LOcalized modes on an Ablowitz-Ladik nonlinear impurity

Lo calized mo des on an A blo witz-Ladik nonlinear impurit y Mario I. Molina Dep artme n t o de F ´ ısic a, F acultad de C iencias, Universidad de Chile, Santiago, Chile Abstract W e study lo ca lized mo des on a single Ablo witz-Ladik impurity em b edded in the bulk or at the surf ace of a one-dimensional linear lattice. Exact expressions are obtained for the b ound state profile and en er gy . Dynamical e xcitation of the lo calized mo de rev eals exp onen tially-high amplitude oscillations of the spatial profile at the impurity location. The pr ese nce of a surface increases the minimum nonlinearit y to effect a dynamical selftrapping. 1 The study of nonlinear dynamics in discrete systems has attr acte d a sp ecial atten tion recen tly due to no v el ph ysics and p ossible interes ting a pplicatio ns [1]. Among these systems, w e find the in tegrable discretized v ersion of the contin uum NLS equation, the so-called Ablo witz-Ladik (AL) equation[2]: i dC n dt + ( V + µ | C n | 2 )( C n +1 + C n − 1 ) = 0 (1) This integrable version supp ort mo ving, nonlinear, spatially-lo calized excitations in the for m of lattice solitons, found through the use of the in ve rse scattering transform method. The AL equations constitute a starting p oin t for man y studies on the in terplay of disorder, nonlinearit y and discretenes s. F or instance, when examining the effects of disorder, a well- kno wn approac h is to assume a p erturbativ e approach and try to compute the ev olution of the soliton parameters[3]. When the scale of the disorder is high, this a pproac h is no longer tenable and one m ust resort to numeric al sc hemes. On the other hand, when nonlinearit y is large, the spatial soliton profile is w ell lo calized in space, meaning that only a small n umber of sites a round the soliton cen ter are effectiv ely nonlinear. The system then lo oks v ery similar to a linear system con taining a small cluster of nonlinear sites, o r ev en a single nonlinear impurit y . This simplified system is now amenable to exact mathematical treatmen t, and the influence of other p oten tially comp e ting effects, suc h as dimensionalit y , b oundary effects, noise, etc., can b e more easily studied without losing the essen tial ph ysics. This approach has been succes sfully used for the DNLS equation[4 ], i dC n dt + V ( C n +1 + C n − 1 ) + γ | C n | 2 C n = 0 , (2) where it w as predicted tha t , for a semi-infinite nonlinear c hain, t he presence of a surface w ould increase the amoun t of no nlinearity required to form a lo calized surface mo de. This w as subseq uen tly observ ed in lat er studies[5, 6]. When used for the tw o- dime nsional semi- infinite square la ttice , this pro cedure predicted that this time, the presence of a b oundary w ould decrease the minim um nonlinearity needed to create a surface lo calized mo de[7]. This w as later fo und to b e the case[8]. In this Letter, we in tro duce a no v el t yp e of nonlinear defect in a one-dimensional discrete c hain, this time using the f r a me work of the AL equation (1). W e consider a one-dimensional arra y of linear sites, con taining a single, Ablowitz-Ladik impurit y lo cated at site n 0 . In the tight-binding f ramew ork, the ev o lution equation for the 2 amplitude is given b y i dC n dt + ( V + δ n,n 0 µ | C n | 2 )( C n +1 + C n − 1 ) = 0 (3) where C n is the complex amplitude at site n , V is the nearest-neigh b or coupling co efficien t, and µ is the Ablo witz-Ladik (AL) parameter. W e will b e intere sted in stationar y -state solutions of the form C n ( t ) = C n exp( iω t ). This leads to the sys tem o f equations: − ω C n + ( V + δ n,n 0 µ | C n | 2 )( C n +1 + C n − 1 ) = 0 . (4) F rom Eq.(3) it can b e easily prov en that the norm N = ( V /µ ) log(1 + ( µ/V ) | C 0 | 2 ) + X n ′ | C n | 2 , (5) is a conserv ed quan tit y , where the prime in the sum indicates that the sum is carried out o v er all sites, ex cepting the impurity site, n = n 0 . W e normalize the time to τ = V t and the pr o babilit y amplitude to φ n = C n / √ N . With these definitions, Eq.(3) simplifies to i dφ n dτ + (1 + δ n,n 0 ν | φ n | 2 )( φ n +1 + φ n − 1 ) = 0 (6) where ν ≡ N µ/V . The normalization condition b ecomes 1 = (1 /ν ) log(1 + ν | φ n 0 | 2 ) + X n ′ | φ n | 2 . (7) The equation for the stat io nary state, acquires now its dimensionles s form: − β φ n + (1 + δ n,n 0 ν | φ n | 2 )( φ n +1 + φ n − 1 ) = 0 , (8) where, β ≡ ω /V . W e will fo cus on tw o sp ec ial cases, (i) Impurit y in the “bulk” and (ii) “surface” impurit y . Impurit y in the “bulk” : In this case, −∞ < n < ∞ and without loss of generalit y , w e c ho ose n 0 = 0. W e p ose a solutio n in the for m φ n = A ξ | n | , w here 0 < | ξ | < 1. After inserting this a nsatz in to Eq.(8), one obtains β = 2 ξ (1 + ν A 2 ) and β = ξ + ( 1 /ξ ). After solving f o r ξ , one obtains ξ 2 = 1 1 + 2 ν A 2 (9) On the other hand, fro m the normalization condition, Eq.(7), one obtains the relation 1 = 1 ν log(1 + ν A 2 ) + 2 A 2 ξ 2 1 − ξ 2 . (10) 3 - 10 - 5 0 5 10 - 0.4 0 0.4 - 10 - 5 0 5 10 0 0.5 1 1.5 - 10 - 5 0 5 10 0 0.2 0.4 - 10 - 5 0 5 10 0 0.5 1 1.5 FIG. 1: Impu rit y in bulk: lo calize d mo des for ν = 1 . 25 (left column) and ν = 1 . 5 (righ t column). The top (b ottom) row sho ws the un sta ggered (staggered) v ersions of th e mo de. After com bining these last tw o equations, one obta ins ξ = ± [2 exp( ν − 1) − 1] − 1 / 2 , and A = ((exp( ν − 1) − 1) /ν ) 1 / 2 , whic h implies φ n = ( ± 1) n exp( ν − 1) − 1 ν ! 1 / 2 (2 exp( ν − 1) − 1) −| n | / 2 . (11) The dimensionle ss b ound state energy is β = ±  [2 exp( ν − 1) − 1] − 1 / 2 + [2 exp( ν − 1) − 1] 1 / 2  . (12) As can b e seen from Eq.(11), a lo calized b ound state is p ossible pro vided ν > ν c = 1, and for a giv en ν , there is an unstagger e d ( stagger e d ) v ersion of the bo und s tat e for β > 2 ( < − 2). F ig.1 shows a couple of profiles φ n and their staggered ve rsions, for t w o differen t dimensionless nonlinearit y parameters ν . In F ig.2 w e sho w ξ and the b ound state energy as a function of nonlinearit y . Standard linear stabilit y analysis rev eals that this stationary lo calized state is stable. An in teresting feature arises when w e consider the dynamical excitation of a lo calized state. In this case, one considers Eq.(6) for a highly lo caliz ed initial condition, c hosen as φ n (0) = δ n, 0 q (exp( ν ) − 1) /ν . This c hoice corresp onds to the one that saturates the nor- malization condition, Eq.(7). Examination of t he ensuing dynamics reve als that at lo w 4 1 2 3 4 Ν - 1 - 0.5 0 0.5 1 Ξ 0 1 2 3 4 Ν - 2 0 2 4 Β FIG. 2: Imp urit y in bulk. Left: ξ as a f unction of ν for lo cali zed mo de. Righ t: Bound state energy of lo ca lized mo de as a f unctio n of n o nlin e arity parameter. Th e shaded area marks the p osition of the linear band , while th e upp er (lo w er) cur v e corresp onds to the unstaggered (staggered) mo de. The blac k dot marks the p osition of ν c = 1. nonlinearit y v a lues, the excitation tends to diffract across the ar ra y , while for higher non- linearities, it tends to selftrap at the impurity site, with a high-amplitude o scillation, as Fig.3 clearly shows . The magnitude and frequency of these oscillations increase as the non- linearit y pa r ame ter ν is increased. W e hav e chec k ed num erically the p ersiste nce of this breathing phenomenon for long times, and b eliev e that it can b e understo o d from the sp e- cial f orm of the normalization condition, Eq.(7 ): A small c hange in the sum of the square amplitudes at sites other than the impurit y site will bring ab out a large c hange of the am- plitude at the impurit y site, due to the logarithmic dep endency of the latter. T o be more precise, let us assume that shortly after la unc hing the initial excitation, a certain amount of radiation is emitted causing P ′ | φ n | 2 → P ′ | φ n | 2 − ∆; then it can be easily pro v en that | φ 0 | 2 → | φ 0 | 2 + (1 /ν )(exp( ν ∆) − 1). Thus , it is the particular f o rm of the AL nonlinearit y that amplifies the breathing oscillations exp onen tially a t the impurit y site. W e ha v e also computed the long-time av erag e probability at the initial site, as a function of nonlinearit y strength. F or our relativ ely short c hain (100 sites), there is no sharp selftrapping threshold, although t he re is an inflexion p oin t around ν = 7, as F ig .3 sho ws. Surface impurity : W e no w consider the case when the impurit y is at the v ery b eginning of a semi-infinite lattice. W e relab el the previous c hain, so that the first site is now at n 0 = 0. 5 0 50 100 Position 0 10 20 z 0 5 10 15 20 z 0 10 20 È Φ 0 È 2 0 2 4 6 8 Ν 0 0.4 0.8 1.2 < È Φ 0 È 2 > 0 50 100 Position 0 10 20 z FIG. 3: Impurity in bu l k. T op left: Long-time a v erage probabilit y at impurit y site. T op right: Ev olution of initial localized excitati on across the lattice for ν = 2. Bottom le ft: Evo lution for ν = 8. Bottom right : Evolutio n of amplitude at impurity site for ν = 8. The dimensionle ss statio na ry - stat e equations read no w − β φ 0 + (1 + ν | φ 0 | 2 ) φ 1 = 0 (13) − β φ n + ( φ n +1 + φ n − 1 ) = 0 , n = 0 , 1 , 2 , . . . (14) W e pro ceed as b efore and p ose a solution of the f orm φ n = Aξ n , where 0 < | ξ | < 1 and n = 0 , 1 , 2 , . . . . After replacing this ansatz in to Eq.(13) and (14), one obtains β = (1 + ν A 2 ) ξ and β = ξ + (1 /ξ ), whic h implies ξ 2 = 1 ν A 2 (15) On the other hand, fro m the normalization condition, Eq.(7), w e ha v e 1 = 1 ν log(1 + ν A 2 ) + A 2 ξ 2 1 − ξ 2 (16) F rom Eqs . (15) and (16) , w e obtain a transcenden tal equation for ξ : ν = log 1 + 1 ξ 2 ! + 1 1 − ξ 2 (17) 6 Simple analysis sho ws that there is a critical nonlinearit y v alue ν c = (3 / 2) + log (4) ≈ 2 . 9, suc h that, for ν < ν c there is no bound state, at ν = ν c there is exactly one b ound state, while for ν > ν c there are tw o b ound states. O ne o f these states, b ecomes more narro w and its energy detac hes from the linear band as nonlinearity is increased, while the second one b ecome s wider and its energy approac hes the linear band up on increase in nonlinearity (see Fig.5 b elo w). Straightforw ard linear stability a na ly sis reve als that the former state is stable, while the latter is unstable. The b ound state mo de is giv en b y φ n = 1 √ ν ξ ( ν ) n − 1 n = 0 , 1 , . . . (18) where ξ has to b e found nume rically fro m Eq.(17), for a given ν > ν c . It is p o s sible, how ev er, to deriv e a v ery simple, y et a c curate, approx imation for ξ = ξ ( ν ) , as follow s: W e start from Eq.(17) re-written as exp( ν ) = 1 + ξ 2 ξ 2 ! exp(1 / (1 − ξ 2 )) (19) No w, since 0 < | ξ | < 1, it mak es sense to expand around ξ = 0. T o fourth-order in ξ , Eq.(19) b ecome s ξ 2 e ν − 1 ≈ 1 + 2 ξ 2 + (5 / 2) ξ 4 (20) whic h implies, ξ ( ν ) ≈ ±  1 5 (exp[ ν − 1] − 2 − (exp[2( ν − 1 )] − 4 exp[ ν − 1] − 6) 1 / 2 )  1 / 2 (21) Numerical comparison with the exact v alue, rev eals that the relativ e p ercen tag e error of appro ximation (21) is less than 3% for ν > 3. Figure 4 shows some amplitude profiles in the vicinit y of the lattice surface fo r a couple of differen t ν v alues. Figure 5 sho ws the n umerical solution for ξ and the lo calized state energy as a function of nonlinearit y . As b efore, v alues of β a b ov e (b elo w) the band giv e rise to unstagg ered (stagg ered) states. Comparison b et we en Figs. 2 and 5 rev eals that, as f ar as stationary lo calized mo des is concerned, the presence of a surface increases the minim um amoun t of nonlinearit y needed to create a b ound state. The b oundary is acting a s a repulsiv e surface, similar to what ha s b een observ ed earlier in semi-infinite DNLS systems [4 , 6] Finally , w e examine the dynamics of an excitation initia lly lo calized at the surface of the system n = 0 . The idea is to determine how the presence of a b oundary affects the dynamical 7 creation of a surface lo calized mo de. As b efore, we take φ n (0) = q (exp( ν ) − 1) /ν δ n, 0 and examine the av erage probability remaining at the initial site for long t imes, as well as the b eha vior of the amplitude at the impurity . Results are displa y ed in Fig.6, whic h is qualitative ly similar to its bulk coun terpart, Fig.3 . As b efore, we o bserv e diffractio n b eha vior for small nonlinearit y v alues and selftrapping at larg e ν v alues (a t appro ximately ν ∼ 14 . 6). In the la s t case, w e also observ e larg e -a mplitud e oscillations at the impurit y site. The main difference with the bulk case, is that w e need no w substan tially larger ν v alues to effect selftrapping. In conclusion, w e ha ve examined the stationary-state a nd dynamical lo calized mo des re- siding on a AL-lik e impurity , em b edded w ell inside and a t the surface of a one-dimensional discrete lattice. F or b oth cases, the dynamical lo c alized mo de displays high-amplitude (ex- p onen tial) oscillations at the impurit y site, due to the particularly asymmetric form of the coupling b et w een the impurit y and its neighbors. T he presenc e of a surface, on the other hand, increases the amoun t of minim um nonlinearity needed to create a lo calized mo de, in agreemen t with previous studies on one-dimensional DNLS systems. 1 5 10 - 1 0 1 2 1 5 10 - 1 0 1 2 1 5 10 0 1 2 1 5 10 0 1 2 FIG. 4: Su rface impurity: localized m odes for ν = 3 (left column) and ν = 4 (righ t column). The top (b otto m) r o w sh o ws the unstaggered (staggered) ve rs ions of the mo de. 8 0 1 2 3 4 5 6 Ν 0 0.2 0.4 0.6 0.8 Ξ 1 2 3 4 5 6 Ν - 4 - 2 0 2 4 Β FIG. 5: S urface impurit y . L e ft: Nu merica l solution for ξ in terms of ν . Righ t: Bound state energies of lo ca lized mo des as a function of nonlinearit y p a rameter. Solid(dashed) cur v e denotes stable(unstable) solution. 0 2 4 6 8 10 Position 0 5 10 15 20 z 0 5 10 15 20 z 0 500 1000 1500 È Φ 0 È 2 0 5 10 15 Ν 0 20 40 60 < È Φ 0 È 2 > 0 2 4 6 8 10 Position 0 5 10 15 20 z FIG. 6: Sur fac e impurit y . T op left: Long-time a verag e probabilit y at impur ity site. T op righ t: Ev olution of initial lo calized excitation across the lattice f or ν = 14. Bottom left: Evolution for ν = 15. Bottom right: Ev olution of amp l itud e at impurit y site for ν = 15. 9 I. A CKNOWLEDGMENTS This w ork has b ee n supp orted b y F ondecyt grant 108 0374 in Chile. The a uthor is grateful to M. J. Ablo witz for useful discussions . [1] See, e.g., D.N. Christo doulides, F. Lederer, and Y. Silb erb erg, Nature 424 , 817 (2003 ) and references therein. [2] M. J. Ablo witz and J. F. Ladik, J. Math. Phys. 16 , 598, (1975 ). [3] J. Gernier, Phys. Rev. E 63 , 0266 08 (200 1). [4] M. I. Molina, Ph ys. Rev. B 71 , 035404 (2005 ). [5] K. G. Makris, S. Suntso v, D. N. Christo doulides and G. I. Stegeman, Opt. Lett. bf 30, 2466 (2005 ). [6] M. I. Molina, R. A. Vicencio and Y. S . Kivshar, Optt. Lett. 31 , 1693 (200 6). [7] M. I. Molina, Ph ys. Rev. B 74 , 045412 (2006 ). [8] H. Susan to, P . G. Kevrekidis, B. A. Malomed, R. Carretero-Gonz´ alez and D. J. F ran tzesk akis, Ph ys. Rev. E. 75 , 05660 5 (2007) . 10 List of Figure Captions Figure 1: Impurit y in bulk: lo calized mo des for ν = 1 . 25 (left column) and ν = 1 . 5 (right column). The top (b ottom) ro w sho ws the unstaggered (staggered) vers ions of the mo de. Figure 2: (Color o nline) Impurit y in bulk. Left: ξ a s a function of ν for lo calized mode. Righ t: Bound state energy o f lo calized mo de as a function of nonlinearity parameter. The shaded area mar k s the p osition of the linear band, while the upp er (low er) curv e corresp onds to the unstaggered (staggered) mo de. The blac k dot marks the p osition of ν c = 1. Figure 3: (Color online) Impurit y in bulk. T op left: Long- time a ve rag e proba bility at impurit y site. T op right: Ev olution of initial lo calized excitation across t he lattice for ν = 2. Bottom left: Ev o lut io n for ν = 8. Bottom righ t: Ev olution of amplitude at impurit y site for ν = 8. Figure 4: Surface impurity : lo calized mo des for ν = 3 (left column) and ν = 4 (righ t column). The top (b ottom) ro w sho ws the unstaggered (staggered) vers ions of the mo de. Figure 5: (Color online) Surface impurity . Left: Nume rical solution for ξ in terms of ν . Righ t: Bound state energies of lo calized mo des as a function of nonlinearity parameter. Solid(dashed) curv e denotes stable(unstable) solution. Figure 6: Surface impurit y . T op left: Long-time av erage probabilit y at impurit y site. T op righ t: Ev olution of initial lo calize d excitation across the lattice for ν = 14. Bot t om left: Ev olution for ν = 15. Bo t t o m righ t: Ev olution o f amplitude at impurity site for ν = 15. 11