Moving discrete breathers in a Klein–Gordon chain with an impurity

Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y J Cue as†, F Palme o†, JFR A chilla†and FR Rome o‡ †ETS Ingenie ´ıa In o m´a ica. Uni e sidad de Se illa. A da Reina Me cedes s/n, 41012-Se illa, Spain ‡Facul ad de F´ısica. Uni e sidad de Se illa. A da Reina Me cedes s/n, 41012-Se illa, Spain Abs ac . We analyze he influence o an impu i y in he e olu ion o mo ing disc e e b ea he s in a Klein–Go don chain wi h non-weak nonlinea i y. Th ee diffe en beha iou s can be obse ed when mo ing b ea he s in e ac wi h he impu i y: hey pass h ough he impu i y con inuing hei di ec ion o mo emen ; hey a e eflec ed by he impu i y; hey a e apped by he impu i y, gi ing ise o chao ic b ea he s, as hei Fou ie powe spec a show. Resonance wi h a b ea he cen ed a he impu i y si e is conjec u ed o be a necessa y condi ion o he appea ance o he apping phenomenon. This pape es ablishes a diffe ence be ween he esonance condi ion o he non-weak nonlinea i y app oach and he esonance condi ion wi h he linea impu i y mode in he case o weak nonlinea i y. PACS numbe s: 63.20.Pw 63.20.Ry 63.50.+x 66.90.+ Submi ed o: J. Phys. A: Ma h. Gen. E-mail: [email p o ec ed] 1. In oduc ion In insic localized modes o disc e e b ea he s esul s om he combina ion o nonlinea i y wi h spa ial disc e eness. They can be ob ained in Klein–Go don la ices as exac solu ions o dynamical equa ions [1, 2, 3]. In addi ion, hese localized oscilla ions, unde ce ain condi ions, can mo e and hey a e usually called mo ing b ea he s [4, 5, 6, 7, 8]. The in e ac ion o nonlinea localized oscilla ions wi h impu i ies in a sys em can play an impo an ole in i s anspo p ope ies. This p oblem has been s udied du ing he las decades wi hin diffe en amewo ks. The sca e ing o kinks and en elope soli ons wi h impu i ies has been s udied in one–dimensional a omic la ices wi h nonlinea in e ac ions [9]. Ano he app oach is conce ned wi h he in e ac ion be ween high– equency con inuous b ea he s and impu i ies in he sine-Go don model [10, 11]. This app oach has been ex ended o he low– equency case in [12], and o he case o kinks in he con inuous sine-Go don and ϕ4models [13, 14]. The sca e ing o a kink by an impu i y in he F enkel–Kon o o a model has been conside ed in [15, 16]. Re e ences [9, 15] desc ibe soli ons ha can be eflec ed by he impu i y o pass h ough i , depending on he eloci y. Howe e , in [10, 11, 13, 14], i is obse ed soli ons can also be apped o in e media e eloci ies. Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 2 The in e ac ion o a mo ing disc e e b ea he wi h an impu i y in a Klein–Go don chain has been conside ed by Fo inash e al [17]. In his case, i is assumed ha he sys em has weak nonlinea i y and h ee diffe en beha iou s can be obse ed: (a) he mo ing b ea he passes h ough he impu i y, (b) i is eflec ed, o (c) i is apped by he impu i y, o igina ing a deposi o y o ene gy. All hese effec s a e ela ed o esonances wi h he impu i y modes. In his pape , we a e in e es ed in he las app oach, i.e., o s udy he ea u es o he in e ac ion o mo ing disc e e b ea he s wi h an impu i y a es in a Klein– Go don chain o oscilla o s wi h non-weak nonlinea i y. Al hough he beha iou in all o hese app oaches is quali a i ely simila , he e a e some significa i e diffe ences conce ning o he appea ance o he apping phenomenon. In he case o soli ons, he diffe en phenomena a e eloci y–dependen . In disc e e la ices, he b ea he and he impu i y mode a e ela ed en i ies [18, 19], i.e., hey can be connec ed h ough a con inuous pa h. The e o e, he in e play be ween a b ea he and an impu i y mode can be much s onge han be ween a soli on and an impu i y. In Fo inash’s app oach, he necessa y condi ion o he appea ance o he apping phenomenon is ha he b ea he equency esona es wi h he linea impu i y mode one. In ou case, ha is, a Klein–Go don chain wi h non-weak nonlinea i y, he necessa y condi ion o he appea ance o apping is ha he e mus exis s a b ea he a he impu i y wi h a equency close o ha o he mo ing b ea he . This ac gua an ees he exis ence o a wide ange o pa ame e s o which apping is possible. Ne e heless, his condi ion is no sufficien , as he apping phenomenon does no occu when he ails o he a b ea he cen ed a he impu i y si e and he linea impu i y mode ha e diffe en ib a ion pa e ns. We p opose he hypo hesis ha bo h ails mus ha e he same ib a ion pa e n in o de ha he apping occu s. 2. Model and solu ions gene a ion 2.1. Fo mula ion o he model In o de o s udy he effec s o impu i ies on he mo emen o b ea he s, we conside a simple model whe e mo ing b ea he s can be gene a ed, ha is, a Klein–Go don chain wi h nea es neighbou s a ac i e in e ac ions [4, 5]. I s Hamil onian is gi en by: H= N ∑ n=1 (1 2˙u2 n+Vn(un) + 1 2C(un−un−1)2),(1) whe e un ep esen s he displacemen o he n- h pa icle wi h espec o i s equilib ium posi ion, Cis a coupling cons an and Vn(un) is he subs a e po en ial a he n- h si e. We choose Vas he Mo se po en ial, i.e., Vn(u) = Dn(e−u−1)2, which p o es o be e y sui able o ob aining mo ing b ea he s [5, 6, 8]. Dn ep esen s he well dep h in he n- h si e. In his model, he inhomogenei y is in oduced assuming a diffe en well dep h in only one si e, i.e., Dn=Do(1 + αδn,0), hen we e e o he pa icle loca ed a n= 0 as an impu i y. αis a pa ame e which unes he magni ude o he inhomogenei y. I akes i s alues in he in e al [−1,∞). He ea e , we will conside Do= 1/2. The esul s p esen ed he e co espond o ee ends bounda y condi ions, al hough pe iodic bounda y condi ions lead o he same esul s. Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 3 This model has been used ex ensi ely in DNA dynamics; in his con ex i is usually e e ed o as he Pey a d-Bishop model [20]. In he amewo k o his model, he a iables un ep esen he s e ching o he hyd ogen bonds connec ing each pai o bases, Dis he dissocia ion ene gy, and Cis he s acking coupling cons an . The inhomogenei y can also be in oduced in a simila way a ying he mass o a pa icle, o he coupling cons an . The esul s in he fi s case a e equi alen o he ob ained o he inhomogenei y in he po en ial well. The e a e, howe e , se e al diffe ences when he inhomogenei y is in oduced h ough he coupling cons an , and we will make some commen s abou hem a he end o his pape . The Hamil onian (1) leads o he dynamical equa ions F({un})≡¨un+V′ n(un) + C(2un−un+1 −un−1) = 0.(2) which ha e wo kinds o solu ions, linea ones, which co espond o oscilla ions o small ampli ude, and nonlinea ones, which co espond o in insic localized modes o disc e e b ea he s. 2.2. Linea modes The dynamical equa ions can be linea ized i he ampli udes o he oscilla ions a e small. Thus, he equa ions (2) a e ans o med in he sys em o coupled equa ions: ¨un+ω2 nun+C(2un−un+1 −un−1) = 0,(3) whe e ωnis he na u al equency o he n- h oscilla o in he ha monic limi . I is gi en by ωn=√2Dn, which implies ha ω2 n=ω2 o(1 + αδn,0), wi h ωo= 1, as Do has been chosen o be 1/2. These equa ions has N−1 non-localized solu ions (being N he numbe o pa icles) co esponding o linea ex ended modes and one localized solu ion, which co esponds o a linea impu i y mode. He ea e , unless s a ed o he wise, he e m mode will be ese ed o linea modes. The equencies o he ex ended modes can be calcula ed supposing ha hey a e plane wa es (un( ) = uoexp(iω(q) −nq)) and ha he impu i y mode decays in he space ollowing a dependence o he o m un( ) = uoexp(iωL ) |n|[17], whe e is a spa ial decay pa ame e . Thus, he equencies o he ex ended modes a e gi en by: ω(q, α) = √ω2 o+ 4Csin2q(α) 2,(4) whe e q(α) is he ex ended modes wa e ec o , which depends in a non- s aigh o wa d way on α. The equency o he impu i y mode is gi en by he ela ion [17]: ω2 L=ω2 o+ 2C+ sign(α)√α2ω4 o+ 4C2.(5) The sign o indica es he ib a ion pa e n o he impu i y mode. Thus, i > 0, he pa icles o he mode ib a e in phase and will ha e a wa e ec o q= 0. On he con a y, i < 0, he mode will ha e a zigzag ib a ion pa e n and a wa e ec o q=π. The pa ame e is gi en by: =−sign(α)αω2 o+√4C2+α2ω4 o 2C.(6) Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 4 −1 −0.5 0 0.5 1 1.5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2α es αc 2ωb ωb Linea modes equencies α Figu e 1. F equencies o he linea modes e sus he pa ame e α, o C= 0.13. The dependence is quali a i ely simila o e e y alue o C. A α=α es and α=αc, wo diffe en bi u ca ions occu , being he fi s one due o he esonance be ween he impu i y mode and he b ea he . The e o e, αand has opposi e sign. This ac also implies ha , o he ex ended modes, q∈(0, π] i α < 0 and q∈[0, π) i α > 0. Figu e 1 shows he dependence o he equencies o he linea modes wi h α, whe e he isola ed equencies co esponds o he impu i y modes. As i is explained below, he alues o he equency and wa e ec o o he impu i y mode will be he key o explain he occu ence o he apping phenomenon. 2.3. S a iona y and mo ing b ea he s A s a iona y b ea he can be ob ained by sol ing he ull dynamical equa ions. I can be achie ed using common me hods based on he an icon inuous limi [21]. The implemen a ion o hese me hods basically consis s in calcula ing he o bi o an isola ed oscilla o a fixed equency ωb, and using his solu ion as a seed o sol e he comple e dynamical equa ions by means o a New on–Raphson con inua ion me hod. I he oscilla o ini ially chosen is he co esponding o he impu i y, a s a ic b ea he cen ed a he impu i y is ob ained. I will be called impu i y b ea he . Once a s a iona y b ea he is ob ained, i can be mo ed unde ce ain condi ions. The e exis s a sys ema ic me hod o calcula ing mo ing solu ions [4, 5] which consis s in adding o he eloci ies o he s a iona y b ea he a pe u ba ion o magni ude λ colinea o he di ec ion o he pinning mode and le ing he sys em e ol e in ime. The pinning mode is an an i-symme ic linea localized mode, which may appea in he se o linea pe u ba ions o he sys em p o ided he coupling is s ong enough [6]. Thus, a pe u ba ion in i s di ec ion b eaks he ansla ional symme y o he sys em and make he b ea he mo e. The esul s p esen ed he e co espond o a equency ωb= 0.8 and a coupling Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 5 Figu e 2. Diffe en egimes in he in e ac ion o a mo ing b ea he wi h an impu i y in oduced as an inhomogenei y in he po en ial well dep h. C= 0.13, al hough o he alues o he same o de gi e quali a i ely simila esul s. In his way, we ob ain mo ing b ea he s wi h low phonon adia ion o alues o he pe u ba ion λ.0.2. I is wo h emaking ha he nonlinea i y o ou sys em is highe han he conside ed in [17], as, in ha pape , he equencies o he localized exci a ions oscilla e be ween 0.922 and 0.980 (in ou equency uni s, i.e. no malized o he linea equency a ze o coupling ωo= 1), which a e e y close o he linea ones. We ha e conside ed diffe en alues o he pa ame e αin he ange −1≤α≤1. 3. In e ac ion o mo ing b ea he s wi h an impu i y 3.1. Nume ical obse a ions We ha e s udied he beha iou o mo ing b ea he s when hey in e ac wi h an impu i y. The s udy has been pe o med a ying he alue o he inhomogenei y pa ame e α. Fou diffe en egimes, sepa a ed by c i ical alues o he pa ame e α, ha e been ound (see figu e 2): (i) Ba ie . The impu i y ac s as a po en ial ba ie . I occu s ei he wi h α > 0 o α∈(−1, α1) wi h α1<0. As he mo ing b ea he eaches he impu i y, i is gene ally eflec ed, lea ing he impu i y exci ed du ing a sho ime, whose ampli ude dec eases wi h |α|. The only excep ion o his beha iou occu s o α&0. In his case, he b ea he can pass h ough he impu i y p o ided he ansla ional eloci y is high enough. (ii) Exci a ion. The impu i y is exci ed and he b ea he is eflec ed. I occu s o α∈(α1, α2). The ene gy o he exci ed impu i y is la ge han he ene gy o he impu i y b ea he . Thus, he exci ed impu i y ib a es wi h a equency lowe han ωbas he on–si e po en ial is so . This beha io is shown in figu e 3. (iii) T apping. The b ea he is apped by he impu i y. I occu s in he in e al α∈(α2, α3). When he mo ing b ea he is close o he impu i y, i becomes apped while i s cen e oscilla es be ween he neighbou ing si es, as figu e 4 shows. Fu he mo e, he apped b ea he emi s a g ea amoun o phonon adia ion and seems o be chao ic, as can be app ecia ed om i s Fou ie powe spec um (Figu e 5). (i ) Well. The impu i y ac s as a po en ial well. I occu s o α∈(α3,0) and consis s o an accele a ion o he b ea he as i app oaches o he impu i y, and a decele a ion a e he impu i y has been passed h ough. Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 6 −20 −10 0 10 20 0 50 100 150 200 250 0 0.2 0.4 Pe iods Posi ion Ene gy 0 50 100 150 200 250 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 Ene gy densi y a he impu i y si e Pe iods EN Figu e 3. In e ac ion o a b ea he wi h an impu i y o α=−0.52 and λ= 0.1, which co esponds o he impu i y exci a ion case. Top: E olu ion o he mo ing b ea he . Bo om: E olu ion o he ene gy densi y o he impu i y. ENis he ene gy o he impu i y b ea he . The ansi ion be ween he diffe en egimes is somehow diffuse, ha means, he alues o he c i ical alues o αcanno be exac ly de e mined. Fu he mo e, hey a e sligh ly dependen on he b ea he eloci y. An es ima ion o he c i ical alues o ωb= 0.8, C= 0.13 and λ= 0.1 leads o: α1≈ −0.54, α2≈ −0.49 and α3≈ −0.02. As commen ed abo e, hese egimes ha e also been ound o diffe en alues o he b ea he equency. Fo s onge coupling, he phonon adia ion is significan and could mask some o he desc ibed effec s. Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 7 −20 −10 010 20 0 100 200 300 400 500 0 0.2 0.4 Pe iods Posi ion Ene gy Figu e 4. E olu ion o he mo ing b ea he o α=−0.3 and λ= 0.1, which co esponds o he apping case. The mo ing b ea he becomes apped by he impu i y; a e wa ds, he b ea he emi s phonon adia ion and i s ene gy cen e oscilla es be ween he si es adjacen o he impu i y, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 0 1 2 3 log|S(ω)| ω 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 log|S(ω)| ω Figu e 5. Fou ie powe spec a o he apped b ea he o figu e 4 ( op) and a s a iona y b ea he wi h he same alues o Cand ωb(bo om). The spec um on he op panel is cha ac e is ic o chaos. Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 8 −1 −0.8 −0.6 −0.4 −0.2 0 −1 −0.5 0 0.5 1 1.5 2 uo( =0) α Figu e 6. Pi ch o k bi u ca ion o ωb= 0.8 and C= 0.13. α es =−0.5328. 3.2. Discussion Some o he esul s in he las subsec ion can be explained om he p ope ies o he impu i y b ea he . Pa icula ly, i a con inua ion o a s a iona y b ea he is pe o med a ying he pa ame e α, a bi u ca ion appea s o α=αc>0 and ano he one o α=α es <0 (see Figu e 1). The fi s one is o igina ed by a localized Floque eigenmode which abandons he uni ci cle and leads o a b ea he ex inc ion, i.e., he impu i y b ea he does no exis o α > αc. In he second case, he b ea he bi u ca es wi h he ze o solu ion h ough a pi ch o k bi u ca ion (see figu e 6) in he space o ime– e e sible solu ions o equency ωb‡. In his bi u ca ion, he ou e b anches co espond o impu i y b ea he s ei he wi h un(0) >0 o un(0) <0, while he cen al b anch co esponds o he ze o solu ion, i.e. all he oscilla o s a e a es . Fo α=α es he equency o he impu i y mode is he same as he equency o he impu i y b ea he wi h he mo ing b ea he equency, i.e., ωL=ωb.α es can be calcula ed om he equa ion (5) as a unc ion o ωband C: α es =−√(ω2 b−ω2 o)(ω2 b−ω2 o−4C) ω2 o (7) Thus, o C= 0.13 and ωb= 0.8, α es =−0.5628, which is lowe han α1. Howe e , he apped b ea he does no exis o α > 0. I indica es ha he condi ion α∈(α es,0) migh hold in o de ha he apped b ea he exis s. The scena io o he apped b ea he s when α < 0 is he ollowing: in his case, he impu i y mode has q= 0, and also all he pa icles o he impu i y b ea he ib a es in phase; his ib a ion pa e n indica es ha he impu i y b ea he bi u ca es om plane wa es wi h q= 0 [22], i.e., he impu i y bi u ca es om he impu i y mode and i will be he only localized mode ha exis s when he impu i y is exci ed ‡No e ha he dynamical equa ions (2) do no co espond o a s anda d dynamical sys em ¨x= (x, ), whe e usually pi ch o k bi u ca ions a e desc ibed Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 9 o α > α es. Thus, when he mo ing b ea he eaches he impu i y, i can exci e he impu i y mode. In ac , we ha e pe o med a success ul con inua ion om he impu i y b ea he o he impu i y mode a cons an ac ion and α[18, 19], and a ying a pa ame e swhich unes he nonlinea i y o he sys em. This pa ame e is in oduced by changing he on–si e po en ial o he exp ession V∗ n(un) = Dn(u2 n−su2 n)+sVn(un), whe e Vn(un) is he o iginal po en ial (1). When α es < α < α2, he impu i y b ea he is unable o c ea e a apped en i y. The ene gy o he impu i y mode dec eases wi h |α|, hus he e mus be a minimum alue o he impu i y b ea he ene gy o he exis ence o apping. The na ow window o impu i y exci a ions obse ed in he in e al (α1, α2) can be due o a esonance o he mo ing b ea he wi h he impu i y b ea he wi h a equency sligh ly smalle han ωb. Fo α < α es, he apped b ea he canno be gene a ed, and he mo ing b ea he is always eflec ed. In addi ion, he impu i y b ea he does no exis . The e o e, he e migh be a connec ion be ween bo h ac s, i.e., he exis ence o he impu i y b ea he seems o be a necessa y condi ion in o de o ob ain a apped b ea he . I α > 0, he scena io is diffe en . In his case, he impu i y mode has q=π bu he impu i y b ea he si es ib a e again in phase, ha is, he impu i y b ea he does no bi u ca e om he impu i y mode. Thus, he e a e wo diffe en localized exci a ions o α > 0: he (linea ) impu i y mode and he (nonlinea ) impu i y b ea he . Bu , ac ually, he equa ions ha go e n he sys em a e nonlinea , so he linea modes can only co espond o low-ampli ude oscilla ions. In he case o he impu i y b ea he , he linea egime co esponds o he ails. Thus, i he mo ing b ea he eaches he impu i y si e, i will exci e he impu i y b ea he and he ails o he impu i y mode. Bu he la e ib a es in zigzag. As a consequence, he e will be wo diffe en linea localized en i ies: he ails o he impu i y mode ( ib a ing in zigzag) and he ails o he impu i y b ea he ( ib a ing in phase). The e o e, we conjec u e ha he exis ence o bo h linea localized en i ies a he same ime may be he eason why he impu i y is unable o ap he b ea he when α > 0. I can be hough whe he esonances wi h he ha monics o he b ea he equency could ha e consequences in he in e ac ion o he mo ing b ea he wi h he impu i y. In ou sys em, hese esonances occu o α > αc, and he e o e, he impu i y b ea he does no exis , so ha no apping effec s ake place, leading only o b ea he eflec ions. We summa ize he e ou hypo hesis o he exis ence o apping: T apping hypo hesis:The exis ence o an impu i y b ea he o a gi en alue o αis a necessa y condi ion o he exis ence o apped b ea he s. Howe e , i he e exis s an impu i y mode wi h a diffe en ib a ion pa e n o he impu i y b ea he one, he apped b ea he does no o exis . 4. Inhomogenei y in he coupling pa ame e In o de o check whe he he hypo hesis p oposed in he las sec ion holds o diffe en si ua ions, we conside a chain o oscilla o s o which he imhomogenei y is in oduced h ough he coupling cons an s. In his case, he Hamil onian can be w i en as: H=∑ n(1 2˙u2 n+V(un) + 1 4Cn[(un−un−1)2+ (un−un−1)]),(8) which leads o he ollowing dynamical equa ions: