Demonstration of the stability or instability of multibreathers at low coupling

Demons a ion o he s abili y o ins abili y o mul ib ea he s a low coupling JFR A chilla 1, J Cue as, B S´anchez–Rey and A Al a ez* Nonlinea Physics G oup o he Uni e si y o Se illa Dep. F´ısica Aplicada I, ETSI In o m´a ica and *Facul ad de F´ısica, p. 5 A da Reina Me cedes s/n, 41012 Se illa, Spain Abs ac Whe eas he e exis s a ma hema ical p oo o one–si e b ea he s s abili y, and an unpublished one o wo–si es b ea he s, he me hods o de e mining he s abili y p ope ies o mul ib ea he s ely in nume ical compu a ion o he Floque mul- iplie s o in he weak nonlinea i y app oxima ion leading o disc e e non–linea Sch ¨odinge equa ions. He e we p esen a se o mul ib ea he s abili y heo ems (MST) ha p o ides wi h a simple me hod o de e mine mul ib ea he s s abili y in Klein–Go don sys ems. These heo ems a e based in he applica ion o degen- e a e pe u ba ion heo y o Aub y’s band heo y. We illus a e hem wi h se e al examples. Key wo ds: Disc e e b ea he s; Mul ib ea he s; In insic localized modes; PACS: 63.20.Pw, 63.20.Ry, 66.90.+ . 1 In oduc ion Disc e e b ea he s a e ime–pe iodic, localized oscilla ions in disc e e sys ems due o a combina ion o nonlinea i y and disc e eness. They ha e become a well unde s ood phenomenon since he publica ion o he p oo o exis ence in Re . [1]. This p oo is he o igin o exac and powe ul nume ical me hods o calcula e hem and o de e mine hei s abili y [2]. A deepe insigh has been achie ed since he in oduc ion o Aub y’s band heo y [3]. The las e e ence, oge he wi h Re . [4] can be conside ed as e iews, al hough a new one is badly needed a e he huge de elopmen o he subjec in he las yea s. 1Co esponding au ho . E-mail: [email p o ec ed] P ep in submi ed o Physica D 9 Feb ua y 2003 The s abili y o one–si e b ea he s is p oo ed in Re s. [2,3] unde a he gene al condi ions, he s abili y o he possible wo–si es b ea he s in analyzed in an unpublished heo em in Re . [5], page 69. O he me hods use he hypo hesis o small ampli ude oscilla ions o he o a ing wa e app oxima ion [6,7] leading o he Nonlinea Sch ¨odinge equa ion (DNLS), which is he ac ual equa ion analyzed, and, he e a e, o cou se, he efficien bu slow nume ical me hods men ioned abo e. He e we p opose a me hod based in he p ope ies o he bands in Aub y’s bands heo y and pe u ba ion heo y, o ob ain he s abili y p ope ies o any mul ib ea he a low coupling. In many cases i in ol es only some modi- fica ion o he coupling ma ix and he knowledge o he ha dness/so ness o he on–si e po en ial. We will b iefly summa ize he band heo y in he nex sec ion, while we in oduce he no a ion and some basic concep s, on which ou heo y is based. Howe e , we e e o Re . [3,1,5] o de ailed explana ions, as i will be long and epe i i e o expose i in de ail. In sec ion 3 we de elop he me hod o symme ic on–si e po en ials o in– phase mul ib ea he s, which is syn hesized in a heo em, commen ed in sec- ion 4. I s scope is enla ged o non–symme ic po en ials in sec ion 3, and o gene alized Klein–Go don sys ems in sec ion 6. The me hod is applied o se e al in e es ing examples in sec ions 7 and 8. In Appendix A we calcula e he alue o a magni ude γused in ou heo y, and in Appendix B we ela e he cu a u e o he bands wi h he cha ac e is ics o he on si e po en ial. 2 Band heo y and no a ion 2.1 The New on ope a o We conside Klein–Go don sys ems wi h linea coupling desc ibed by dynam- ical equa ions o he o m: ¨un+V′(un) + ε N ∑ m=1 Cn mun= 0 n= 1, . . . , N (1) whe e he a iables una e unc ions o ime ,V(un) is an homogeneous on– si e po en ial, V′i s de i a i e, ˙undeno es de i a ion wi h espec o ime, Nis he numbe o oscilla o s, Cis a coupling cons an ma ix, which can desc ibe nea es neighbo o long– ange in e ac ion, and includes he bounda y condi ions, and εis he coupling pa ame e . We use a no a ion simila o quan um mechanics, i.e., |u⟩ ≡ [u1( ), . . . , uN( )]†(†meaning he anspose ma ix). Defining V(u) = [V(u1), . . . , V (uN)]†and analogously i s de i a i es, 2 equa ion (1) can be w i en as: |¨u⟩+|V′(u)⟩+ε C |u⟩= 0 .(2) We suppose ha he unc ions un( ) a e ime–pe iodic (wi h pe iod Tand equency ωb), ime– e e sible solu ions and, he e o e, hey can be w i en as cosine Fou ie se ies wi h eal coefficien s (some o hese assump ions will be elaxed la e ). The (linea ) s abili y p ope ies o a gi en solu ion |u⟩depend on he p ope ies o he cha ac e is ic equa ion o he New on ope a o Nε gi en by Nε(u)|ξ⟩≡|¨ ξ⟩+V′′(u)·|ξ⟩+ε C |ξ⟩=E|ξ⟩,(3) whe e ·p oduc is he lis p oduc , i.e., (u)·|ξ⟩is he column ma ix wi h elemen s (un( )) ξn( ). I E= 0, his equa ion desc ibes he e olu ion o small pe u ba ions |ξ⟩o |u⟩. 2.2 The Floque ma ix Any solu ions o Eq. (3) can be de e mined by he column ma ix o he ini ial condi ions o posi ions and momen a Ω(0) = [ξ1(0), . . . ξN(0), π1(0), . . . πN(0)]†, wi h πn( ) = ˙ ξn( ), and Ω( ) desc ibes i s e olu ion in he space o coo dina es and momen a. A base o solu ions is gi en by he 2 N unc ions wi h ini ial condi ions Ων(0), ν= 1, . . . , 2N, wi h Ων l(0) = δν l. We will iden i y o en a gi en solu ion ξ( ) o he New on equa ion wi h he co esponding ma ix o ini ial condi ions Ω(0). As he New on ope a o depends on he T–pe iodic so- lu ion u( ), i is also pe iodic, and he e olu ion o he solu ions o Eq. (3) can be s udied by means o he Floque ope a o , which maps ξ( ) in o ξ( +T). In a fini e sys em his is equi alen o he Floque ma ix FEbeing gi en by:   {ξn(T)} {πn(T)}  =FE  {ξn(0)} {πn(0)}  (4) These ma ices can be easily cons uc ed nume ically, by in eg a ing Eq. (3) 2N imes om = 0 o =T, wi h ini ial condi ions Ων(0). Then, he ν column o FEis gi en by Ων(T). 2.3 S abili y and bi u ca ions The eigen alues {λl}2N l=1 o F0, called he Floque mul iplie s, de e mine he linea s abili y o he solu ion u( ). I he e is any eigen alue wi h |λl|>1, he co esponding eigen unc ion ξl( ) g ows wi h ime and u( ) is uns able; i λl= 1, ξl( ) is T-pe iodic; i λl=−1, ξl( ) is 2 T–pe iodic. Howe e , F0(and 3 any FE) is a symplec ic ma ix, as i is de i ed om he symplec ic sys em Eq. (3), which implies ha i λlis a non–ze o eigen alue, so i is 1/λ.FEis also eal, which in u n implies ha λ∗ land 1/λ∗a e also mul iplie s. The e o e, he eigen alues o FEcome in g oups o ou , i hey a e no eal, o in pai s, i hey a e eal o i |λl|= 1. We can w i e he Floque mul iplie s as {exp(iθl)}, wi h θl, in gene al, complex numbe s. The complex numbe iθla e called he Floque exponen s, and θl he Floque a gumen s. The only possibili y o he s abili y o u( ) is ha all he eigen alues o F0 ha e moduli 1, i.e., hey a e a he uni ci cle. The e o e, he condi ion o linea s abili y o u( ) can be desc ibed as all he Floque a gumen s o F0 being eal. Mo eo e , i u( ) is s able and, hen, e e y |λl|= 1, he Floque mul iplie s come in complex conjuga e pai s wi h a gumen s (±θl) o as double 1 o −1 (θ= 0 o θ=±π). I a pa ame e like he coupling εis changed he mul iplie s o F0change con inuously. The e o e, a bi u ca ion o an uns able solu ion can only ake place in h ee diffe en o ms: a) Ha monic ins abili y: wo complex eigen alues mo ing along he uni ci cle collide a λ= 1 (θ= 0); b) Subha monic ins abili y: wo complex eigen alues mo ing along he uni ci cle collide a λ=−1 (θ=±π); c) Oscilla o y o Hop ins abili y: wo pai s o complex eigen alues mo ing along he uni ci cle collide a ±θ= 0 and abandon he uni ci cle as a quad uple (λ, 1/λ, λ∗,1/λ∗). A Floque mul iplie o F0is always known. Calcula ing he de i a i e o Eq. (2) wi h espec o ime we ob ain ha N˙u= 0. ˙uis also T–pe iodic, he e o e, i is an eigen unc ion o F0wi h mul iplie 1, and as hey come in pai s, he e is always a double 1 mul iplie . ˙uis called he phase mode because i s meaning is ha i ˙u( ) is a solu ion o Eq. (2), ˙u( +d ) is also a solu ion. While he solu ion u( ) exis s, his double eigen alue is always he e. Due o he possible o ms o he bi u ca ions, i all he mul iplie s a e isola ed excep he double 1, he sys em is s uc u ally s able, i.e., he e is a neighbo hood in he space o he pa ame e s whe e u( ) is s able. 2.4 Aub y’s band heo y Howe e , i u ns ou ha much in o ma ion can be ob ained by s udying also he Floque a gumen s o E= 0, wi h is known as Aub y’s band heo y [3]. The se o poin s (θ, E), wi h θa eal Floque a gumen o FEha e a band s uc u e. The ac ha he Floque mul iplie s come in pai s o complex conjuga e pai s b ings abou ha i (θ, E) belongs o a band, (−θ, E) does i oo, i.e., he bands a e symme ic wi h espec o θand dE/ dθ(0) = 0. As ˙u has eigen alue 1 (θ= 0), he e is always a band angen o he axis E= 0 a θ= 0. The e a e a mos 2 Npoin s o a gi en alue o Eand, he e o e, he e a e a mos 2 Nbands c ossing any ho izon al axes in he space o coo dina es 4 (θ, E). The condi ion o linea s abili y o u( ) is equi alen o he exis ence o 2 Nbands c ossing he axis E= 0 (including angen poin s wi h hei mul iplici y). I a pa ame e like he coupling εchanges, he bands e ol e con inuously, and hey can loss c ossing poin s wi h E= 0, b inging abou he ins abili y. We know mo e abou he eigen unc ions o he New on ope a o . The ac ha i is pe iodic, o , in o he wo ds, ha i commu es wi h he ope a o Po ansla ion in ime a pe iod, P ( ) = ( +T), implies ha i s eigen unc ions can be chosen simul aneously as eigen unc ions o P, which by Bloch heo- em, a e gi en by ξ(θ, ) = χ(θ, ) exp(i θ /T), χ(θ, ) being a column ma ix o T–pe iodic unc ions. I is s aigh o wa d o check ha exp(iθ) is he co - esponding Floque mul iplie , and θi s Floque a gumen (FEis simply he ep esen a ion o Pin he base Ωl). 2.5 Bands a he an icon inuous limi The key concep on he demons a ion o b ea he exis ence [1], single b ea he s abili y [3] and he p esen pape is he an icon inuous limi , i.e., he sys em wi h all he oscilla o s uncoupled, ε= 0, in Eq. (1–2). A he an icon inuous limi , Eq. (1) educes o Niden ical equa ions: ¨un+V′(un) = 0 .(5) Supposing ha we conside ime– e e sible solu ions a ound a single minimum o V, he e a e only h ee diffe en solu ions: a) oscilla o s a es un= 0; b) exci ed oscilla o s wi h iden ical un( ), he ea e deno ed u0( ); c) exci ed oscilla o s wi h a phase diffe ence o πwi h he p e ious ones, gi en by un( ) = u0( +T/2). Each si e index is gi en a code σn, which akes elemen s in {0,1,−1}, whe e σn= 0 ep esen s an oscilla o a es , un= 0; σn= 1, an oscilla o wi h solu ion u0( ); and σn=−1, he solu ion u0( +T/2). The ma ix o codes σ= [σ1, . . . , σN]† ep esen s he s a e o he sys em a he an icon inuous limi . We suppose ha he e a e poscilla o s a es and N−pexci ed oscilla o s. Equa ion (3) a ε= 0 becomes N0(u)|ξ⟩≡|¨ ξ⟩+V′′(u)·|ξ⟩=E|ξ⟩,(6) o , equi alen ly, N−piden ical equa ions: ¨ ξn+V′′(un)ξn=E ξn,(7) wi h only a pe iodic eigen unc ion o E= 0, he phase mode ˙un( ) o he iso- la ed, exci ed oscilla o s. The o he one is he g ow h mode, (see Appendix B) 5 which is no bounded and will no be used in his wo k. They gi e ise o N−p bands angen o he axis E= 0, shown in Fig. 1. They ha e posi i e cu - a u e a (θ, E) = (0,0) i he on–si e po en ial is so ( dH/ dωb<0) and nega i e i i is ha d ( dH/ dωb>0), as i is demons a ed in Appendix B. The possibili y o dH/ dωb= 0 is excluded by he condi ions o he b ea he exis ence heo em [1], which means ha he on-si e po en ial is uly nonlin- ea o he isola ed oscilla o s wi h he b ea he equency ωb. I , as we a e supposing he e, he on–si e po en ials V(un) a e iden ical, he solu ions un( ) a e also iden ical excep o a change o phase and he bands a e supe posed, i no , we migh ha e diffe en bands bu wi h he same gene al shape. The emaining pequa ions co esponding o he oscilla o s a es a e o he o m: ¨ ξn+ (ω0)2ξn=E ξn,(8) wi h ω0=√V′′(0). They ha e only a T-pe iodic solu ion o E= 0 , he null solu ion, due o he non– esonance condi ion o b ea he exis ence nωb=ω0 (p∈N) [1], i.e., none o he ha monics o he b ea he esona es wi h he oscil- la o s a es . They p o ide piden ical bands which a e easily calcula ed sol - ing he equa ion abo e. I s solu ions a e ξ= exp(±i√ω2 0−E ), wi h Floque mul iplie s exp(±i√ω2 0−E T) and Floque a gumen s θ=±√ω2 0−E T. Tha is, he bands a e gi en by E=ω2 0−ω2 b(θ/T)2, whe e θcan be e- duced o he fi s B illouin zone [−π, π] by he addi ion o 2πp,p∈Z. I he oscilla o s ha e diffe en es equencies, he es bands a e no supe posed bu hey ha e he same cha ac e is ics. The es bands a e also shown in Fig. 1. No e ha his figu e is only a ske ch, o cla i y, as e y o en some exci ed oscilla o s bands a e e y fla and di - ficul o app ecia e a he same scale. This ske ch ep oduces, howe e , he basics ac s o he band s uc u e. When he coupling is swi ched on, he degene acy o he bands is gene ically aised. All he angen bands a (θ, E) = (0,0) excep one, han con inues he e, mo e upwa ds o downwa ds. I he on–si e po en ial is so , Fig. 1 (le ), and a band mo es upwa ds, a double angen poin wi h he axis E= 0 is los , o , in o he e ms, a pai o Floque a gumen s o F0becomes complex and he solu ion u( ) is uns able. I he on si e po en ial is ha d, Fig. 1 ( igh ), he same occu s when a band mo es downwa ds. 3 Mul ib ea he s s abili y In his wo k use degene a e pe u ba ion heo y [8] o demons a e he s abil- i y o ins abili y o he b ea he s o any code. Degene a e pe u ba ion heo y 6 es ablishes ha i N0is a linea ope a o wi h a degene a e eigen alue E0, wi h eigen ec o s {| n⟩}, which a e o ono mal wi h espec o a scala p od- uc , i.e., ⟨ n| m⟩=δn m, and i ε˜ Nis a pe u ba ion o N0, wi h εsmall; hen, o fi s o de in ε, he eigen alues o N0+ε˜ Na e E0+ε λi, wi h λibeing he eigen alues o he pe u ba ion ma ix Qwi h elemen s Qn m =⟨ n|˜ N| m⟩. No e ha pe u ba ion heo y as desc ibed in he e e ence ci ed, is ime inde- penden pe u ba ion heo y, bu ou ime a iable is hei spa ial coo dina e x. Le us conside again Eq. (3) wi h ze o coupling, ε= 0. As explained in he p e ious sec ion, i he e a e N−pexci ed oscilla o s, he e N−pze o eigen- alues co esponding o he T–pe iodic eigen unc ion ˙un( he phase modes o he isola ed oscilla o s), o in o he wo ds, a N−p imes degene a e eigen alue E0= 0 i we es ic he domain o N o pe iodic unc ions. Wha we need o know is he sign o his degene a e eigen alues when he coupling is swi ched on. To apply degene a e heo y o mul ib ea he s abili y we need o iden i y he pe u ba ion ope a o , a sui able scala p oduc and a o ono mal basis o he eigenspace wi h eigen alue E0= 0. The scala p oduc is defined as: ⟨ξ1|ξ2⟩= N ∑ n=1 ∫T/2 −T/2 ξ∗ 1( )ξ2( ).d . (9) Le us suppose ini ially ha all he exci ed oscilla o a e iden ical and ib a e in phase. We will deno e by u0( ) hese iden ical solu ions. In his case, all he phase modes a e also iden ical and will be deno ed as ˙u0. The N−pelemen s o he basis a e |n⟩=1 µ[0, . . . , 0,˙u0,0,...,0]†,(10) wi h he non-ze o elemen a he posi ion n,nbeing he index o he exci ed oscilla o s, and µ=√∫T/2 −T/2( ˙u0)2d . I is s aigh o wa d o check ha hey a e o ono mal. I is enough o know he eigen alues co esponding o pe iodic and eal so- lu ions, i.e., wi h Floque a gumen θ= 0, as he in e sec ions o he bands wi h he axis θ= 0 co espond o pe iodic solu ions o (3). The e o e he N−p|n⟩’s o m he basis o he degene a e eigen alue E0= 0 needed o apply pe u ba ion heo y. To ob ain he pe u ba ion ope a o , we expand in Taylo se ies Eq. (3) a ε= 0 and ob ain o fi s o de in ε: Nε(u)|ξ⟩=|¨ ξ⟩+V′′(u)·|ξ⟩+ε(V′′′(u)·uε·|ξ⟩+C|ξ⟩)= = (E0+ε λi)|ξ⟩,(11) 7 −2 0 2 −1 −0.5 0 0.5 1 1.5 θ E −2 0 2 −1 −0.5 0 0.5 1 1.5 θ Fig. 1. Ske ch o he band s uc u e o a so on–si e po en ial (le ) and a ha d on–si e po en ial ( igh ). The con inuous lines co espond o exci ed oscilla o s and he dashed ones o oscilla o s a es . wi h uε=(∂u ∂ε )ε=0 and uis also he ε= 0 solu ion. The e o e he pe u ba ion ope a o o N0is ˜ N |ξ⟩=V′′′(u)) ·uε·|ξ⟩+C|ξ⟩,(12) whe e uand uεa e calcula ed a ε= 0. We do no know uε, bu de i ing wi h espec o ε, a ε= 0, he dynamical equa ions (2) we ob ain: |¨uε⟩+V′′(u)·|uε⟩+C|u⟩= 0 o N0|uε⟩=−C|u⟩.(13) The pe u bed eigen alues Eia e ε λi,λibeing he eigen alues o he pe u - ba ion ma ix Q, wi h elemen s ⟨n|˜ N|m⟩. The ma ix ˜ Co elemen s ⟨n|C|m⟩ is simply he ma ix Cwi hou he columns and ows co esponding o he oscilla o s a es . The o he e ms a e: ⟨n|V′′′(u)·uε|m⟩= 1 µ2∫T/2 −T/2 [. . . , 0,˙u0 n,0, . . . ] [. . . , 0, V ′′′(u0)um,ε ˙u0 m,0, . . . ]†d = δn m µ2∫T/2 −T/2 ˙u0V′′′(u0)un,ε ˙u0d , (14) wi h un,ε =(∂un ∂ε )ε=0. Thus, only he diagonal elemen s ⟨n|V′′′(u)·uε· |n⟩ 8 a e non–ze o. To calcula e he las in eg al in (14) we will in eg a e by pa s and use ha he in eg al in a pe iod o he de i a i e o a pe iodic unc ion is ze o. Besides, he unc ions un,ε a e pe iodic as he coefficien s o hei Fou ie se ies a e gi en by he de i a i es wi h espec o εo he Fou ie coefficien s o un. In he deduc ion below, all he in eg al limi s a e −T/2 and T/2, and he e ms be ween b acke s om in eg a ion by pa s will be ze o. The las in eg al in Eq. (14) becomes: [˙u0un,ε V′′(u0)]T/2 −T/2−∫V′′(u0) ˙u0˙un,ε d −∫V′′(u0) ¨u0un,ε d = −[V′(u0) ˙un,ε,]T/2 −T/2+∫V′(u0) ¨uε,n,d −∫V′′(u0) ¨u0un,ε d = −∫¨u0(¨un,ε +V′′(u0)un,ε) d (15) The e m be ween pa en heses, is he ncomponen o he lhs o Eq. (13), i.e., i becomes −∑mCn m u0 m, whe e u0 m=u0, i he oscilla o mis exci ed, and ze o o he wise, i.e., i is −∑m˜ Cn m u0=−(∑m˜ Cn m)u0. Equa ion (15) becomes: (∑ m ˜ Cn m)∫¨u0u0d = (∑ m ˜ Cn m)([˙u0u0]T/2 −T/2−∫( ˙u0)2d )=−(∑ m ˜ Cn m)µ2.(16) Tha is, equa ion (14), leads o: ⟨n|V′′′(u)·uε·|n⟩=−∑ m ˜ Cn m a ε= 0 .(17) The e o e he diagonal elemen s o he pe u ba ion ma ix Qa e Qn n =−∑ m ˜ Cn m +˜ Cn n =−∑ ∀m=n ˜ Cn m =∑ ∀m=n Qn m .(18) To summa ize, he pe u ba ion ma ix Qis gi en by: Qn m =˜ Cn m, n =m , Qn n =−∑ ∀m=n Qn m ,(19) ˜ Cbeing he coupling ma ix wi hou he p ows and columns co esponding o oscilla o s a es . This esul can be e y easily ex ended, o he case when he e a e oscilla o s ou o phase, a leas o an e en po en ial V(u). In his case u0( +T/2) = −u0( ) and ˙u0( +T/2) = −˙u0( ). Le us define a new code ˜σ, which is equal o σ, bu wi h +1 ins ead o ze os, and pe o m he ansa z un= ˜σn˜un. By 9 2δn m −δ|n−m|,1and CN1=C1N=−1, ha is: C=           2−1 0 . . . 0−1 −1 2 −1 0 . . . 0 . . . . . . . . . . . . . . . . . . −1 0 . . . 0−1 2           .(34) I he wo oscilla o a e in phase, he pe u ba ion ma ix is Q=   1−1 −1 1   , wi h eigen alues (λ1, λ2) = (0,2), he fi s one co esponds o he phase mode, he second o he an isymme ic eigenmode. Thus, one o he wo bands an- gen o he axis E= 0 a ze o coupling will emain he e, and he o he will mo e upwa ds. I he on–si e po en ial is so , i.e., wi h posi i e cu a u e, a in e sec ion/ angen poin will be los , and he wo si e b ea he is uns able. Fo he ha d on–si e po en ial, he cu a u e o he bands is nega i e, and when he band mo ed upwa ds, he in e sec ion poin s will be kep , and he sys em is be s able. I ε < 0 he signs o he eigen alues ε Eia e e e sed and so a e he conclusions. Fo he ou –o –phase, wo-si e b ea he , wi h code σ= [1,−1], he pe u ba- ion ma ix is: Q=  −γ γ γ−γ  , wi h eigen alues (λ1, λ2) = (0,−2γ). The conclusions abo e a e e e sed, being he ins abili y mode, when app op ia e, he symme ic one. No e ha he bounda y condi ions in Cha e no effec . Theo em 4 (Two–si e b ea he s) Conside ime– e e sible, wo–si e b ea he s in an homogeneous Klein-Go don sys em wi h a ac i e linea coupling, hen: a) The wo-si e b ea he s wi h codes ±[1,1] and so on–si e po en ial a e un- s able and wi h ±[−1,1] a e s able. b) The wo-si e b ea he s wi h code ±[1,1] and ha d on–si e po en ial a e s a- ble and wi h ±[−1,1] a e uns able. c) I he coupling is epulsi e, he conclusions a e e e sed. 16 7.2 Th ee–si e b ea he s Le us enumbe he si es 1,2,3 and suppose ha he code o he si e in he middle is σ2,= +1 as a e e ence, i.e., σ= [σ1,1, σ3]. We define αi= 1 i σi= 1 and αi=−γi σi=−1, o i= 1,2. Using C om Eq. (34) he pe u ba ion ma ix is: Q=       α1−α10 −α1α1+α3−α3 0−α3α3        .(35) I s nonze o eigen alues a e λ±=α1+α3±√(α1+α3)2−3α1α3. In o de o ha e bo h eigen alues wi h he same sign, we need ha α1α3>0, i.e., ei he α3=α1= 1, o α3=α1=−γ. Tha is, i σ1=σ3= 1, λ±= 1,3>0, i σ1=s3=−γ,λ±=−γ, −3γ < 0, and i σ1σ2=−1, hen λ±ha e diffe en signs. The e o e: Theo em 5 (Th ee–si e b ea he s) Conside ime– e e sible, h ee–si e b ea he s in an homogeneous K-G sys em wi h nea es –neighbo , linea coupling, a ac- i e o ε > 0. Then: a) The h ee–si e b ea he s wi h codes σ=±[−1,1,−1] a e s able i ε > 0 (ε < 0)and he on–si e po en ial is so (ha d). b) The h ee–si e b ea he s wi h code σ=±[1,1,1] a e s able i ε > 0 (ε < 0) and he on–si e po en ial is ha d (so ). c) All o he h ee-si e b ea he s a e uns able o any sign o εand ype o on–si e po en ial. We can compa e he p edic ions o he alues gi en by he heo y wi h he nume ically calcula ed alues o he in e sec ion poin s o he bands wi h he axis θ= 0. As an example, he nume ically calcula ed eigen alues and he p edic ions gi en by heo ems 1 and 2 o he h ee–si e b ea he wi h code [−1,1,−1] a e plo ed in Figu e 3. The e y good accu acy is e iden . 7.3 A model wi h long– ange in e ac ion An example o sys em wi h long ange in e ac ion is he wis model used in Re . [9]. The dynamical equa ions a e: ¨un+V′(un) + ε n+N/2 ∑ m=n−N/2 cos[θ w(n−m)] |n−m|3um= 0 (36) 17 0.01 0.02 0.03 0.04 0.05 0.06 0.07 −0.25 −0.2 −0.15 −0.1 −0.05 0 ε E Fig. 3. Compa ison be ween he eigen alues Eo he New on ope a o ob ained nume ically (poin s), p edic ed by he symme ic MST, heo em 1, (dashed lines) and by he non–symme ic MST, heo em 2, (con inuous line), which is he igh one o be used. Mo se po en ial, code [−1,1,−1], equency ωb=γ= 0.8. Vbeing he Mo se po en ial, and θ w he angle be ween wo dipole momen s co esponding o neighbo ing base pai s in a simplified model o DNA. The coupling ma ix elemen s a e: Cn m =cos[θ w(n−m)] |n−m|3,(37) Some o he esul s, also checked nume ically and wi h good accu acy wi h espec o he eigen alues E, gi en by heo em 1, in spi e o Vbeing non– symme ic a e: Code θ w = 0 θ w =π 11 S able Uns able 1 -1 Uns able S able 101 S able S able 111 S able Uns able 1-1 1 Uns able S able 11-1 Uns able S able As an example o he effec o he symme y coefficien γin heo em 2, we can conside he code [1,−1,1] and plo he New on eigen alues Eas a unc ion 18 0 0.5 1 1.5 2 2.5 3 −4 −3 −2 −1 0 1 2 3 4 θ w λ Fig. 4. Eigen alues λo he pe u ba ion ma ix Q o a h ee-si e b ea he wi h code [−1,1,−1] on he wis model. Con inuous line: γ=ωb= 1; dashed line: γ=ωb= 0.8; and do ed line: γ=ωb= 0.1. Only alues o γ e y a om 1 lead o a diffe en s abili y p edic ion. o θ w. The pe u ba ion ma ix is: Q=       γa −b−γa b −γa 2γa −γa b−γa γa −b        ,(38) wi h a= cos(θ w) and b= cos(2θ w)/8. Figu e 4 shows he dependence o he eigen alues λon he wis angle θ w o h ee alues o he symme y coefficien γ. I can be seen ha only alues o γ e y a om 1 lead o a change on he s abili y p edic ion. 8 Mul ib ea he s, phonob ea he s and da k b ea he s. Pa i y in- s abili ies. Fo la ge , ime– e e sible mul ib ea he s in homogeneous Klein–Go don sys- ems wi h nea es –neighbo , linea coupling, i is i ial o ob ain he s abili y p ope ies o he wo diffe en solu ions wi h wa e numbe q= 0, all he os- cilla o s in phase, o q=π, neighbo ing oscilla o s ou o phase. Conside any numbe N′o con iguous oscilla o s in phase in a sys em coupled by he 19 ma ix Cin Eq. (34). The N′×N′pe u ba ion ma ix is gi en by: Q=               1−1 0 . . . 0 0 −1 2 −1 0 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . −1 2 −1 0 0 . . . 0−1 1               .(39) The cha ac e is ic equa ion o Q x =λ x co esponds o he ollowing equa- ions: x1−x2=λ x1, 2xn−xn+1 −xn−1=λ xn,∀n= 1,2, xN′−xN′−1=λ xN′.(40) Subs i u ion o he ial solu ion xn= cos(q n −ϕq), leads o ϕq=q/2, q= m π/N′, wi h m= 0, . . . , N′−1, and λm= 4 sin2(m π/(2 N′)), i.e., posi i e eigen alues wi hou a degene a e ze o. In case all he oscilla o s o he g oup a e ou o phase, Q→ −γQ and λm→ −γλm. The e o e: Theo em 6 I ε > 0, he ime– e e sible, in-phase, mul ib ea he s a e un- s able (s able) wi h so (ha d) on–si e po en ial. Fo any ou –o –phase mul i- b ea he s, and ε < 0 he conclusions a e e e sed. Al hough, we ha e no ma hema ical p oo , acco ding o nume ical calcula ions o he eigen alues o he pe u ba ion ma ices co esponding o g oups wi h diffe en codes, he numbe s o nega i e and posi i e eigen alues a e equal o he numbe s o −1 and +1 in {σnσn+1}N′−1 n=1 . The e o e being uns able o any on–si e po en ial. No e ha i he e a e se e al non–con iguous mul ib ea he s, he pe u ba ion ma ix i composed o independen blocks o pe u ba ion subma ices o he same ype, each one wi h a ze o eigen alue by cons uc ion as seen in subsec- ion 4.1. The s abili y o such a sys em canno be de e mined by he MST. The only possible conclusion is ha i he e a e eigen alues o diffe en sign his sys em would be uns able. Phonob ea he s a e mul ib ea he s wi hou oscilla o s a es . Fo ime– e e sible phonob ea he s in a sys em ei he wi h ee–ends, o fixed-ends bounda y condi ions, he pe u ba ion ma ix Qis he same as in Eq. (39). I he bounda y condi ions a e pe iodic, he only change is ha Q1 1 =QN,N = 2, QN1=Q1,N =−1, and he eigen alues a e he same. Tha is, he mode q= 0 wi h a ac i e coupling and ha d po en ial is s able. Changing he ha dness, 20 he ype o coupling, o he mode, changes each ime he s abili y. In his way, we ep oduce he esul s in Re s. [7,6] as a consequence o modula ional ins abili y and using he DNLS app oxima ion, espec i ely, and in heo em 9 in Re . [3], based in he p ope ies o he ac ion. Pa i y ins abili ies a e a in e es ing effec o phonob ea he s wi h pe iodic bounda y condi ions. I is a consequence o he pa i y o he o al numbe o si es N. I , o example, Vis so , he coupling is a ac i e, and Nis e en, all he eigen alues o he pe u ba ion ma ix a e nega i e, i.e., he sys em is s able. Bu i Nis odd, he e is an isola ed posi i e eigen alue, b inging abou he ins abili y. The e is a clea physical o igin o his phenomenon, as he bounda ies a e equi alen o an inhomogenei y. Acco ding o ou heo y i is also ob ious: we ha e, in ac wo oscilla o s a he ends coupled and in phase, which gi es ise o a posi i e eigen alue. Da k b ea he s a e mul ib ea he s wi h only one o a ew oscilla o s a es . Thei s abili y depends on he cha ac e is ics o he con iguous g oups o ex- ci ed oscilla o s, as a consequence o heo em 6. We ha e o be ca e ul abou he numbe o pa icles, o example he πmode wi h code a ound he da k si e [. . . , 1,−1,0,1,−1, . . . ], needs an odd numbe o si es o a oid ha ing wo in–phase oscilla o s a he ends, and a pa i y ins abili y. The one wi h [. . . , −1,1,0,1,−1, . . . ] needs an e en numbe o si es. I he sys em has ee o fixed pe iodic condi ions, he pe u ba ion ma ix is decoupled in wo and he s abili y is undefined as commen ed abo e. We ha e compa ed he p edic ions o he ins abili ies and he alues o he in e sec ion poin s o he bands wi h he axis θ= 0, in many cases, as o example in Re . [10], wi h e y good esul s. The ques ion o up o which alues o he coupling pa ame e ε he heo y is alid, is howe e unsol ed. Gene ically speaking alues up o ε= 0.1 (compa ed o he es equency ω0= 1), a e sa e, bu he e a e some excep ions. In he same e e ence, o he da k b ea he ha d on–si e po en ial and a ac i e coupling, a mode no p edic ed by he MST gi es ise o an ins abili y as soon as ε= 0.022. As he wid h o he phonon band can be calcula ed, and he wid h o he backg ound is gi en by he eigen alues o he pe u ba ion ma ices, i is also possible o p edic he alues o he ε, o which oscilla o y and subha monic ins abili ies occu , bu we do no ex end u he he e. 9 Summa y and conclusions We ha e de eloped using degene a e pe u ba ion heo y a me hod o ob- aining he s abili y p ope ies o mul ib ea he s o any code a low coupling. 21 The me hod is syn hesized in h ee diffe en e sions o a heo em, e e ed o as he mul ib ea he s abili y heo em o MST o sho . The wo simples e - sions co espond o ime– e e sible mul ib ea he s, wi h linea coupling and homogeneous on–si e po en ials. I he on–si e po en ial is symme ic o we conside only in–phase mul ib ea he s, i in ol es only some simple modifica- ions o he coupling ma ix. I none o hose condi ions a e ulfilled hen i in ol es he analy ical o nume ical calcula ions o a magni ude γ, called he symme y coefficien , as i s alue is 1 o symme ic on–si e po en ials. Fo so po en ials and alues o he equency un il abou one hal o he es equency, a good app oxima ion is he mul ib ea he equency, esul which we demons a e is exac o he Mo se po en ial. Fo ha d po en ials i s alue i s close o uni y. The gene alized e sion o he MST is o no so s aigh o - wa d applicabili y, al hough i s complexi y depends on he cha ac e is ics o he sys em and mul ib ea he conside ed. We gi e some examples o applica ion o he me hod and compa e i wi h nu- me ical esul s. The sys ems conside ed a e wo and h ee–si e b ea he s, wi h nea es neighbo o long– ange in e ac ions, mul ib ea he s, phonob ea he s and da k b ea he s. A pa i y ins abili y can appea in fini e sys ems depending on he pa i y o he numbe o oscilla o s. All hese examples a e in e es ing in hemsel es, bu also illus a e he use o he MST, and show ha i p o ides a powe ul me hod o easy applicabili y o de e mine mul ib ea he s s abili y. Some o he applica ions unde s udy a e mul ib ea he s in 2D and 3D sys ems, p edic ion o he oscilla o y and subha monic ins abili ies and he peculia i ies o some diso de ed sys ems Acknowledgemen s This wo k has been suppo ed by he Eu opean Union unde he RTN p ojec , LOCNET, HPRN–CT–1999–00163. J Cue as acknowledges an FPDI g an om ‘La Jun a de Andaluc´ıa’. A Calcula ion o he symme y pa ame e γ o a Mo se po en ial In his appendix, we calcula e he alue o he symme y pa ame e γin heo em 2 o he pa icula case o a Mo se po en ial. The choice o his kind o po en ial elies in he ac ha he exp essions o he o bi s a e easy o manage. We will p oo ha , o his po en ial, γ=ωb. Le us suppose an isola ed oscilla o submi ed o a Mo se po en ial. The 22 ene gy o he sys em is: E=1 2˙x2+1 2(exp(−x)−1)2.(A.1) We look o solu ions wi h pe iod T= 2π/ωb ha a e ime– e e sible, i.e, wi h ˙x(0) = 0, o , in o he wo ds, a = 0 he sys em is a a u ning poin . The e a e wo o hem, ob ained om he equa ion abo e: x1=−log(1 + √2E)<0 and x2=−log(1−√2E)>0. I we conside he solu ion wi h x(0) = x1, (x) is gi en by: (x) = ∫x x1 dx √2(E −(exp(−x)−1)2/2) ,(A.2) which leads o: (x) = 1 √1−2E(π 2−a csin [1 √2E(1 + 2E −1 exp(−x))]).(A.3) By subs i u ion o =T/2 = π/ωband x=x2we ob ain ωb=√1−2E. The in e sion o Eq. (A.2) leads o: x( ) = log 1−√1−ω2 bcos ωb ω2 b .(A.4) To calcula e γ, we need he Fou ie coefficien s o x( ). I he solu ion x( ) is exp essed as: x( ) = z0+ 2 ∞ ∑ k=1 zkcos(k ωb ),(A.5) he coefficien s {zk}∞ k=0 a e gi en by: zk=1 T∫T/2 −T/2 x( ) cos(k ωb ) d . (A.6) The calcula ion o hese in eg als is s aigh o wa d and leads o: z0= log 1 + ωb 2ω2 b ;zk=−1 n(1−ωb √1−ω2 b)k .(A.7) The pa ame e γis gi en by: γ=−∑k≥1(−1)kk2z2 k ∑k≥1k2z2 k .(A.8) By subs i u ion o he Fou ie coefficien s in Eq. (A.7), we ob ain γ=−∑k≥1(− )k ∑k≥1 kwi h =1−ωb 1 + ωb .(A.9) 23 As | |<1, γcan be easily calcula ed using he exp essions o he sum o an infini e geome ic p og ession: ∑ k≥1 k= 1− ,∑ k≥1 (− )k=− 1 + (A.10) leading o γ=ωb, as we wan ed o p oo . B Cu a u e o he bands In his sec ion we demons a e a key poin used in ou heo y: he ac ha he band co esponding o an isola ed exci ed oscilla o s has nega i e cu a u e i he on si e po en ial is ha d and posi i e i i is so . Un o una ely, he demons a ion is qui e long i we wan i o be sel –consis en . A simila one can be ound in Re . [11] and diffe en ones in Re . [5,12]. Le us conside he dynamical equa ion o an isola ed oscilla o wi h Hamil- onian H= 1/2p2+V(u), wi h ua single– alue, eal unc ion o ime and p= ˙u: ¨u+V′(u) = 0 (B.1) The cha ac e is ic equa ion o he New on ope a o Nco esponding o a gi en pe iodic solu ion o Eq. (B.1)is Nξ≡¨ ξ+V′′(u)ξ=Eξ , (B.2) whe e ξis a C2 unc ion o . Fo each eigen alue E he e a e only 2 independen solu ions, de e mined, o example, by he alues o ξ(0) and ˙ ξ(0). We can ob ain hem o E= 0, by de i ing Eq. (B.1) wi h espec o and wi h espec o ωb: ... u+V′′(u) ˙u= 0 ⇔ N ˙u= 0 ; ¨uw+V′′(u)uw= 0 ⇔ Nuw= 0 .(B.3) As we ha e a single oscilla o we can suppose ha u( ) is ime symme ic wi h a sui able o igin o ime, i.e., ˙uis a ime-an isymme ic unc ion o wi h he same pe iod T= 2 π/ωbas u( ): ˙u( +T) = ˙u( ), ˙u(0) = ˙u(T) = 0. The p ope ies o uw=du/dωb, can be also ob ained. We can w i e uas a cosine Fou ie se ies u( ) = ∑∞ k=0 ckcos(ωb ), wi h he coefficien s ckdepending on ωb. The e o e, du/dωb=∑∞ k=0(dck/dωb) cos(ωb ) + ∑∞ k=0 −k sin(ωb ), o : uw≡du dωb =γ( ) + ωb ˙u( ) (B.4) γ( ) being a ime symme ic, pe iodic unc ion o ime wi h he same pe iod Tas u. This unc ions cons i u e a base o he eigenspace co esponding o E= 0. 24 On he o he hand, N(B.2) is a pe iodic ope a o because i depends on a pe iodic unc ion u. I s eigen unc ions a e gi en by he Bloch heo em: ξ(θ, ) = χ(iθ, ) exp(θ /T),(B.5) wi h χa T–pe iodic unc ion o ime and θ=θ(E). I we conside only alues o E o which θis eal, he se o poin s (θ, E(θ) cons i u es a band. I ξis a solu ion o Nξ=Eξ, he ac ha Nis a eal ope a o implies ha ξ∗is ano he solu ion, i.e., (−θ, E) also belongs o he band and his is symme ic wi h espec o θand, he e o e, dE/ dθ(0) = 0. As Nis ime–symme ic ξ(θ, − ) is ano he , and has o be p opo ional o ξ∗as he e can be only wo independen solu ions. The Floque ma ix FEis defined as:    ξ(T) ˙ ξ(T)  =FE   ξ(0) ˙ ξ(0)   .(B.6) The eigen alues {λl}o FEa e he Floque mul iplie s, i we w i e hem as λl= exp iθl, wi h θ eal o complex, iθla e he Floque exponen s, and θl, he Floque a gumen s. An eigen unc ion wi h θl eal is bounded, i θl= 1 he eigen unc ion has pe iod T. The condi ion o linea s abili y o he solu ion uis ha all he Floque a gumen s o F0a e eal. I is easy o check some p ope ies. The Bloch unc ions Eq. (B.5) a e eigen- unc ions o FEwi h Floque exponen θ, he e o e FEis diagonalizable (o e C) wi h Floque mul iplie s exp(±iθ) and Floque a gumen s ±θ. Howe e , a E= 0, he e is only one eigen unc ion o F0, ˙u, he o he independen unc ion o he subspace Nξ= 0, uw, i is no an F0–eigen unc ion. Tha is F0has a degene a e Floque mul iplie 1, o Floque a gumen 0 and can be ans o med in o a Jo dan block. Ano he consequence o he la e is ha (θ, E) = (0,0) belongs o he band. We need o calcula e he cu a u e o he band a his poin . Le us come back o he Bloch unc ions ξ(θ, ) in Eq. (B.5) and ξ∗(θ, ) co esponding o a pai o symme ic poin s (±θ, E) o he band, and sup- pose ha θ→0 (and hen E→0). A θ= 0 he wo unc ions collide in ξ(0, ) = χ(0, ) = χ∗(0, ), he e o e ξ(0, ) is a eal, T–pe iodic unc ion. Bu we al eady now ha a E= 0, he e is a T–pe iodic unc ion solu ion o Nξ= 0, ˙u. The e o e, ξ(0, ) = ˙u(Nis lineal, so we can adjus he no m o ξin o de ha he la e equa ion is ulfilled). We can eob ain he missing unc ion ha spans ke (N). I we de i e Eq. (B.2) wi h espec o θ, we ob ain Nξθ=Eθξ+Eξθand a E= 0 (Eθ(0) = 0) we 25