 63.20.Ry , 05.45.-a A w ealth of ph ysial systems from natural rystals to the utting-edge te hnology pro duts lik e miro and nanome hanial system arra ys are spatially extended and disrete. In teration b et w een their elemen ts is a k ey soure for a n um b er of fundamen tal dynamial and statis- tial ph ysial phenomena inluding thermal ondutivit y , w a v e exitation and propagation, eletron and phonon sattering. T o pro vide with a full understanding of these pro esses the theory of olletiv e vibrational mo des is in demand. The prinipal question to b e answ ered is the eet of the t w o fundamen tal features of an y lattie: nonlinearit y and disorder. In reen t deades w e ha v e witnessed a remark able progress in studying their impats separately . Nonlinear- it y indues in teration b et w een linear normal mo des and energy sharing if it is strong enough (the F ermi-P asta- Ulam (FPU) problem) [1, 2℄, or exp onen tial lo alization of exat p erio di solutions (disrete breathers) in spae [3℄. Indep enden tly , disorder leads to exp onen tially lo al- ized linear vibrational mo des (Anderson mo des) [ 4 ℄. Ho w ev er, the onurren t eet of nonlinearit y and disorder has not reeiv ed a satisfatory full desription y et. Str ongly disor der e d and we akly nonline ar latties enjo y in tensiv e resear h, in partiular, on on tin uation of Anderson mo des in to nonlinear regime [5℄, w a v epa k et spreading [6℄, ligh t propagation in photoni latties [ 7 ℄, and Bose-Einstein ondensate (BEC) lo alization in ran- dom optial p oten tials [8℄. In on trast, little is kno wn in ase of pr onoun e d nonline arity and we ak disor der . Im- p ortan tly , this situation is realized in miro and nano- me hanial osillatory arra ys that are often driv en in to nonlinear regime, while the spatial disorder is onstan tly redued b y impro ving fabriation te hniques [9, 10 ℄. On the atomi sale, the surfae vibrational mo des are though t to b e a soure of seletiv e atalyti prop erties of three-dimensional gold nano-lusters for a v ariet y of hemial reations [11 ℄. Ligh t propagation and BEC dy- namis in random optial media are equally strong mo- tiv ating problems. q -Breathers (QBs) presen t a reen tly diso v ered funda- men tal lass of nonlinear osillatory mo des. They are ex- at time-p erio di solutions to nonlinear lattie equations, on tin ued from linear normal mo des and exp onen tially lo alized in the linear mo de spae. In tro dued to explain the FPU parado x (energy lo king in the lo w-frequeny part of the sp etrum, reurrenies, and size-dep enden t sto hastiit y thresholds) [12 ℄, they ha v e b een found in t w o and three dimensional FPU arra ys and disrete non- linear S hrö dinger (DNLS) latties [ 13 ℄; last y ear quan- tum QBs w ere observ ed in the Bose-Hubbard hain [ 14 ℄. QBs ha v e b een suggested as k ey ators in a BEC pulsat- ing instabilit y [15 ℄ and a four-w a v e mixing pro ess in a nonlinear rystal [16 ℄. In this pap er w e extend the onept of q -breathers to random arra ys, implemen ting the paradigmati DNLS mo del as an example. The ornerstones of our approa h are on tin uation of QBs in to non-zero 'frozen' disor- der, taking a nonlinear lo alized solution as a seed, and analysing statistis then. W e sho w that QBs displa y the rosso v er from the exp onen tial lo alization near the en- tral mo de to the p o w er-la w dea y at a distane. Their a v- erage linear stabilit y threshold in nonlinearit y k eeps the same v alue in the rst order appro ximation. The v ari- ane inreases linearly with disorder, manifesting high sensitivit y on partiular realizations. Finally , w e demon- strate, that the sup erimp osed p erio di mo dulation of the linear oupling strength an b e a means of the energy o w on trol. The DNLS lattie is represen ted b y the Hamiltonian H = X n ((1 + D κ n ) ψ n − 1 ψ ∗ n + (1 + D κ n +1 ) ψ n +1 ψ ∗ n + + µ 2 | ψ n | 4 ) , (1) 2 and the equations of motion are i ˙ ψ n = ∂ H /∂ ψ ∗ n : i ˙ ψ n = (1+ D κ n ) ψ n − 1 +(1 + D κ n +1 ) ψ n +1 + µ | ψ n | 2 ψ n (2) Here ψ is a omplex salar that ma y desrib e the slo w small-amplitude dynamis of a lassial nonlinear osil- lators arra y [17 , 18 ℄, probabilit y amplitude of an atomi loud on an optial lattie site [19 ℄, or the amplitudes of a propagating eletromagneti w a v e in an optial w a v eg- uide [20 ℄. Zero b oundary onditions apply: ψ 0 = ψ N +1 = 0 . µ and D are the nonlinearit y and disorder parameters, random κ n ∈ [ − 1 / 2 , 1 / 2] are uniformly distributed and unorrelated: h κ n κ m i = σ 2 κ δ n,m , σ 2 κ = 1 / 12 . Beside the total energy , the norm B = P n | ψ n | 2 is onserv ed. Chang- ing µ is stritly equiv alen t to hanging the norm B , th us w e x B = 1 further on. The anonial transformation to the reipro al spae of normal mo des with new v ariables Q q ( t ) is giv en b y ψ n ( t ) = r 2 N + 1 N X q Q q ( t ) sin π q n N + 1 , (3) and the dynamis in this spae reads: i ˙ Q q + Ω q Q q = ρ 2 X p,r,s G q,p,r,s Q p Q r Q ∗ s + d X p V q,p Q p , (4) where ρ = µ N +1 , d = D √ N +1 , Ω q = − 2 cos π q N +1 are the normal mo de frequenies for the linear disorder- free system with µ = D = 0 . The nonlin- ear in termo de oupling o eien ts are G q,p,r,s = P ± ( − 1) ( ± p )( ± r )( ± s ) δ q ± p ± r ± s, 0 + δ q ± p ± r ± s, ± 2( N +1) and the disorder indued ones read V q,p = 2 √ N +1 × N − 1 P n =1 κ n (sin π qn N +1 sin π p ( n +1) N +1 + sin π q ( n +1) N +1 sin π pn N +1 ) . In the disorder-free ase QBs are time-p erio di sta- tionary solutions ψ n ( t ) = φ n exp( i Ω t ) with the frequeny Ω and time-indep enden t amplitudes φ n lo alized in nor- mal mo de spae. In the q -spae they ha v e the form Q q ( t ) = A q exp( i Ω t ) , the amplitudes of the mo des A q b eing time-indep enden t and related to the real-spae am- plitudes b y the anonial transformation (3); the mo de energies are dened as B q = | A q | 2 . Here w e fo us on time-rev ersible p erio di orbits and, th us, onsider A q to b e real n um b ers. The amplitudes satisfy a losed system of algebrai equations: (Ω q − Ω) A q = ρ 2 X p,r,s G q,p,r,s A p A r A ∗ s + d X p V q,p A p , X q | A q | 2 − B = 0 (5) Our metho dology onsists of t w o steps. Firstly , w e tak e a kno wn QB solution for non-zero nonlinearit y [ 13 ℄. A partiular realization of { κ n } is hosen and d regarded 10 20 30 40 50 60 70 80 90 100 −30 −20 −10 0 q log 10 〈 B q 〉 (a) 10 20 30 40 50 60 70 80 90 100 −30 −20 −10 0 q log 10 〈 B q 〉 (b) D=0 D=10 −5 D=10 −2 D=0 D=10 −5 D=10 −2 FIG. 1: The a v erage mo de energy distribution in QBs with inrease of disorder, where µ = 0 . 1 , N = 100 : (a) the lo w frequeny mo de q 0 = 11 and (b) the middle frequeny mo de q 0 = 53 . Filled irles are analytial estimates (11) as the disorder parameter. T ogether with the nonlin- earit y parameter ρ , it is assumed to b e small ρ, d ≪ 1 . Then, an asymptoti expansion in p o w ers of { ρ, d } is de- v elop ed. Subsequen t linear stabilit y analysis emplo ys the onstruted solution. Seondly , statistial prop erties of the QB solution and the instabilit y threshold are ana- lyzed. Con tin uation of QBs from µ 6 = 0 , D = 0 to µ, D 6 = 0 exploits the same ideas as from µ = D = 0 to µ 6 = 0 , D = 0 [ 13℄. F or small amplitude exitations the non- linear and disorder terms in (4) an b e negleted and the q -osillators get deoupled, their harmoni energy B q = | Q q | 2 b eing onserv ed in time. Single q -osillator exitations ( B q 6 = 0 for q ≡ q 0 only) are trivial stationary and q -lo alized solutions for β = D = 0 . In the disorder-free ase su h p erio di orbits an b e on tin ued in to the nonlinear ase at xed total energy [13 ℄ b y solving the system of algebrai equations (5), gran ted b y the impliit funtion theorem [21 ℄, as the non- resonane ondition Ω q 0 6 = Ω q 6 = q 0 holds. This is v alid for d ≪ 1 as w ell, for the sp etrum remains non-resonan t with the probabilit y 1 [5℄. Numerially , w e w ere able to on tin ue QBs in to the β , D 6 = 0 domain for all parameters tak en. T ypial results for the lo w-frequeny and middle- frequeny QBs are sho wn in Fig.1 . They demonstrate the rosso v er b et w een the exp onen tial lo alization and the disorder indued ba kground. The disorder-free ex- p onen tial lo alization p ersists in some neigh b orho o d of the en tral mo de for suien tly small disorder, but is range shrinks as disorder gro ws. High-frequeny QBs b e- ha v e analogously . Let us onstrut an asymptoti expansion for the QB solution. W e assume ρ, d ≪ 1 and start from 3 the disorder-free QB prole A N L q for the mo des q 0 , 3 q 0 ,. . . , (2 n + 1 ) q 0 ,. . . ≪ N in the leading order of ρ [ 13 ℄: A N L (2 n +1) q 0 = ( − 1) n γ n A q 0 , γ = µ ( N + 1) 16 π 2 q 2 B q 0 , Ω N L = Ω q 0 − ρ 2 A 2 q 0 (6) W e seek an asymptoti expansion in p o w ers of d ≪ 1 : ˆ A q = A (0) q + dA (1) q + O ( d 2 , ρd ) , ˆ Ω = Ω (0) + d Ω (1) + O ( d 2 , ρd ) , where A (0) q = A N L q , Ω (0) q = Ω N L q . Substitu- tion in to (5) giv es A (1) q = V q,q 0 Ω q − Ω q 0 A q 0 , q 6 = q 0 , Ω (1) = − V q 0 ,q 0 (7) The ensem ble a v erage of the "disorder on tribution" to the energy B DO q = dA (1) q 2 is B DO q = 2 d 2 σ 2 κ (1 + Ω q Ω q 0 / 4) (Ω q − Ω q 0 ) 2 B q 0 , (8) that appro ximates w ell the disorder-dominated part of the n umerially obtained QB proles in dieren t parts of the linear sp etrum (Fig.1). The p o w er-la w dea y B DO q ∝ ( q − q 0 ) − 2 ts in the large part of the q - spae. One an estimate the rosso v er lo ation b et w een the exp onen tial dea y and the p o w er-la w, in partiular, when the mo des next the to the en tral one b eome ex- ited almost equally w ell. Letting q = q 0 + 1 w e obtain the "small" σ κ D ≪ π / (2( N + 1) 3 / 2 ) and the "large" σ κ D ≫ π/ p ( N + 1) disorder riteria for delo alization of the least robust mo des q 0 = 1 , N and the most ro- bust one q 0 = N / 2 . Th us, single-ite en tered mo des do not exist ab o v e the size-dep enden t threshold in disorder magnitude. (Note, that delo alization in the mo de spae appro ximately orresp onds the onset of the Anderson lo- alization in the diret spae.) The linear stabilit y of QBs is determined b y on- sidering the ev olution of small omplex-v alued p ertur- bations ζ q ( t ) to the stationary solution [13 ℄: Q q ( t ) = ( ˆ A q + ζ q ( t )) exp ( i Ω t ) . In linearized equations the stabil- it y requires all the eigen v alues b e negativ e. Numerially w e solv e the orresp onden t problem in the diret spae (2). In the follo wing w e restrit our atten tion to the lo w and middle-frequeny QBs, lea ving the more om- plex ase of q 0 > N / 2 (when for D = 0 the instabilit y threshold b eha v es erratially vs. q 0 [13 ℄) for the future study . W e nd, that the instabilit y dev elops similarly for zero and non-zero disorder, the inrease or derease of the threshold µ ∗ sensitiv ely dep ending on a partiular real- ization (Fig.2). The a v erage h µ ∗ i remains v ery lose to the zero-disorder v alue µ ∗ 0 . In on trast, the v ariane σ µ ∗ is signian tly gro wing, dep ending on D almost linearly (Fig.2, deviations b eing observ ed when the probabilit y of µ ∗ b eing next to zero b eomes substan tial). 0.02 0.04 0.06 0.08 0.1 0.12 0.14 1 2 3 4 5 x 10 −4 µ θ (a) 0 1 2 3 4 5 6 x 10 −3 0 0.02 0.04 D σ µ * (b) D=0 D=0.004 D=0.008 D=0.012 q 0 =5, N=32 q 0 =7, N=64 q 0 =9, N=128 FIG. 2: (a) The maximal eigen v alues θ of QBs with q 0 = 15 , N = 128 and t w o dieren t sets of { κ n } vs. the nonlinearit y o eien t for sev eral v alues of disorder strength D . F or one realization of disorder the instabilit y threshold is inreasing with D , for another dereasing. (b) The v ariane of the QB instabilit y threshold σ µ ∗ vs. D . Solid lines are analytial estimates (11) The analyti study of the QB stabilit y has not b een done b efore (ev en for D = 0 ) and w e presen t it here for the rst time (restriting to q 0 < N / 2 as ab o v e). Linearized equations for small p erturbations read: i ˙ ζ q = ( ˆ Ω − Ω q ) ζ q + ρ 2 B q 0 X G q,q 0 ,q 0 ,p ( ζ ∗ p + 2 ζ p ) + d X V q,p ζ p (9) In analogy to the FPU hain [12 ℄ w e suggest (and v erify that b y omparison with the n umerial results) that the eigen v etors for the main instabilit y b e almost parallel to the subspae { ζ q = 0 : q 6 = q 0 ± 1 } . Th us w e arriv e at a simpler task of nding eigen v alues of the system of t w o omplex-v alued linear equations (retaining O ( ρ, d ) terms only): i ˙ ζ q 0 − 1 = ( ˆ Ω − Ω q 0 − 1 + dV q 0 − 1 ,q 0 − 1 ) ζ q 0 − 1 + + 1 2 ρB q 0 ( ζ ∗ q 0 +1 + 2 ζ q 0 +1 ) + dV q 0 − 1 ,q 0 +1 ζ q 0 +1 , i ˙ ζ q 0 +1 = ( ˆ Ω − Ω q 0 +1 + dV q 0 +1 ,q 0 +1 ) ζ q 0 +1 + + 1 2 ρB q 0 ( ζ ∗ q 0 − 1 + 2 ζ q 0 − 1 ) + dV q 0 − 1 ,q 0 +1 ζ q 0 − 1 (10) After an extensiv e algebra one nally gets the bifuration p oin t: µ ∗ ≈ µ ∗ 0 1 − d π 2 | Ω q 0 | ∆ V q 0 ,q 0 , h µ ∗ i ≈ µ ∗ 0 , σ µ ∗ ≈ D σ κ p 3( N + 1 ) B q 0 , (11) 4 where ∆ V q 0 ,q 0 = V q 0 − 1 ,q 0 − 1 − 2 V q 0 ,q 0 + V q 0 +1 ,q 0 +1 , and the disorder-free µ ∗ 0 = π 2 | Ω q 0 | 2 B q 0 ( N +1) . It sho ws a go o d oin- idene with the n umerial results (Fig.2). Note, that in- reasing the hain length dereases the instabilit y thresh- old and inreases its v ariation. Th us, in suien tly large arra ys the solution will lo ose stabilit y at v ery small non- linearities with the probabilit y , almost equal to that of ∆ V q 0 ,q 0 b eing negativ e, whi h is 0 . 5 in our ase. If the instabilit y dep ends that sensitiv ely on the disorder realization, there m ust b e ertain lasses of inhomogeneities that augmen t it or suppress. Iden- tifying them oers the p ossibilit y of on trolling the energy o w in the mo de spae b y designing sp ei impurities and, further, hanging them in time. The disorder indued orretion in (11 ) redues to ∆ V q 0 ,q 0 = 8 √ N +1 N − 1 P n =1 κ n cos π q 0 (2 n +1) N +1 sin 2 π (2 n +1) 2( N +1) + O ( N − 2 ) . Note, that it is linear in κ n , and, therefore, one an represen t κ n as a sum of spatial F ourier omp onen ts, their on tributions b eing additiv e to ∆ V q 0 ,q 0 . Let us onsider a harmoni inhomogeneit y κ n = 1 2 cos ( π p ( n +1 / 2) N +1 + ϕ ) , where ϕ is the phase shift. It is natural to exp et the absolute extrema of ∆ V q 0 ,q 0 ( p, ϕ ) (and the maximal gain or loss in stabilit y) to b e rea hed for p = 2 q 0 . This ase yields ∆ V q 0 ,q 0 ≈ √ N + 1 cos ϕ . Th us, the bifuration p oin t rea hes its maxim um and minim um for ϕ = π and ϕ = 0 resp etiv ely , giving µ ∗ ≈ µ ∗ 0 1 ± D ( N +1) 2 π 2 | Ω q 0 | . A t the same time one gets a zero shift for ϕ = ± π 2 . Analogously , for p = q 0 ± 1 one gets µ ∗ ≈ µ ∗ 0 1 ± 8 D ( N +1) 2 3 π 3 | Ω q 0 | sin ϕ , whi h is − π 2 shifted in ϕ and has a bit smaller amplitude. F or p = q 0 ± 2 it reads µ ∗ ≈ µ ∗ 0 1 + D ( N +1) 2 2 π 2 | Ω q 0 | cos ϕ , whi h is π shifted in ϕ and has t wie a smaller amplitude. Larger deviations from 2 q 0 lead to progressiv ely dereasing shifts. These results are illustrated in Fig.3 , and sho w a go o d orresp ondene to the n umerially determined QB stabilit y . Summing up, the spatial F oirier omp onen ts p ∈ [2 q 0 − 2 , 2 q 0 + 2 ] of { κ n } are deisiv e for the q 0 -QB stabilit y . The dep endene is notably dieren t and m u h more ompliated than the p ossible "naïv e" exp etation that harmoni inhomogeneities with p = q 0 will most ef- fetiv ely stabilize or destabilize q 0 -QBs. A remark able fat is the sensitiv e dep endene on the phase of the im- purit y harmonis: ev en for a xed p opp osite shifts in the threshold o ur. Presumably , this is the onsequene of the deformation of the linear sp etrum due to inho- mogeneities, as ∆ V q 0 ,q 0 is, atually , the dierene in the frequeny shifts of linear mo des ( 7 ). In its turn, this is determined b y the b oundary onditions, whi h also aet the nonlinearit y indued in teration. It learly highligh ts one of the future diretions of study . These ndings suggest a p ossibilit y of on trolling the 5 10 15 20 25 30 0.24 0.245 0.25 0.255 0.26 0.265 p µ * ϕ=π ϕ=π/2 FIG. 3: The instabilit y threshold for QBs with q 0 = 10 , N = 32 and κ n = 1 2 cos “ πp ( n +1 / 2) N + 1 + ϕ ” . Dash-dotted and mark ed lines are n umerial results for (2), solid lines are analytial estimates energy o w b et w een mo des. Indeed, b y imp osing a prop er spatially p erio di mo dulation of the linear ou- pling one an destabilize ertain QB exitations and (i) sp eed up equipartition or (ii) stabilize others, where the energy will b e radiated to and trapp ed. New QBs ma y also b e destabilized to arrange a further energy o w. Ex- p erimen tally , in miniature me hanial latties inhomo- geneities an b e reated, for example, b y laser heating, either as harmoni or sp ot impurities, lik e it w as designed to on trol disrete breathers relo ation in an tilev er ar- ra ys [9 ℄. In optial latties one an implemen t the same te hnique that has b een reen tly used for generating dis- ordered p oten tials in studies of the Anderson lo alization of ligh t [7 ℄ and matter (BEC) [8℄ w a v es. In summary , w e ha v e generalized the onept of QBs to the ase of non-zero disorder and analyzed these non- linear vibrational mo des in w eakly disordered DNLS ar- ra ys. W e demonstrated, that QBs remain exp onen tially lo alized in the mo de spae and stable, if the disorder is suien tly small. Their stabilit y dep ends sensitiv ely on a partiular realization of disorder, and ma y b e en- haned or undermined. The prev ailing on tribution to the stabilit y is made b y the spatial harmonis of disor- der whi h w a v e n um b ers are lose to t wie of that of the QB seed mo de. Th us, inhomogeneities design app ears to b e a promising te hnique of on trolling the energy o w b et w een nonlinear mo des. 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