Discrete breathers in Φ4 and related models

Disc e e b ea he s in φ4and ela ed models Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis Abs ac In his Chap e , we ouch upon he wide opic o disc e e b ea he (DB) o ma ion wi h a special emphasis on he p o o ypical sys em o in e es , namely he φ4model. We s a by in oducing he model and discussing some o he ap- plica ion a eas/mo i a ional aspec s o explo ing ime pe iodic, spa ially localized s uc u es, such as he DBs. Ou main emphasis is on he exis ence, and especially on he s abili y ea u es o such solu ions. We explo e hei spec al s abili y nume i- cally, as well as in special limi s (such as he icini y o he so-called an i-con inuum limi o anishing coupling) analy ically. We also p o ide and explo e a simple, ye powe ul s abili y c i e ion in ol ing he sign o he de i a i e o he ene gy s. equency dependence o such solu ions. We hen u n ou a en ion o nonlinea s abili y, b inging o h he impo ance o a opological no ion, namely he K ein signa u e. Fu he mo e, we b ie ly ouch upon linea ly and nonlinea ly uns able dy- namics o such s a es. Some special aspec s/ex ensions o such s uc u es a e only ouched upon, including mo ing b ea he s and dissipa i e a ia ions o he model and some possibili ies o u u e wo k a e highligh ed. While his Chap e by no means aspi es o be comp ehensi e, we hope ha i p o ides some ecen de elop- men s (a la ge ac ion o which is no included in ime-hono ed DB e iews) and associa ed u u e possibili ies. Jes´ us Cue as–Ma a e G upo de F´ ısica No Lineal, Uni e sidad de Se illa, Depa amen o de F´ ısica Aplicada I, Escuela Poli ´ ecnica Supe io . C/ Vi gen de ´ A ica, 7, 41011-Se illa, Spain, Jes´ us Cue as–Ma a e Ins i u o de Ma em´ a icas de la Uni e sidad de Se illa (IMUS). Edi icio Celes ino Mu is. A da. Reina Me cedes s/n, 41012-Se illa, Spain e-mail: jcue [email p o ec ed] Panayo is G. Ke ekidis Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s, Amhe s , MA 01003- 4515, USA e-mail: ke[email p o ec ed] 1 a Xi :1901.00545 2 [nlin.PS] 28 May 2019 2 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis 1 A b ie desc ip ion o disc e e b ea he s: de ini ion, his o ical pe spec i e and applica ions Dynamics o localized exci a ions is, undoub edly, one o he mos impo an opics wi hin he ealm o nonlinea science. In disc e e sys ems, he localized exci a ions ha can be a gued o be mos gene ic [1, 2, 3] a e he so-called disc e e b ea he s (DBs). These can be de ined as ime-pe iodic spa ially-localized cohe en s uc u es eme ging a coupled nonlinea oscilla o la ices. This e m was coined by Campbell and Pey a d [4] in o de o dis inguish hem om he “con inuous” b ea he s ound as exac solu ions h ough he in e se sca e ing machine y in he sine-Go don PDE [5]. [I is wo hwhile o men ion in passing ha con inuous b ea he s ha e a pa icu- la ly in e es ing his o y associa ed wi h hem in φ4models [6, 7], which is desc ibed in de ail in a di e en Chap e o his special olume.] Such DBs a e also known as in insic localized modes; his name was in oduced in o de o emphasize ha hei o igin was in he in insic nonlinea i y o he sys em, con a y o he case o Ande - son modes, whe e localiza ion s ems om la ice diso de [8] and can exis e en in he linea limi . A majo de elopmen ha sp ingboa ded he s udy o DBs ook place in 1988, h ough he nume ical s udy o DBs in some p o o ypical la ice models as epo ed in wo pionee ing wo ks [10, 11] by Takeno, Sie e s and Kisoda. In hese pape s DBs a e calcula ed o he i s ime in he wo basic kinds o la ices whe e hey can eme ge, namely Fe mi–Pas a–Ulam–Tsingou (FPUT)1[10] and Klein–Go don (KG) [11]. Addi ionally, some o hei s abili y p ope ies we e also de e mined. FPUT la ices a e cha ac e ized by he absence o subs a e (on-si e) po en ial and he nonlinea i y o he in e -si e o ces, whe eas KG la ices possess nonlinea on- si e po en ial and ( ypically) linea in e -si e o ces. In e es ingly, in hose e e - ences, po en ials o he φ4 o m and ela ed models we e conside ed. The e a e also mixed cases o la ices whe e nonlinea i ies a e p esen in bo h he subs a e and in- e si e po en ials (see e.g. [12]); hese la ices a e some imes deno ed as KG/FPUT la ices. While hese ea ly wo ks p o ided c edible nume ical e idence and plan ed he seed o s udying DBs, hey did no igo ously p o e hei exis ence. The la e came in a celeb a ed 1994 pape by MacKay and Aub y [13] and was based on he concep o he so-called an i-con inuous (AC) limi . They basically demons a e in a igo ous ashion, by making use o he implici unc ion heo em, ha DBs can gene ically exis in nonlinea KG la ices as a esul o unobs uc ed (when some sui able esonance es ic ions a e a oided) con inua ion o pe iodic o bi s o indi- idual (uncoupled) oscilla o s. Mackay–Aub y’s heo em, oge he wi h he igo - ous p oo o s abili y by Aub y in 1997 [14] and he nume ical me hods de eloped by Ma ´ ın among o he s [15] led o an in ense in e es on DBs in he la e 1990’s and he beginning o he 21s Cen u y, no only om he heo e ical bu also om he 1These la ices ha e been adi ionally deno ed as Fe mi–Pas a–Ulam, o ge ing he ou s anding ole o Ma y Tsingou who was he pe son esponsible o all he nume ical simula ions in hese i s compu a ions o nonlinea la ice dynamics [9] Disc e e b ea he s in φ4and ela ed models 3 expe imen al poin o iew. The ele an ac i i y has been e y well summa ized by now in a se ies o e iews; see, e.g., [1, 2, 3, 16, 17]. DBs ha e been sough o in many ields o Physics. They ha e been expe imen- ally gene a ed in a ays o Josephson junc ions [18, 19], mechanical [20] and mag- ne ic pendula [21] and mic ocan ile e s [22], nonlinea elec ical la ices [23, 24] and g anula media [25, 26]. They ha e been obse ed in molecula and ionic c ys- als like he so-called P Cl [27] and an i e omagne s [28], and ha e been a gued as plausible explana ions o obse a ional indings in sys ems such as α-U anium [29] o NaI [30]. They ha e also been specula ed o play an impo an ole in DNA dena u a ion in an ex ensi e li e a u e e iewed, e.g., in [31], as well as o a ise in econs uc i e chemical eac ions [32] and in he slow decay o luminiscence in Pb-doped alkali-halide c ys als as KB [33], and p edic ed o exis in c ys als o Niobium and Nickel [34] o ca bon ma e ials like g aphene, ca bon nano ubes, ulle enes o hyd oca bons (see [35] o a e iew). Unde sui able condi ions, DBs can mo e along he la ice and a e known in ha se ing as mo ing b ea he s. As such, hey ha e been a gued o be esponsible o he obse ed acks in musco i e mica shee s [16, 36] and o he in ini e cha ge mobili y (also dubbed as hype con- duc i i y) expe imen ally e idenced in such c ys als [37, 38]. Admi edly, hese a e only some examples o an e e inc easing lis o applica ions which is by necessi y, due o he limi ed scope o his Chap e , a he incomple e. Ne e heless, i se es o illus a e he gene ic na u e and wide impac o such s uc u es and hei b oad ele ance o s udy. The p esen Chap e is de o ed o e iewing mo e conc e ely some esul s on he exis ence, s abili y and dynamics o DBs in KG la ices wi h φ4on-si e po en ial, and some o he miscellaneous opics ela ed o DBs in such la ices; many o hese esul s ha e been ound o gene ic KG la ices, bu we will ocus he e on he p in- cipal heme o his special olume, namely he φ4po en ial. Mo eo e , by choice, he emphasis will be on some ecen esul s, no only due o hei connec ions o he esea ch in e es s and ecen wo k o he au ho s, bu also because, o he bes o ou knowledge, he associa ed indings and he esul ing o e -a ching s abili y pe spec i e ha e no been collec ed in such a summa izing body o wo k o da e elsewhe e. 2 The Klein-Go don la ice and he an i-con inuous limi A e he b ie p esen a ion o he DB concep , in his Sec ion we will in oduce some de ini ions. Fi s o all, we need o add ess he concep o Klein-Go don non- linea dynamical la ice; om a ma hema ical poin o iew, i can be de ined as a sys em o coupled second-o de o dina y di e en ial equa ions o he o m: Fn(u)≡¨un+V0(un) + X m Cm(un−un+m−un−m)=0 (1) 4 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis whe e nand ma e D-dimensional indices, V(un)is he on-si e po en ial, which is no necessa ily homogeneous, and Cmis he coupling cons an , depending on he dis ance om he neighbo s. In mos case examples, he in e -si e o ce is nea es - neighbou wi h Cm=Cδm,1, in which case he p e ious equa ion can be w i en e.g. o he one-dimensional la ice as Fn(u)≡¨un+V0(un) + C(2un−un+1 −un−1) = 0.(2) This dynamical equa ion de i es om he ollowing Hamil onian: H=X n hn=X n 1 2˙u2 n+V(un) + C 4(un−un−1)2+ (un−un+1)2(3) wi h hnbeing he ene gy densi y. As men ioned abo e, he an i-con inuous (AC) limi in oduced by MacKay–Aub y’s heo em [13] is o pa amoun impo ance in o de no only o p o e he exis ence o disc e e b ea he s in nonlinea KG la ices, bu also o gi e a hin on how o ob ain hem nume ically. The AC limi is ha o all he la ice oscilla o s being uncoupled (i.e. C= 0) and ei he oscilla ing wi h he same equency ωbo emaining a es . By i ue o he implici unc ion heo em, MacKay and Aub y es ablished ha he solu ion can be con inued om he AC limi o a ini e alue o C→0whene e wo condi ions a e ul illed: (1) he po en ial o he isola ed oscilla o s is anha monic and (2) no in ege mul iples o he DB equency ωb esona e wi h he linea modes equency (i.e. he so-called phonon band which we will quan i y u he below). I hese condi ions a e ul illed, a cohe en (in he sense ha all he si es oscilla es wi h he same equency) and exponen ially localized s uc u e o a DB o m will exis . These disc e e b ea he s a e exac pe iodic o bi solu ions (up o machine p ecision) o he nonlinea KG equa ion. This heo em is no only o heo e ical alue, bu also o p ac ical use ulness as i p o ides a s a egy on how o nume ically calcula e DBs ha was exploi ed by Ma ´ ın and collabo a o s [15]. Me hods based on he AC limi a e qui e simple: as a he AC limi he oscilla o s a e uncoupled, i su ices o ge a pe iodic o bi o single oscilla o s subjec ed o po en ial V(u)and con inue his solu ion by means o ixed-poin me hods (like New on-Raphson) up o he desi ed coupling. Among he nume ical me hods used o a aining DBs, we can highligh wo: (1) Fou ie space me hods in which DBs a e ep esen ed by a Gale kin unca ion up o index kmin a Fou ie se ies expansion o he o m: un= km X k=−km zkeikωb ,(4) ans o ming he coupled ODE sys em (1) in o a se o nonlinea algeb aic equa- ions; and (2) shoo ing me hods, whe e DBs a e ixed poin s o he map: ({un(0)},{˙un(0)})→({un(T)},{˙un(T)}),(5) Disc e e b ea he s in φ4and ela ed models 5 wi h T= 2π/ωbbeing he b ea he pe iod. Fou ie me hods ha e he ad an age o using an analy ical Jacobian bu one has o pay he p ice o handling wi h a la ge numbe o equa ions. In addi ion, he e a e pa hological po en ials [23, 39] o which he con e gence o Fou ie se ies is e y slow and his me hod canno be used. The e is a g ea numbe o po en ials ha can be ound in he DB li e a u e. They can be classi ied as so o ha d, i he ene gy dec eases o inc eases, espec i ely, wi h he equency; a simple way o disce n i a po en ial is so o ha d is by making use o he ha dening coe icien h= 3V0000(0) −5V000(0) [40] so ha he po en ial is so (ha d) when h < 0(h > 0). Mo eo e , a so (ha d) oscilla o ib a es wi h a equency ωbwhich is smalle (g ea e ) han i s na u al equency ωo=pV00(0) (no ice ha in mos cases, ωo= 1). Typical examples o so po en ials include he Mo se, Lenna d-Jones, cubic (φ3), sine-Go don and double-well po en ials, many o which a ise in applica ions [1, 2]. Ha d po en ials a e usually pa icula cases o polynomials po en ial which, depending o pa ame e s, can be ei he so o ha d, namely cubic-qua ic o pu ely qua ic (φ4) anha monici ies, whe e he qua ic e m a ises wi h a posi i e sign; see below. This la e po en ial, on which we will ocus in he p esen Chap e , is gi en by V(u) = 1 2u2+1 4su4(6) When s= 1 (s=−1), he po en ial is ha d (so ). The o bi s u( )o an isola ed oscilla o can be exp essed in e ms o Jacobi ellip ic unc ions and he modulus m is ela ed o he oscilla ion equency ωb h ough a anscenden al equa ion. Thus, i s= 1, u( ) = 2m 1−2mcn 2K(m)ωb π, m, ωb=π 2√1−2mK(m)(7) and, i s=−1 u( ) = 2m 1 + m2cd 2K(m)ωb π, m, ωb=π 2√1 + m2K(m)(8) No ice ha in he so case, he e a e he e oclinic o bi s sepa a ing oscilla ing s a es om unbounded ones ep esen ing escape om he po en ial [41]. Ano he in e es ing ea u e ha dis inguishes so and ha d po en ials is ela ed o he oscil- la ion pa e n o he ails. When he po en ial is ha d, he ails a e s agge ed and hey a e uns agge ed in he so case [42]. An example o disc e e b ea he s in so and ha d po en ials is shown in Fig. 1 whe e he p o ile and he ime-e olu ion is displayed. A he AC limi , i is possible o cons uc DB-like solu ions wi h mo e han one exci ed si es, dubbed as mul ib ea he s. When sui ably cons uc ed ( ypically wi h he “exci ed” oscilla o s in- o ou -o -phase as summa ized, e.g., in [43]), hese a e also ound o pe sis when he coupling is swi ched on. O e he yea s, some o he DB s uc u es ha e been endowed wi h dis inguishing names. The Sie e s-Takeno 6 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis Fig. 1 P o ile ( op panels), ime-dependence o he cen al si es (middle panels) and Floque mul i- plie spec um (bo om panels) o a 1-si e b ea he in a so (le panels) and in a ha d ( igh panels) po en ial. In he o me case, pa ame e s a e ωb= 0.85 and C= 0.3, whe eas in he la e case, ωb= 2.5and C= 1. In he igh panels, mul iplie s wi h posi i e (nega i e) K ein signa u es a e depic ed wi h ed c osses ×(blue pluses +); see he associa ed discussion a ound Eq. (15) below. Disc e e b ea he s in φ4and ela ed models 7 mode co esponds o a DB wi h only one exci ed si e (i can also be deno ed as single si e o 1-si e b ea he ). Also, he Page mode is a mul ib ea he whe e wo adjacen si es a e exci ed; no ice ha he o me is a si e-cen e ed b ea he whe eas he la e co esponds o a bond-cen e ed (in e -si e-cen e ed) one. When all he si es a he AC limi a e exci ed, we a e dealing wi h a nonlinea phonon o phonob ea he ; a da k b ea he [44] is a phonob ea he wi h one (o a ew) non-exci ed si e(s), esem- bling he unc ional o m o a da k soli on o he nonlinea Sch ¨ odinge equa ion. Mul ib ea he s a e usually o med by ime- e e sible oscilla o s. In hese cases, hey can be cha ac e ized by a coding sequence σ≡ {σn}indica ing he phase and he exci a ion s a e o he ele an si es a he AC limi . This code is σn= 0 i he n- h oscilla o is a es , σn= 1 i i oscilla es wi h a equency ωband phase 0 (i.e. un(0) >0) and σn=−1i oscilla es wi h ini ial phase π(i.e. un(0) <0). Fo ins ance, he Sie e s-Takeno and Page modes a e ep esen ed by σ={1}and σ={1,1}( o so po en ials) o σ={1,−1}( o ha d po en ials), espec i ely. The e a e some cases whe e he code can be mo e complex as in sine-Go don po- en ials, whe e o o s can coexis wi h oscilla o s; in such cases, s uc u es called o ob ea he s (see [14, 45]) can eme ge i he exci ed si e co esponds o a o o ; hey ha e been expe imen ally gene a ed in Josephson junc ion a ays [18, 19]. The e a e also, howe e , some special cases whe e mul ib ea he s a e cons i u ed by non- ime- e e sible oscilla o s, as demons a ed in [14]. One can ind such kind o solu ions in 1D chains wi h pe iodic bounda y condi ions in he o m o phono- b ea he s wi h phase o sion [46, 47], o in 2D pe cola ing clus e s o so-called disc e e o ices [48, 49, 43]. They can also eme ge in 1D la ices wi h long- ange in e ac ions [50, 51]. Addi ionally, hey cons i u e a po en ial a ac o in pe iodi- cally o ced and damped oscilla o ne wo ks [52]. The non- esonance condi ion o MacKay-Aub y’s heo em clea ly es ablishes ha he b ea he equency ωbmus lie in he gaps be ween he linea modes (phonons) band; addi ionally ha monics o his equency mus a oid esonances wi h he band o ensu e he absence o ene gy dispe sing mechanisms a ec ing he DB. In he case o 1D KG la ices, he e is an op ical band o phonons gi en by ω2 ph =ω2 o+ 4Csin2q 2(9) whe e qis he phonon wa enumbe and ωph he equency o he associa ed e ec- i ely plane wa e exci a ions ∼ei(qn−ωph ). In ini e la ices (which a e needed o nume ical compu a ions), he alue o qis quan ized wi h he quan iza ion being de e mined by he na u e o he imposed bounda y condi ions and he numbe o la ice nodes N. Consequen ly, b ea he s in ha d po en ials, can be con inued un il ωbcollides wi h he uppe edge o he phonon band (q≈π), an e en occu ing when C≈(ω2 b−ω2 o)/4( he app oxima ion symbol is used o ake in o accoun he ac ha , depending on he quan iza ion o q, i could happen ha q=πis no in he band). Resonances in so po en ials a e caused by in ege mul iples o he b ea he equency colliding wi h he equencies o he phonon band. The ele an c i ical poin eme ges when he second ha monic in asymme ic po en ials and he hi d one in symme ic ones (like he pu ely qua ic φ4analyzed in his Chap e ) collides 8 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis once again wi h he uppe edge o he phonon band; ha is, when C≈ω2 b−ω2 o/4 o C≈(9ω2 b−ω2 o)/4, espec i ely. In addi ion, in a ini e la ice and so po en- ials, he e a e gaps in he phonon band and b ea he s can “bypass” he esonance equency, by exis ing wi hin hese ini e gaps. In his case, he b ea he hyb idizes wi h he bi u ca ing phonon c ea ing a s uc u e called phan om b ea he s; s ic ly speaking, hey a e non-exponen ially-localized b ea he s as hey ha e a ail oscilla - ing wi h a equency nωbwi h ndepending on he mul iple o he b ea he equency ha esona es [53]. These s uc u es a e he disc e e analogue o he nanop e a ob- se ed a he con inuum limi o KG la ices wi h he φ4double well po en ial [7]. As shown in [54] o sine-Go don and φ4double well po en ials, con inua ion up o he con inuum limi is simila o a Wannie -S a k ladde . This phenomenon (i.e. b ea he -phonon hyb idiza ion) akes place because he s agge ing cha ac e o he phonons is di e en han ha o he b ea he ; i he b ea he ails had he same s ag- ge ing cha ac e o he bi u ca ing phonon, he b ea he would be smoo hly con- inued om he phonon and i s ampli ude would be ze o a he phonon equency. No ice ha gaps in phonon band appea s na u ally in diso de ed la ices because o Ande son localiza ion; b ea he s in such sys ems can delocalize [55] o emain lo- calized [56, 57]. The abo e scena io is ypical o Sie e s-Takeno and Page modes. When o he mul ib ea he s a e conside ed, b ea he s canno each he phonon band and he bi u ca ion scena io is mo e complex (see e.g. [54]). Le us also men ion ha decay o b ea he s is no necessa ily exponen ial. In la - ices wi h long- ange in e si e in e ac ions, he decay can be algeb aic [58], possibly ea u ing a ansi ion om exponen ial o algeb aic; o a ecen expe imen al eal- iza ion o his ansi ion in sys em based on magne s, see, e.g., [59]. In addi ion, in some KG/FPUT la ices wi h φ4in e si e po en ial, b ea he s can decay supe expo- nen ially being dubbed as (nea ly) compac DBs [60, 61]. Finally, we mus ema k ha he e a e se e al mo e exis ence p oo s. Fo in- s ance, a a ia ional p oo was in oduced in [62]; his is alid only o ha d po en- ials. A p oo based on he cen e mani old heo em was in oduced in [40] and is alid o DBs whose equency is close o he phonon band edge. 3 S abili y o disc e e b ea he s This Sec ion can be conside ed as he co e o he p esen Chap e . The ea men o he opic will be as ollows: i s o all, we will p esen an in oduc ion o he Floque heo y applied o he linea s abili y o disc e e b ea he s; hen, we will show di e en app oaches o he linea s abili y o mul ib ea he s in he icini y o he AC limi and in oduce he ene gy- s- equency mono onici y c i e ion o he linea s abili y o b ea he s and mul ib ea he s a a bi a y coupling; a e wa ds, nonlinea s abili y c i e ia will be summa ized oge he wi h he dynamical e olu ion o some examples o uns able solu ions. Disc e e b ea he s in φ4and ela ed models 9 3.1 Floque analysis In o de o assess he dynamical obus ness o he iden i ied DB solu ions and hei po en ial accessibili y in physical expe imen s, a key s ep is he de e mina ion o hei s abili y. In he pape whe e MacKay and Aub y demons a e he exis ence o b ea he s [13], i is specula ed ha 1-si e b ea he s a e likely o be s able. This was inally p o en by Aub y in [14] o ini e la ices and MacKay and Sepulch e in [63] o in ini e la ices. As b ea he s a e ime-pe iodic solu ions o he equa ion sys em (1), hei spec al s abili y can be de e mined by means o a Floque analysis. To his aim, we need o e alua e he e olu ion o a pe u ba ion ξn( ) o a solu ion n( ). We hus in oduce —in e.g. he 1D equa ion (2)— he solu ion un( ) = n( ) + ξn( )wi h being a small cons an . Then, he equa ion ha he pe u ba ion sa is ies o O()is ¨ ξn+V00( n)ξn+C(2ξn−ξn+1 −ξn−1)=0.(10) This equa ion can be w i en in a mo e compac o m as N( ( ))ξ= 0 (11) whe e ξ≡ {ξn( )}and ( )≡ { n( )}.Nis known as he linea iza ion ope a o . I ξ∈ C2, he s udy o his ope a o spec um o e s in o ma ion abou s abili y. I , mo eo e , ξ∈ E2 s(ωb)(i.e. ξbelongs o he space o ime- e e sible solu ions wi h equency ωb), his ope a o can be iden i ied as he Jacobian (F ´ eche de i a i e) o he dynamical equa ions, i.e. ∂ F( ( )) = N( ( )). Equa ion (11) can be iewed as a he pa icula case E= 0 o he eigen alues equa ion o he New on ope a o N( ( ))ξ=Eξ (12) The s udy o he spec um o Nis ela ed o Aub y’s band heo y [14] as we will see u he in he p esen sec ion. In Hamil onian dynamical sys ems, his lin- ea iza ion ope a o is ime-symme ic, eal, symplec ic and He mi ian. In addi ion, i is in a ian on ime ansla ions o pe iod T, so by i ue o Bloch’s heo em, he co esponding eigen unc ions ξ( )can be exp essed as Bloch unc ions: ξ( ) = eiθ /T υ( ).(13) To pe o m Floque analysis, we need o s udy he spec um o he Floque ope - a o F, de ined om he ollowing map: Ω(T) = FoΩ(0),wi h Ω( )=[ξ( ),˙ ξ( )] (14) The ep esen a ion o Foin R2Nis deno ed as he so-called monod omy ma ix. The eigen unc ions o Foa e hose o he New on ope a o wi h E= 0. Thus, as a consequence o Bloch’s heo em (13), Ω(T) = exp(iθ)Ω(0). In o he wo ds, monod omy eigen alues (also known as Floque mul iplie s) a e o he o m λ= exp(iθ)wi h θ∈C.θis dubbed as Floque a gumen . 16 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis Fig. 3 Dependence wi h espec o he coupling cons an Co he a gumen (le panels) and modulus ( igh panels) o he Floque mul iplie s co esponding o he ollowing con igu a ions: ( op panels) σ={1,1}mul ib ea he wi h ωb= 0.8in so po en ials; (middle panels) σ= {1,−1}mul ib ea he wi h ωb= 2.5in ha d po en ials; (bo om panels) σ={1,1}mul ib ea he wi h ωb= 1.5in ha d po en ials. No ice ha in he igh panels, he mode wi h posi i e (nega i e) K ein signa u e is depic ed in ed (blue). Disc e e b ea he s in φ4and ela ed models 17 ope a ion has a ixed poin (i he o iginal la ice has a genuine a eling wa e). The pe iod o he associa ed pe iodic o bi is di ec ly associa ed wi h he speed o he a eling wa e (TW) since ωb= 2πs/h whe e sis he speed o he wa e and h he spacing o he la ice. We hus gi e a e y sho p oo o he heo em o he case o a eling wa es and, by di ec analogy, o DBs. Conside he dynamical sys em: U =u p =J∇H(U)whe e J=0I −I0. We seek a TW in he o m: u=u0(n−s ) = u0(ξ)[sol ing −sU0,ξ =J∇H(U0)] and linea ize a ound i acco ding o: u(ξ, ) = u0(ξ) + W(ξ, )and p(ξ, ) = p0(ξ) + P(ξ, ). Then, he linea iza ion ope a o and i s adjoin can be ound ex- plici ly as: M:= s∂ξ+J∇2H(U0),M∗= (−∇2H(U0)J−s∂ξ) = JMJ. Sim- ila ly o he b ea he p oblem, he ime ansla ion in a iance induces a eigen ec o and a gene alized eigen ec o wi h 0eigen alue o he o m: ∂ξU0( he eigen ec o ) and M(−∂sU0) = ∂ξU0( he equa ion o he gene alized eigen ec o ). Then he s aigh o wa d p oo o he s abili y heo em is as ollows: I an ex a eigen ec o ˜ Ywi h λ= 0 exis s, hen i mus sa is y M˜ Y=∂sU0(yielding an ad- di ional gene alized eigen ec o ). This, howe e , imposes he ollowing symplec ic o hogonali y condi ion: 0 = hJ∂ξU0,M˜ Yi=Z(J∂ξU0)·(∂sU0)dξ =Z1 s∇H(U0)·∂U0 ∂s dξ =1 sZ∂H(U0) ∂s dξ =1 sH0(s). Consequen ly, a his c i ical poin he de i a i e H0(s)mus anish, and simila ly o DBs H0(ωb)=0. As explained in [76] o DBs and [78, 79] o TWs, one can ake he calcula ion u he p o iding es ima es o he bi u ca ing ins abili y- inducing mul iplie s on he wo sides o he c i ical poin . 3.4 Nonlinea s abili y As illus a ed in [80] o a wide ange o NLS ype models, linea s abili y is no he las wo d, as he e a e cases o e.g. spa ially-an isymme ic soli ons, which a e lin- ea ly s able bu he dynamics is can be nonlinea ly uns able, i i is e ol ed o e su - icien ly long ime scales. In bo h la ice and con inuum NLS modes, i was shown in his wo k ha spec al s abili y may be inconclusi e i modes o nega i e K ein signa u e exis in he spec um and i hei (nonlinea i y induced) ha monics a e in esonance wi h he con inuous spec um. In pa icula , an o dina y di e en ial equa ion (ODE) was de i ed sugges ing ha o posi i e ene gy modes, hei en- e ge ic con en is deple ed due o dispe si e wa e adia ion. Howe e , o nega i e K ein signa u e modes, he e e se p ocess occu s, e en ually pumping (o e a slow, powe -law in ime p ocedu e) he in e nal mode and ul ima ely leading o he demise o he cohe en s uc u e due o his genuinely nonlinea mechanism. 18 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis Fig. 4 The op panels show he ene gy- equency dependence o disc e e b ea he s in 2D la ices wi h a so po en ial and C= 0.1(le panels) and a ha d po en ial and C= 0.3( igh panels). Bo om panels show he imagina y pa o he a gumen o he Floque mul iplie s o hese solu ion amilies. Based in such indings, in [73] we explo ed he nonlinea ins abili y o linea ly s able mul ib ea he s. We ound ha nonlinea ins abili y is possible when an in ege mul iple o he equency o an in e nal (i.e. localized) eigenmode esona es wi h he phonon band and he K ein signa u e o such a mode and he phonon a c a e opposi e o each o he . No ice ha he equency Ωo he in e nal eigenmode and i s Floque a gumen ollows he ela ion θ=±ΩT mod 2π; consequen ly, i he coupling cons an is small enough, he in e nal mode has no collided wi h he phonon a c (in ac , his condi ion is necessa y in o de o he b ea he o be s able agains po en ial Hamil onian Hop bi u ca ions), hen Ω=θ/T =ωbθ/(2π). In his case, howe e , a ha monic o such a mode can be loca ed inside he a cs. I i is he 2nd ha monic, hen he slow g ow h ollows a −1/2law, i he 3 d, a −1/4law e c., as de i ed by he co esponding ODE [80, 73]. Ha ing in mind he s abili y p ope ies o mul ib ea he s nea he AC limi , he only s able solu ions among hem a e hose whose codes σa e in phase i he on-si e po en ial is ha d and in an i-phase i he po en ial is so . As demons a ed in [73], he in e nal modes ha de aches om θ= 0 ha e, in he uppe hal -ci cle (i.e. o Disc e e b ea he s in φ4and ela ed models 19 θ∈[0, π], K ein signa u e κ= 1 i he po en ial is so and κ=−1i i is ha d. On he o he hand, in o de o ha e nonlinea ins abili y, i is needed ha he K ein signa u e o he phonon a cs in he uppe hal -ci cle a e he opposi e o he in e nal modes. Because o his, as explained a he beginning o Subsec ion 3.2, nonlin- ea ins abili y can only be possible i ωo< ωb≤2ωoi he po en ial is ha d and 2 2k+1 ωo< ωb<1 kωowi h k∈Ni he po en ial is so . These condi ions we e co - obo a ed in he de ailed nume ical compu a ions o [73]. In he simple case o he NLS and DNLS models, no such condi ions need o be imposed, howe e nume ical compu a ions e i ied he exis ence o he ins abili y in [80]. He e, we only p o ide a p o o ypical case example o he associa ed phenomenology in he igh panel in Fig. 3. This shows he pa icula case o a he σ={1,1}mul ib ea he in a ha d po en ial. This solu ion is linea ly s able be o e he Hop bi u ca ion akes place, al hough p esen s nonlinea ins abili y as he condi ions o he pa ag aph abo e a e ul illed. The dynamics e en ually mani es s his ins abili y al hough he la e only a ises o e ex emely long imes, much longe han he linea ly uns able cases o he le and middle panels. 3.5 Dynamics In his subsec ion we will u he discuss some p o o ypical examples o uns able dynamics s emming om linea and nonlinea ins abili ies o he (2-si e) mul i- b ea he s shown p e iously (especially in Fig. 3). Fi s o all, we conside linea ly uns able mul ib ea he s in he so φ4po en ial. In ha case, he main beha io is a blow-up caused by a phenomenon simila o he escape obse ed in [41]. Le panels o Fig. 5 illus a e he blowing-up dynamics obse ed in an uns able σ={1,1}mul ib ea he . No ice ha blow-up is caused by bo h exponen ial and oscilla o y ins abili ies. I.e., he linea ly uns able g ow h leads he oscilla o s o exi he ini e heigh po en ial ba ie and hence end o ±∞. In he case o ha d po en ials, he main dynamical ea u es consis o he ans o - ma ion o he mul ib ea he in o a less ene ge ic 1-si e b ea he . In his ans o ma- ion, some ene gy is shed om he b ea he and, in some cases, i can be localized. The cen al panels o Fig. 5 shows he dynamics o he σ={1,−1}mul ib ea he . I is na u al o he con igu a ion o end o he si e-cen e ed a ian as ha is gene - ically s able, as we discussed abo e. Finally, he igh panels o he igu e conside a linea ly s able bu nonlinea ly un- s able mul ib ea he wi h σ={1,1}displaying a simila beha iou . The nonlinea ly uns able dynamics ul ills he condi ions o he heo em o [73] as he equency o he in e nal mode is Ω= 0.2073 and he spec al bands expands in [0.32,0.5]. As a esul , he second ha monic o he in e nal mode lies in he phonon a c, ul ima ely (a e y long imes) slowly eeding he nonlinea g ow h and e en ually leading o he des uc ion o he wo-si e con igu a ion. No ice, howe e , as also indica ed abo e, he cha ac e is ically longe (by a leas an o de o magni ude) ime needed o he mani es a ion o his (nonlinea ) ins abili y. 20 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis Fig. 5 Dynamical e olu ion o he uns able con igu a ions displayed in Fig. 3. Top panels show he space- ime dependence o he ene gy densi y, middle panels depic he ene gy densi y e olu ion o he cen al si es and bo om panels compa e o he p o iles o he pe u bed b ea he a he beginning and a he end o he simula ion. The mul ib ea he s conside ed a he igu es a e: (le panels) σ={1,1}in a so po en ial wi h ωb= 0.8and C= 0.2; (cen al panels) σ={1,−1} in a ha d po en ial wi h ωb= 2.5and C= 0.5; ( igh panels) σ={1,1}in a ha d po en ial wi h ωb= 1.5and C= 0.1. 4 Some glimpses on o he b ea he ea u es Al hough he co e o he p esen chap e ocuses on he s abili y p ope ies o dis- c e e b ea he s in φ4la ices, we men ion in passing some de elopmen s ela ed o a judicious selec ion o aside opics, namely mo ing b ea he s and he gene a ion o disc e e b ea he s in la ices wi h dissipa ion. Disc e e b ea he s in φ4and ela ed models 21 4.1 Mo ing b ea he s As men ioned in he p e ious sec ion, when an an an i-symme ic eigenmode de- aches om he phonon a c, i can each θ= 0 b inging abou a angen bi u ca ion. Jus a θ= 0, he mode is ma ginal and esembles a ansla ional mode, so ha a pe - u ba ion along i can se he b ea he in o mo ion. Con a y o con inuous b ea he s, mo ing disc e e b ea he s gene ically adia e phonons (see e.g. [72]) and hey e en- ually s op. Due o he special ea u es o mo ing b ea he s, he e does no exis a sys ema ic unde lying ma hema ical heo y ha can clea ly cha ac e ize hem. Some a emp s o de ining a Peie ls-Naba o ba ie simila o kinks ha e been pe o med, bu hey only seem o wo k in FPUT la ices close o he con inuum limi [81]. In any case, he e mus be a mechanism alike o Peie ls-Naba o ba ie which is ela ed o he exis ence o angen bi u ca ions o he ansla ional mode ha de aches om he phonon a c. Because o his, mo ing b ea he s can only be obse ed in la ices o which he b ea he s expe ience such bi u ca ions. The e a e only a ew epo ed cases o one-dimensional KG la ices, namely, wi h Mo se, sine-Go don and double- well po en ials [42]. We ha e also been able o gene a e mo ing b ea he s in wo- dimensional KG la ices wi h Mo se po en ial, a esul ha ha e no been published ye . Mo ing b ea he s (wi h high mobili y) has been obse ed in wo-dimensional la ices wi h in-plane deg ees o eedom modeling e.g. musco i e mica [16] which a e also known as quodons and, as men ioned in he In oduc ion, a e specula ed o play an impo an ole in cha ge anspo p ope ies in such ma e ials. Mo ing b ea he s do no exis in KG la ices wi h he φ4po en ial, as he angen bi u ca ion o he ansla ional mode does no ake place. Howe e , such a bi u ca- ion was obse ed o KG/FPUT la ices wi h on-si e and in e ac ion po en ials o he ha d φ4 o m [64]. Such a la ice has been used o modeling mic omechanical can ile e a ays [22]. In [82] we gene a ed mo ing b ea he s in such a model and analyze hei in e ac ion wi h geome ical de ec s. 4.2 Dissipa i e la ices Mos o he expe imen al indings o disc e e b ea he s ha e been achie ed on la - ices wi h dissipa ion and ex e nal d i ing, such as mic omechanical [22], pendula [20] and Josephson junc ions [18, 19] a ays, nonlinea elec ical la ices [23, 24] o g anula media [25, 26]. Such classes o sys ems emain qui e popula o his day wi h nume ous a ia ions con inuously a ising including, e.g., piecewise-linea sys- ems emula ing he β- o m o he celeb a ed FPUT la ice [83], o elec ical sys ems in ol ing beyond-nea es -neighbo in e ac ions [84]. As demons a ed in [85], disc e e b ea he s in dissipa i e la ices can also ex- is away om he an icon inuum limi . Con a y o Hamil onian la ices, he e a e no esonances wi h phonons ( he spec um o plane wa e exci a ions is pushed o he le hal o he complex spec al plane); as a esul such s a es a e now po en- 22 Jes´ us Cue as–Ma a e and Panayo is G. Ke ekidis ial a ac o s o he sys em. The wo k o [52] shows he complex phenomenology ha is obse ed in d i en and damped F enkel–Kon o o a la ices. I he sys em is d i en wi h a equency ωb, disc e e b ea he solu ions acqui e he same equency as he d i ing o ce. In gene al, all he la ice si es oscilla e wi h he same equency; howe e , we ha e ound an elec ical la ice whe e subha monic esonance eme ges ( ha is, he exci ed si es o he b ea he oscilla e wi h hal o he equency o he low ampli ude si es) [86]. Recen ly, mul is able a ia ions o pendula, po en ially applicable o SQUID me ama e ials, ha e been mani es ed as po en ial sou ces o mo e complex b ea hing pa e ns such as he celeb a ed chime a s a es [87]. Disc e e b ea he s ha e been s udied in d i en and damped one-dimensional KG la ices wi h ha d φ4po en ials in [88]. Such a la ice is de ined by equa ion: ¨un+α˙un+V0(un) + C(2un−un+1 −un−1) = Fn( ) + ηn( )(21) wi h Fn( )being a pe iodic unc ion o equency ωband ηn( )is a Gaussian whi e noise wi h ze o mean and au oco ela ion < ηn( )ηm( 0)>= 2Dδnmδ( − 0). In he de e minis ic case (D= 0) and o a s agge ed d i ing o he o m Fn( ) = (−1)n sin(ωb ), he phenomenology is qui e simple: a gi en damping and e- quency, disc e e b ea he s only exis abo e a h eshold h. Howe e , i he noise is in oduced in he la ice, he e a e wo in e es ing phenomena: i he d i ing ampli- ude is sup a h eshold ( > h), b ea he s wi h equency ωbcan be gene a ed e en i he ini ial condi ion is uni o m; i he d i ing ampli ude is sub h eshold ( < h), b ea he s a e s ill p oduced by he conce ed ac ion o noise and he d i ing o ce, in a way ha noise, on he one hand, enables sys em ansi ions be ween he uni o m and he cohe en localized s a es and, on he o he hand, des oys any deg ee o o - de o he sys em i i s ampli ude is la ge enough: in o he wo ds, we a e dealing wi h a s ochas ic esonance phenomenon. 5 Ou look and u u e di ec ions F om he abo e discussion, i is clea ha he heme o disc e e b ea he s is one ha is g adually ma u ing and eme ging in a wide ange o applica ions and a di- e se a ay o sys ems. In pa icula , we a e g adually depa ing om he simples nea es -neighbo scena ios o ei he jus KG o jus FPUT ypes and mo ing on o a new, mo e elabo a e phase whe e sys ems can be designed wi h beyond-nea es [84] and e en long- ange in e ac ions [59] and also wi h mul iple and po en ially com- pe ing [12] in e ac ions, o wi h ones ha a e p og essi ely mo e amenable o ana- ly ical conside a ions [83]. This sugges s ha he e is a signi ican need o u he heo e ical and compu a ional de elopmen s o suppo he co esponding eme ging expe imen al pla o ms. While he ea ly s ages o de elopmen o DBs a o ed analy ical p oo s o ex- is ence and associa ed echniques o nume ical exis ence and s abili y, subsequen ones a o ed a mo e sys ema ic explo a ion o spec al p ope ies and an a emp o Disc e e b ea he s in φ4and ela ed models 23 classi y he di e en mul ib ea he s and o e sys ema ic guidelines abou when hey may be expec ed o be dynamically obus . In his Chap e , we summa ized some o his sys ema ic e o in he p e ious decade and some o i s c ys allized esul s and con e gence o di e en me hods o e he pas ew yea s. Mo e ecen ly, u he ools ha e a isen in p obing spec al and dynamical ea u es o DBs. Among o he s, we ha e explo ed and summa ized he e he ene gy- e sus- equency mono onici y c i e ia and how hey ela e o linea ins abili ies and gi en connec ions be ween hese and he s abili y o a eling wa es in la ices. Addi ionally, we ha e wa ned he eade agains he nai e expec a ion ha spec al s abili y is he ull s o y, p e- sen ing case examples whe e his ails o be ue due o he nonlinea ins abili y o in e nal modes wi h opposi e K ein signa u e han ha o he phonon a cs. The slow, powe -law na u e o he la e ins abili ies, as opposed o he exponen ial g ow h o linea ins abili ies was highligh ed. Las ly, some possibili ies o u he de elop- men s owa ds mo ing b ea he s o dissipa i e la ices we e b ie ly ouched upon. Clea ly, he e is need o u he heo e ical sys ema ics. Many o he ele an poin s we e b ough up in pa s o ou discussion. Unde s anding he s abili y o phase-shi mul ib ea he s and o ex b ea he s in highe -dimensional sys ems is an impo an open opic. Ca ying ou he associa ed s abili y compu a ions o highe o de is pa icula ly ele an . Recen wo k, in ac , b ings up he possibili y ha ele an solu ions may ail o exis a highe o de s in some impo an case exam- ples [89]. S udying also long- ange in e ac ions may b ing abou su p ises and p o- duce gaps in he spec um whe e no el DBs may exis , as pe he ecen wo k o [90]. Again, his is a opic me i ing u he explo a ion. The s udy o sys ems wi h non- i ial ails (nanop e a) and he examina ion o whe he he s abili y ea u es/c i e ia p esen ed he ein apply o hem is also an open opic. The same holds ue o sys- ems wi h ex e nal d i e and damping: can we o e some guidelines o cha ac e ize hei s abili y cha ac e is ics, sui ably adap ing wha we know in he mo e s uc u ed Hamil onian cases o pe haps no ? Plus hen he e a e opics which, while ouched upon, s ill seem ai ly poo ly unde s ood o wide open o new insigh s: among hem mo ing b ea he s, o quasi-pe iodic solu ions and hei exis ence and s abili y, as well as he ole o DBs in asymp o ic dynamics and he maliza ion (see, e.g., [91] o a ecen summa y in he disc e e NLS case) a e only some ha come o mind. In summa y, disc e e b ea he s may ha e ma u ed bu ha e many mo e challenges o o e o he yea s o come bo h a he undamen al, a he compu a ional and a he expe imen al le el... Acknowledgemen s This ma e ial is based upon wo k suppo ed by he Na ional Science Foun- da ion unde G an No. DMS-1809074 (P.G.K.). J.C.-M. hanks inancial suppo om MAT2016- 79866-R p ojec (AEI/FEDER, UE). 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