Discrete peakons

Disc e e peakons A. Comech1, J. Cue as2, and P.G. Ke ekidis3 3Ma hema ics Depa men , Texas A&M Uni e si y, College S a ion, TX 77843-3368, USA, 2G upo de F´ısica No Lineal, Depa amen o de F´ısica Aplicada I, ETSI In o m´a ica, Uni e sidad de Se illa, A da. Reina Me cedes, s/n. 41012-Se illa, Spain, and 1Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s, Amhe s , MA 01003-4515, USA We demons a e he possibili y o explici cons uc ion in a disc e e Hamil onian model o an exac solu ion o he o m exp(−|n|), i.e., a disc e e peakon. These disc e e analogs o he well- known, con inuum peakons o he Camassa-Holm equa ion [Phys. Re . Le . 71, 1661 (1993)] a e ound in a model di e en om hei con inuum siblings and om ea lie s udies in he disc e e se ing [Phys. Re . Le . 83, 248 (1999)]. Namely, we obse e disc e e peakons in Klein-Go don- ype and nonlinea Sch ¨odinge - ype chains wi h long- ange in e ac ions. The in e es ing linea s abili y di e ences be ween hese wo chains a e examined nume ically and illus a ed analy ically. Addi ionally, in e -si e cen e ed peakons a e also ob ained in explici o m and hei s abili y is s udied. We also p o e he global well-posedness o he disc e e Klein-Go don equa ion, show he ins abili y o he peakon solu ion, and he possibili y o a o ma ion o a b ea hing peakon. I. INTRODUCTION In he las wo decades, in insic localized modes (ILMs), also e med disc e e b ea he s (DBs), ha e become a opic o in ense heo e ical and expe imen al in es iga ion; see, e.g., [1] o a numbe o ecen e iews on he opic. Pe hei inhe en abili y o bo leneck and po en ially anspo he ene gy in a cohe en ashion, such exponen ially localized in space and pe iodic in ime en i ies ha e come o be o in e es in a a ie y o con ex s. These ange om nonlinea op ics and a ays o wa eguides [2] o Bose-Eins ein condensa es (BECs) inside op ical la ice po en ials [3] and om p o o ypical models o nonlinea sp ings [4] o Josephson junc ions [5] and dynamical models o he DNA double s and [6]. The ubiqui ous na u e o nonlinea la ice sys ems (i.e., a ays o coupled nonlinea oscilla o s) has p omp ed he examina ion o he beha io o nonlinea wa es in hese sys ems, pa icula ly o he wa es ha a e well-known and unde s ood in he con inuum analogs o hese equa ions i.e., in nonlinea pa ial di e en ial equa ions. In his di ec ion o ac i i y, many o he cohe en , nonlinea wa e s uc u es ha ha e p e iously been disco e ed in con inuum se ings such as egula soli ons [7], compac ons [8], shock wa es [9] o gap soli ons [10] ha e been ecen ly examined in he disc e e se ing; see, e.g., he wo ks o [1] o egula soli a y wa es, [11] o disc e e compac ons, [12] o disc e e shock wa es o [13] o la ice gap soli ons. On he o he hand, a con inuum nonlinea wa e ha was ecen ly ob ained in he con ex o shallow wa e wa e equa ions, namely he peakon (a peaked soli on solu ion wi h a discon inui y in he i s de i a i e a i s peak) only e y ecen ly s a ed o be conside ed in he disc e e con ex [14]. Peakons we e gi en hei name (in he con ex o he so-called Camassa-Holm equa ion, which is a dispe si e, in eg able model [15, 16]) due o hei discon inuous de i a i e a peak ampli ude. I is wo h no ing, howe e , ha such “sha ply poin ed” s uc u es we e also ob ained much ea lie in he con ex o nonlocal models o plasmas (see, e.g., [17]). While, a i s glance, i may no appea pa icula ly na u al o ha e disc e e analogs o hese solu ions, as de i a i es a e no , s ic ly speaking, de ined in he con ex o he spa ial la ice, ou aim in he p esen wo k is o examine he “disc e e peakon”. This will be a la ice p o ile in he o m (in ac , exac ly) ∼exp(−a|n|) ha in he con inuum limi will asymp o ically app oach he con inuum peakon solu ion. A numbe o wis s accompany his disc e e peakon solu ion in oduced he e. In he discussion, we will illus a e ha glimpses o solu ions ha can be ca ego ized unde his new “species” may ha e al eady appea ed in a somewha gene alized o m in ea lie wo ks. We hope his may ini ia e a mo e gene al examina ion o he p esen ly sugges ed la ice peakons. We belie e ha we ha e encapsula ed he key mechanisms whose in e play can gi e ise o he exis ence o peakons (in bo h he disc e e and he con inuum se ing) in a model o he Klein-Go don (KG) a ie y and i s nonlinea -Sch ¨odinge (NLS) ype analog. I is in e es ing o no e ha bo h he dispe si e and he nonlinea e ms ha we ha e combined in he p esen se ing ha e appea ed p e iously in di e se con ex s ( ha will be men ioned below); howe e , hey we e ne e combined o allow o he exis ence o peakons. Ano he in iguing bi conce ns he s abili y p ope ies o he peakons which a e also examined in wha ollows. We ind ha he peakons, while uns able in he KG a ian o ou model, become s able in i s NLS e sion. This is because he nega i e ene gy di ec ion p esen he o me model is p ohibi ed by he cha ge conse a ion in he la e model. In he KG case, disc e e b ea hing peakons a e possible ins ead. In e es ingly, such solu ions we e also iden i ied ea lie in closed o m in a class o models e y di e en han he ones examined he ein [mo e speci ically in a class o homogeneous, nea es 2 neighbo , Klein-Go don Hamil onians] in [18]. Finally, i is ema kable ha he p esen ly examined class o models and i s s abili y ea u es can be ea ed by me hods a ailable o he con inuous nonlinea ield equa ions as illus a ed below. Ou p esen a ion will be s uc u ed as ollows. In Sec ion II, we will discuss he model and i s mo i a ion. In Sec ion III, he nume ical esul s will be p esen ed o si e-cen e ed, as well as o in e -si e cen e ed peakons. In Sec ion IV, we will examine he s abili y o such s uc u es h ough analy ical conside a ions based on cons ained ene gy minimiza ion, as well as on unc ional analy ic a gumen s. In Sec ion V, he exis ence o disc e e b ea hing peakons is discussed. Finally, in Sec ion VI, we will summa ize ou indings and p esen ou conclusions, as well as some open ques ions o u u e s udies. II. MODEL AND MOTIVATION Examining he peakon om he “ e e se enginee ing” (o he in e se p oblem) poin o iew, we would like o use he p ope ies o a solu ion o he o m u(x)∼exp(−|x|), o cons uc a (con inuum as well as a) disc e e la ice wi h dynamics ha suppo s such peaked solu ions. Adop ing his iewpoin , some o he key p ope ies o u(x) = exp(−|x|) a e ha u0(x) = −u(x)sgn(x),(1) (1 −∂2 x)u(x) = 2δ(x)u(x).(2) This second p ope y (c . also [15]) jus i ies why a s ongly localized impu i y (o he o m o a δ- unc ion) may be a ele an con ex in which such peaked solu ions could a ise. We will e u n o his poin in Sec ion VI. Ano he , pe haps mo e in e es ing o ou pu poses, p ope y is he esul o con olu ion o such a peaked unc ion wi h a Kac-Bake exponen ial in e ac ion ke nel J(|x−y|) = exp(−|x−y|) [20, 21]. The con olu ion yields: J ? u ≡Z∞ −∞ exp (−|x−y|) exp(−|y|)dy = (1 + |x|) exp (−|x|).(3) This sugges s immedia ely om a ma hema ical pe spec i e a Klein-Go don (KG), as well as a nonlinea Sch ¨odinge (NLS) model wi h long- ange in e ac ions ha would suppo exac peakon solu ions o he o m u(x) = Aexp(−a|x|). In pa icula he KG model would ead: u =aZ∞ −∞ exp(−a|x−y|)u(y)dy −µ1−1 2ln µu2 A2¶¶u. (4) The peakons in his case would ep esen s a ic solu ions o he KG equa ion. The co esponding NLS model would be o he o m iu =−aZ∞ −∞ exp(−a|x−y|)u(y)dy −1 2ln µ|u|2 A2¶u, (5) whe ein he peakons co espond o s anding wa e solu ions o he o m u(x) = Aexp(i ) exp(−a|x|). Con inuing along his lane o e e se cons uc ion ( he mo i a ion o each o he ele an e ms will be gi en below), we now explain ha an in e es ing ea u e o he abo e conside a ions is ha hey can also be ca ied h ough in he disc e e se ing. In pa icula , o un= exp(−a|n|), we sum up he geome ic se ies es ablishing ha : X m∈Z exp(−a|n−m|)um=µexp(2a)+1 exp(2a)−1+|n|¶un.(6) Consequen ly, o he disc e e Kac-Bake in e ac ion ke nel Jnm = exp(−|n−m|),(7) we can de ise a disc e e KG, as well as a disc e e NLS model ha ha e disc e e analogs o peaked solu ions. The disc e e KG model is o he o m: ¨un=X m∈Z Jnmum−·exp(2a)+1 exp(2a)−1−1 2aln µu2 n A2¶¸un,(8) 3 wi h he exac disc e e peakon solu ion πn=Aexp(−a|n|), while he co esponding disc e e NLS chain can be o mula ed as: i˙un=−X m∈Z Jnmum+·2 exp(2a)−1−1 2aln µ|un|2 A2¶¸un,(9) whe e a disc e e peakon gi en by he s anding wa e exp(i )πnis he exac solu ion o he model. I is hese la e equa ions [Eqs. (8) and (9)] and hei dependence on he pa ame e a ha de e mines he in e ac ion “ ange” ha we plan on in es iga ing in wha ollows. No ice ha acan be conside ed as a na u al spacing pa ame e . I is in e es ing o no e as a side ema k ( o which we will e u n in la e sec ions) ha his model no only suppo s an exac “on-si e” disc e e peakon solu ion such as he one gi en abo e, bu addi ionally suppo s exac “in e -si e” peakon solu ions o he o m: un=Bexp(−|n−1/2|) o he Klein Go don (in he DNLS case i is un=Bexp(i ) exp(−|n−1/2|)). The alue o he p e ac o Bis gi en by he ela ion ln(B/A) = a·(exp(a)−1) 2(exp(a) + 1)¸.(10) Such explici solu ions (especially in e -si e ones) a e a ely a ailable in non-in eg able disc e e models. In e -si e solu ions a e ypically less s able han hei on-si e siblings [19]. In he p esen se ing, we will s udy in de ail he beha io o such wo-si e peakons nume ically as well as analy ically. In mo i a ing he model, aside om i s in insic ma hema ical in e es due o he exis ence o he peaked solu ions (bo h in he disc e e case and in he con inuum limi ), we should ema k ha bo h he dispe si e and he nonlinea e ms included he e ha e appea ed in a a ie y o se ings be o e. The Kac-Bake ype ke nel [20, 21], aside om i s ele ance in models o s a is ical physics, has been used qui e ex ensi ely in ecen nonlinea s udies o la ice models emula ing biopolyme dynamics including DNA; see, e.g., [22–27]. Hence, his ype o in e ac ion is a he ubiqui ous and can be con ollably uned (depending on he alue o a), o be p ac ically nea es neighbo ( o la ge a) o much longe ange ( o a→0). Le us no e also ha in he pas and in he amewo k o con inuum equa ions simila o ms o his ke nel had been examined in he wo k o [28] (bu in a a he di e en dynamical model, namely one o he KdV ype) whe ein i was ound ha a eling wa es acqui ed a peaked wa e o m. In con ex s mo e closely ela ed o he ones o he p esen wo k, le us also men ion ha “cusp soli ons” (i.e., peakons) we e also ound in con inuum models o he NLS ype wi h Kac-Bake in e ac ions in [29] (howe e , hey we e uns able) and in he case o a nonlocal Klein-Go don ield heo y in [30]. The loga i hmic nonlinea i y in nonlinea model equa ions was o iginally in oduced in he con ex o quan um ield heo y [31]. I eappea ed in [32], whe e i was p oposed as an equa ion o gene alized quan um mechanics. La e in [33], i was sugges ed as a desc ip ion o ex ended objec s in nuclea ma e , while mo e ecen ly i was examined in he con ex o a scala ield model in in la iona y cosmology [34]. These s udies ha e also igge ed a mo e ma hema ically o ien ed in e es in his nonlinea i y and he p ope ies o he solu ions o he co esponding nonlinea wa e equa ions [35]. Pe haps, mos closely o he pu poses o in e es o his s udy, his ype o loga i hmic ield heo y has appea ed in sa u able nonlinea op ical media. The ini ial in es iga ions o [36] in he la e con ex , in he amewo k o he “migh y mo phing” spa ial soli ons, we e la e placed in a mo e physically ealis ic amewo k in connec ion wi h pho o e ac i e ma e ials in [37]. The wo k o [37] sugges s ha o he nonlinea wa eguide e olu ion, he loga i hmic nonlinea i y p o ides an accessible model ha o e s aluable insigh , while main aining he cha ac e is ic ea u es o he unde lying physical p ocess. A no e o cau ion should howe e be made in his connec ion in ha he nonlinea i y o Eqs. (8)-(9) should be iewed as a mo e easonable physical model o la ge ampli udes (whe e i can be conside ed as an app oxima ion o a mo e physical nonlinea e m such as ln(1 + |un|2)). Fo ampli udes ending o 0, he di e gence o ln(|un|2) appea s o be somewha unphysical and leads o he absence o a small ampli ude exci a ion (so-called “phonon”) spec um. The combina ion o he ea u es o he dispe si e in e ac ion (i s con ollable ange and wide applicabili y) and o he loga i hmic nonlinea i y (an accessible one ep esen ing adequa ely a numbe o physical p ocesses) ende s ou model a possibly good playg ound o s udy, e.g., an a ay o coupled sa u able nonlinea (loga i hmic) wa eguides. Bo h he po en ial ele ance o ou esul s in his con ex , as well as hei inhe en ma hema ical in e es in es ablishing he disc e e p ope ies and beha io o he peaked solu ions, lead us o examine Eqs. (8) and (9) in wha ollows. 4 III. NUMERICAL RESULTS A. Gene al Se up The equa ions ha we will examine can be e-w i en in a mo e gene al o m: ¨un−X m∈Z Jnmum+F(un) = 0 (11) o he KG la ice and i˙un=−X m∈Z Jnmum+G(un, u? n) (12) o he DNLS chain; ecall ha Jnm is gi en by Eq. (7). Fo Eqs. (8) and (9), he espec i e on-si e e ms a e: F(un) = ·exp(2a)+1 exp(2a)−1−1 2aln µu2 n A2¶¸un,(13) G(un, u? n) = ·2 exp(2a)−1−1 2aln µunu? n A2¶¸un.(14) Wi hou loss o gene ali y, we se A= 1. The exac solu ions o in e es o Eq. (11) a e o he amilia peakon o m men ioned p e iously: πn= exp(−a|n|). We examine he linea s abili y o hese solu ions by using in Eq. (11) un=πn+²exp(iω ) n,(15) whe e πnis he o iginal peakon and ωa e he eigen equencies o linea iza ion a ound he solu ion ( na e he co esponding eigen ec o s). The esul ing linea s abili y equa ion (ob ained by using he ansa z o Eq. (15) o O(²) in Eq. (11)) eads: −ω2 n=X m∈Z Jnm m−F0(πn) n.(16) This is an eigen alue p oblem o he ma ix Jnm −δnmF0(πn). The disc e e peakon is linea ly uns able i he e a e eigen equencies ωwi h he nega i e imagina y pa . Since he ma ix elemen s Jnm a e bounded and ansla ion- in a ian (only depend on n−m) while F0(πn) exponen ially decays as n→ ∞, one can show ha he eigen alues o he unca ed ma ix, wi h |m|,|n| ≤ N, will end o he eigen alues o (16) as N→ ∞. The eigen alue o he unca ed ma ix can easily be sol ed using nume ical linea algeb a packages; his gi es he app oxima e eigen equencies ω and he co esponding eigen ec o s n. Fo he DNLS la ice, he s abili y can be pe o med in he “co- o a ing” ame [39], using he ansa z un= exp(i ) [πn+²(anexp(−iω ) + b? nexp(iω? ))] .(17) Then, he esul ing linea s abili y equa ions will be o he o m: ωµak b? k¶=J·µak b? k¶, whe e Jis he linea s abili y (Jacobian) ma ix o he o m J=Ã∂Fi ∂uj ∂Fi ∂u? j −∂F? i ∂uj−∂F? i ∂u? j!, and Fn=−Pm∈ZJnmum+G(un, u? n) + un( he Jacobian should be e alua ed a he peakon p o ile, un=πn). We now p oceed o examine s abili y and dynamics p ope ies o peakons in Klein-Go don and DNLS sys ems. 5 −50 0 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 un n −5 0 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Im(ω) Re(ω) FIG. 1: Le panel: Spa ial p o ile o an exac disc e e peakon (a= 1). Righ panel: Spec al plane (Re(ω),Im(ω)) o he s abili y ma ix o his solu ion in a Klein–Go don chain. B. Klein–Go don 1-si e peakons The p o ile o a peakon is shown in Fig. 1, oge he wi h he spec um o eigen equencies o he equa ion linea ized a he peakon. Fo he KG chain, we ind ha he solu ions a e uns able ( o all alues o a). This is because o a nega i e ene gy di ec ion ha leads o an imagina y pai o eigen equencies. The co esponding eigen ec o has he same shape as he peakon i sel . This ac can be obse ed in mo e de ail in Fig. 2, whe e he dependence o he imagina y pa o he uns able eigen alue o he s abili y ma ix is shown as a unc ion o a. This igu e also gi es he dependence o he ene gy o he peakon on a, which can be analy ically calcula ed: E(a) = A2co h(a)/(2a). F om hese igu es, i can be deduced ha he solu ion becomes less uns able wi h (inc easing) a, o , in o he wo ds, when he wid h o he peakon dec eases. This can be equi alen ly in e p e ed as a weake ins abili y as he solu ion app oaches i s an i-con inuum limi , single-si e peakon sibling. This is a a he na u al ea u e o spa ial disc e eness which ypically se es o s abilize cohe en s uc u es ha a e uns able in he con inuum limi (e.g., due o collapse) o e en ones ha do no exis in ha limi [1]. In o de o examine he dynamical e olu ion o he ins abili y o he KG la ice peakon, we used di ec nume ical simula ions, pe o med wi h a 5 h o de Cal o’s symplec ic in eg a o [38] wi h a ime s ep ∆ = 0.001, which p ese es he ene gy up o a ac o 10−15. We in oduce a pe u ba ion ξn=επn( o he exac peakon solu ion πn) wi h ε= 0.1 o exci e he uns able eigendi ec ion. The exponen ial g ow h o he peakon ins abili y is shown in he le panel o Fig. 3. I he pe u ba ion we e ξn=−επnwi h ε= 0.1 again, he peakon does no g ow. Ins ead, i e ol es o a b ea hing s a e (see e.g. igh panel o Fig. 3), which will be analyzed in Sec ion V. C. DNLS 1-si e peakons As explained abo e, he (spa ial dependence o he) p o ile o a DNLS peakon is he same as ha o i s Klein-Go don analog (see Fig. 1). Howe e , in he DNLS se ing, he peakon is s able o all alues o a. This ac has i s o igin in he gauge in a iance o he solu ions o DNLS- ype equa ions. In pa icula , an in e es ing ea u e o he disc e e peakons is ha he U(1) symme y o he DNLS chain p ohibi s he single nega i e ene gy di ec ion ha was p esen in he KG la ice. Essen ially, he uns able di ec ion o he KG la ice is ans e sal o he same-cha ge hype su ace in he DNLS case. As a esul , pe u ba ions along his po en ially uns able di ec ion a e banned by he p esence o he ex a symme y. Figu e 4 shows he spec al plane o a ypical case oge he wi h he dependence on ao he cha ge (also e e ed o as powe in op ics) o he peakon. The cha ge is de ined as Q(u) = Pn|un|2/2 and i can be obse ed ha i s alue dec eases wi h aand ends o Q= 1/2 (as should be expec ed as a→ ∞). This dependence can be analy ically 6 0 2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 |max(Im(ω))| a 0 2 4 6 8 10 1 2 3 4 5 6 7 8 Ene gy a FIG. 2: Le panel: Dependence o he maximum imagina y eigen equency (i.e., maximum eal eigen alue) o he s abili y ma ix on a. Righ panel: Dependence o he ene gy o he peakon on a. −10 −5 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 un n 0 20 40 60 80 100 120 140 160 180 −5 0 5 Displacemen es FIG. 3: Le panel: Ins abili y e olu ion o a disc e e peakon a di e en imes (a= 1). Exponen ial g ow h can be obse ed, while he ene gy o he sys em is p ese ed up o a 10−15 p ecision. The di e en snapsho s o he solu ion co espond ( om inne o ou e ) o imes 0, 1.8, 2.7 and 3.6. Righ panel: Time e olu ion o a pe u bed peakon ha de elops in o a b ea hing s a e. The displacemen o he cen al si es o he peakon a e shown as a unc ion o ime. calcula ed: Q(a) = A2co h(a)/2. The s abili y o he solu ion can be e i ied in he ime e olu ion nume ical expe imen shown in Fig. 5, which has been pe o med h ough a 4 h o de Runge-Ku a in eg a o wi h ime s ep ∆ = 0.01. The phase space plo a he cen al si e shows ha a andomly pe u bed solu ion emains o bi ally close o he exac disc e e peakon solu ion. 7 −30 −20 −10 0 10 20 30 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Im(ω) Re(ω) 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 P a FIG. 4: Le panel: Spec al plane o he s abili y ma ix o he peakon o Fig. 1 in he case o he DNLS chain. Righ panel: Dependence o he peakon cha ge (powe ) Q=Pnπ2 n/2 as a unc ion o a. −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Im(u( )) Re(u( )) 0.85 0.9 0.95 1 1.05 1.1 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 Im(u( )) Re(u( )) FIG. 5: Le panel: Phase space diag am o he cen al si e o a pe u bed ( ull line) and an unpe u bed (dashed line) DNLS peakon. Righ panel: A blow-up o he le panel ha illus a es he o bi al s abili y o he peakon solu ion (since he pe u bed solu ion emains in i s icini y). D. 2-si e Peakons We now examine he beha io o wo-si e (i.e., in e -si e) peakons, bo h in he KG, as well as in he DNLS chain. The ene gy and cha ge o 2-si e peakons can be analy ically calcula ed as E(a) = B2/(4asinh(a)) and Q(a) = B2/(2 sinh(a)). Con a y o hei single si e coun e pa s, wo-si e solu ions a e uns able (also in he DNLS model). In his case, wo nega i e ene gy di ec ions and hence wo imagina y eigen equencies could be iden i ied in he spec al plane o he linea iza ion eigen equencies in he case o he KG la ice, while one such eigen equency was p esen in he DNLS se ing (see Fig. 6). The eigenmode co esponding o he KG case is an isymme ic. We also simula ed he dynamical de elopmen o hese ins abili ies, obse ing ha KG wo-si e peakons a e com- 8 −5 0 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Im(ω) Re(ω) −30 −20 −10 0 10 20 30 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Im(ω) Re(ω) FIG. 6: Spec al plane o a he wo-si e peakon (wi h a= 1). Le panel: KG peakon. Righ panel: DNLS peakon. 0 50 100 150 200 250 300 350 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 |un( )|2 n −20 −15 −10 −5 0 5 10 15 20 0 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 FIG. 7: Time e olu ion o he wo-si e peakon in he DNLS chain (wi h a= 1) induced by pe u bing he cen al pa icle. Le panel: The ull line ep esen s |u0|2, he dashed line ep esen s |u1|2. Righ panel: Cha ge densi y in space- ime. ple ely des oyed (as hei one-si e coun e pa s a e also no s able). Fo he wo-si e DNLS peakons exci ed wi h he pe u ba ion ²δn,0wi h ²∼10−4, he solu ion oscilla es be ween he one-si e and wo-si e peakons. These esul s a e shown on Fig. 7. IV. ANALYTICAL RESULTS The abo e esul s mo i a e us o examine he s abili y o he Klein-Go don and DNLS peakons om an analy ical pe spec i e and, in pa icula , using ene ge ic conside a ions. We now p oceed o s udy he s abili y o he uni o m s eady s a e (“ acuum”) and o one-si e peakons in each o hese se ings. The conclusions abou s abili y o ins abili y do no depend on he alues o Aand a, so we se A= 1, a = 1, 9 so ha he peakon p o ile is gi en by πn= exp(−|n|). A. Klein-Go don 1. Hamil onian o mula ion. We can ew i e he Klein-Go don equa ion (11) as ¨un+∂unT(u) + ∂unW(u) = 0 (18) whe e T(u) = −1 2X (n,m)∈Z2 e−|n−m|unum,(19) W(u) = X n∈ZµµΛ 2+1 4¶u2 n−1 4u2 nln u2 n¶.(20) Abo e, Λ is a posi i e cons an aken o be Λ = (π, π) = X n∈Z e−2|n|=e2+ 1 e2−1,(21) whe e (u, u) = Pn∈Zu? nun(in he Klein-Go don case, we assume ha he componen s una e eal- alued). Rema k 1 La e we will show ha he Klein-Go don equa ion (18) is globally well-posed in l2(Z)×l2(Z): i bo h u(0) and ˙u(0) belong o l2(Z), hen he e is a global solu ion u( )wi h ku( )kl2<∞,k˙u( )kl2<∞ o 0≤ < ∞. See Theo em 4. A he same ime, he no ms ku( )kl2,k˙u( )kl2could g ow unboundedly la ge wi h ime. We can ew i e (18) as ¨u+T0(u) + W0(u) = 0,(22) whe e T0(u), W0(u) may be in e p e ed as a ia ional de i a i es wi h espec o uo he unc ionals Tand W. The alue o he ene gy unc ional EKG(u, ˙u) = X n∈Z ˙u2 n 2+T(u) + W(u) (23) is conse ed along he ajec o ies o (22). 2. S abili y o acuum. Fi s , le us no ice ha he ze o solu ion is s able wi h espec o l2-pe u ba ions o he ini ial da a. We bound T(u) by |T(u)| ≤ 1 2¯¯¯¯¯ sup nX m∈Z e−|n−m|¯¯¯¯¯ (u, u)≤1 2 e+ 1 e−1(u, u).(24) This inequali y is due o he Schu es applied o he ma ix Jnm =e−|n−m|, which yields kJukl2≤Ãsup n∈ZX m∈Z|Jnm|! 1 2Ãsup m∈ZX n∈Z|Jnm|! 1 2 kukl2=e+ 1 e−1kukl2.(25) 16 Fo L≥1, he igh -hand side is mono onically inc easing (and exceeds i s ange o 0 <L<1). The e o e, 0≤L( )≤Z( ), whe e Z( ) is a unc ion ha sa is ies Z00( ) = C(1 + Z+ 2Zln Z) (58) and he ini ial da a Z(0) = max(1, L(0)) = max(1,(u0, u0)), Z0(0) = max Ã1,dL d ¯¯¯¯ =0 != max (1,2(u0, 0)) . We can ew i e (58) as Z00 +C∂Z(−Z−Z2ln Z) = 0; (59) mul iplying by Z0and in eg a ing in , we ge (Z0)2 2−CZ −CZ2ln Z=E, whe e E=(Z0(0))2 2+C(−Z(0) −Z2(0) ln Z(0)) is a cons an o in eg a ion. Exp essing Z0and sepa a ing a iables, we ge =ZZ( ) Z(0) dZ √E+CZ +CZ2ln Z. Since he in eg al R∞ Z(0) dZ √E+CZ+CZ2ln Zdi e ges a he uppe limi (as ln1 2Z), we conclude ha Zcan no become in ini e in ini e ime. The ini eness o l2-no m o ˙u ollows om ela ion (54), bounds (55) and (56), and he ini eness o L=kuk2 l2 ha we al eady p o ed. This inishes he p oo o Theo em 4. C. Linea ized s abili y o B ea hing Peakons We will now analyze he linea s abili y o he b ea hing peakon, showing he absence o he exponen ial ins abili y o small pe u ba ions o he ini ial da a. We ew i e Eq. (18) as he i s o de sys em, ½˙u= ˙ =−∂u(T(u) + W(u)),(60) and conside he pe u ba ion o he solu ion (u0( ), 0( )) = (g( )π, ˙g( )π) ha co esponds o he peakon: u( ) = u0( +γ( )) + δu( ), ( ) = 0( +γ( )) + δ ( ). The unc ion γ( ) adjus s he loca ion o he b ea he (g( )π, ˙g( )π) so ha i is close (in a ce ain sense) o he pe u bed solu ion (u( ), ( )). We conside he linea iza ion o sys em (60). Fo his, we i s compu e T00(gπ) + W00(gπ) = T00(π) + W00(π)−ln g2 2=H+−ln g2 2,(61) whe e H+was in oduced in (46), and hen we can w i e ½g0( +γ( ))˙γ( )π+∂ δu( ) = δ ( ), g00( +γ( )) ˙γ( )π+∂ δ ( ) = −(H+−ln g2 2)δu( ).(62) We spli δu( ) = a( )π+φ( ), δ ( ) = b( )π+ψ( ), 17 wi h φ,ψ∈l2(Z) bo h o hogonal o π. P ojec ing sys em (62) on o π, gi es he ollowing sys em: ½g0( +γ( ))˙γ( ) + ˙a( ) = b( ), g00( +γ( )) ˙γ( )π+˙ b( ) = −(−1−ln g2 2)a( ),(63) whe e we used he ac ha πis an eigen ec o , H+π=−π. P ojec ion o (62) on o he di ec ion no mal o πgi es he ollowing sys em: (˙ φ( ) = ψ( ), ˙ ψ( ) = −(H+−ln g2 2)φ( ).(64) The analysis o sys em (63) is s aigh o wa d. We would a i e a his sys em i we pu sued he s abili y analysis o Eq. (52). A he same ime, we could analyze ha equa ion opologically. I co esponds o an unha monic oscilla o ; i s phase po ai in he (g, ˙g) plane con ains a se o closed ajec o ies ci cling a ound he o igin ( hey co espond o he ini ial da a (g, ˙g) such ha |g|<1 and he alue o Ein (53) is smalle han 1/4). Each o hese closed ajec o ies is s able wi h espec o small pe u ba ions o he ini ial da a: The e is a neighbo ing closed ajec o y passing h ough he poin ha co esponds o he pe u bed ini ial da a. Le us now analyze sys em (64). We can ew i e i as ˙ Φ = JHΦ,whe e Φ = ·φ ψ¸,J=·0 1 −1 0 ¸,H=·H+−ln g2 20 0 1 ¸, wi h φ,ψo hogonal o π. The ma ix H+(see (46) and he ea e ) has eigen alues σ(H+) = {−1,Λ−1, . . .}, whe e he only nega i e eigen alue −1 co esponds o πand he nex eigen alue, Λ−1, is posi i e. The e m −ln g2 2in (61) u he shi s he spec um upwa ds as long as |g|<1. The e o e, His posi i e-de ini e on he space π⊥×l2(Z). Thus, √His well-de ined; he ma ix JH is simila o √H J√Hand hence has pu ely imagina y spec um. This shows ha he b ea hing peakon solu ions g( )π o (18) a e spec ally s able. Rema k 5 This linea ized app oach o he s abili y does no p o e he dynamic o bi al s abili y o he b ea hing peakons. The nonlinea e ms may ans e he ene gy be ween “b ea hing” oscilla ions in π-di ec ion and he pe - u ba ions in he space π⊥, pumping he ene gy om sys em (63) in o (64). I is he e o e possible ha he ene gy o he b ea hing peakon, a e a small pe u ba ion, would wind up ans e ed, pa ially o comple ely, in o di ec ions o hogonal o π(and hen maybe back). We do no ha e a sa is ac o y desc ip ion o his p ocess, e en hough we belie e ha echniques such as he Hamil onian dispe si e no mal o ms o [48] could be ele an in add essing i . While ou side he scope o he p esen s udy, his may be an in e es ing ques ion o u u e in es iga ions. D. Nume ical Resul s An o bi o equency ωp o g( ) can be de e mined by sol ing Eq. (52). This can be done h ough a a ie y o me hods such as, e.g., a shoo ing me hod in eal space o using quad a u es. He e we ha e chosen a Chebyshe quad a u e me hod in o de o in eg a e he equa ion. Figu e 9 shows he dependence o g(0) and Eo he peakon equency ωp. I can clea ly be obse ed (see he le end o he g aphs) ha as ωp→0, he ene gy app oaches 1/4 and he ampli ude is 1, hence he solu ion is e y close o he uns able c i ical poin o E= 1/4 and g= 1. We can also obse e ha he peakon equency can ha e any alue as he e do no exis esonances wi h he con inuous spec um (ac ually, such small ampli ude, ex ended wa e exci a ions do no exis since V00(0) = ∞). Figu e 10 shows he ime e olu ion o a b ea hing peakon. I is wo h poin ing ou ha he main di e ence be ween DNLS peakons and KG b ea hing peakons is ha , in he i s case, only he i s Fou ie coe icien is di e en om ze o, whe eas in he second case, he e a e mo e non-ze o coe icien s. We ha e con i med in he nume ical simula ions ha ini ial condi ions co esponding o a pe u bed disc e e b ea hing peakon s ays o bi ally close o he exac solu ion. Figu e 11 shows he di e ence be ween he e olu ion o he cen al pa icle o he peakon in a pe u bed and an unpe u bed case. The pe u ba ion used is ξn=εδn,0wi h ε= 0.01. The pe u bed peakon appea s o be o bi ally s able. Con a y o he s a ic case, b ea hing wo-si e peakons a e no des oyed by pe u ba ions. Ins ead, he ene gy densi y oscilla es as shown in Fig. 12. 18 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 g(0) ωp 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 ε ωp FIG. 9: Dependence o g(0) (le ) and Ewi h espec o he peakon equency. 0 2 4 6 8 10 −4 −2 0 2 4 Displacemen Pe iods FIG. 10: Time e olu ion o he b ea hing peakon ( he displacemen o each o he i s ew si es is shown as a unc ion o ime). VI. DISCUSSION In his pape , we ha e enginee ed a ma hema ical model ha has disc e e peakons as exac solu ions. The con inuum e sion o he model was also gi en and i s abili y o suppo he con inuum analog o he solu ions was highligh ed. Bo h he dispe si e and he nonlinea pa o he ele an in e ac ions we e connec ed o ea lie wo ks. Fu he mo e, i was ad oca ed ha o sui able choice o he in e ac ion ange, his can be a ele an model in nonlinea op ics wi h he cha ac e is ic ea u es o pho o e ac i e ma e ials [37], while being a la ice dynamical model o in e es in i s own igh . We ha e iden i ied he disc e e peakon solu ions analogous o hei (discon inuous in he i s de i a i e) con inuum limi , i.e., πn∼exp(−|n|), as well as hei in e -si e siblings. Howe e , a na u al ques ion o in e es would be whe he he e is a mo e gene al way o de ining such solu ions in he disc e e se ing. This is pa icula ly ele an as solu ions simila o he ones ob ained he e ha e appea ed in o he con ex s. Such examples consis o , e.g., he wa e o m o Fig. 1 in [40] (a ising om he p esence o an impu i y) o ha o Fig. 7 in [27] (a ising because o he in e play 19 −0.4 −0.2 0 0.2 0.4 0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0( ) x0( ) 0.4 0.41 0.42 0.43 0.44 0.45 0.46 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0( ) x0( ) FIG. 11: Le panel: Phase space diag am o he cen al si e o a pe u bed ( ull line) and an unpe u bed (dashed line) b ea hing peakon wi h ωp= 1. Righ panel: Blow-up o he le panel. The c oss indica es he ini ial poin o he simula ion. 0 50 100 150 200 250 300 350 0 0.05 0.1 0.15 0.2 en( ) n −20 −15 −10 −5 0 5 10 15 20 0 50 100 150 200 250 300 350 400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 FIG. 12: Time e olu ion o he wo-si e b ea hing peakon chain (wi h a= 1 and ωp= 1) pe u bed wi h he an isymme ic mode. Le panel: The ull line ep esen s e0, he dashed line ep esen s e1, whe e enis he ene gy densi y o he n- h si e. Righ panel: Ene gy densi y in he space- ime e olu ion. o nonlinea i y wi h long ange in e ac ions, as is he case in his pape ). Pe haps, an al e na i e c i e ion possibly in ol ing he sign o he second di e ence close o he cen e o he wa e could be used o a mo e gene al de ini ion o he disc e e peakon. This would be an in e es ing opic o u u e s udies. We ha e also in es iga ed he s abili y o such disc e e wa es and ha e ound disc e e peakons o be pa icula ly in e es ing om his aspec as well. We ha e nume ically obse ed ha in he Klein-Go don la ice se ing such solu ions a e always uns able; howe e he nega i e ene gy di ec ion esponsible o his ins abili y is elimina ed due o an addi ional symme y ( he phase in a iance ha leads o he l2-no m conse a ion) in he case o he DNLS chain. We showed (using a De ick- ype a gumen ) ha he peakon is no a local minimize o he ene gy in he Klein- Go don case and mo eo e indica ed he di ec ion o he pe u ba ion ha leads o a linea ins abili y. We also showed ha in he U(1)-in a ian DNLS equa ion he peakon is a local minimize o he ene gy unde he cha ge cons ain 20 and hence is o bi ally s able. Con a y o hei one-si e coun e pa s, wo-si e peakons p o ed o be uns able, as was explici ly demons a ed ia a well-known unc ional analy ic c i e ion ele an o DNLS ype equa ions. Finally, using a sepa a ion o a iables app oach, we we e able o show he exis ence o he exac pe iodic (b ea hing) peakon solu ions in he Klein-Go don la ice. The b ea hing peakons we e shown o co espond o he subc i ical ini ial condi ions, while supe c i ical ini ial condi ions lead o he ampli ude o he solu ion ending o in ini y (in in ini e ime), wi h he uns able s a ic peakon being he sepa a ix be ween he wo ypes o beha io . We also sys ema ically ackled he ini ial alue p oblem o he Klein-Go don la ice and showed ha he ini e- ime blow-up o he l2-no m o he solu ion is no possible, so ha he sys em is globally well-posed. We hen p o ed he absence o he linea ins abili y o he b ea hing peakons. Ye , he ques ion o he long- ime beha io o pe u bed b ea hing peakons emains open. I may be in e es ing o y o ex end his class o models o highe dimensional se ings and obse e how hei dynamical beha io is a ec ed by he dimensionali y o he unde lying la ice. AC was suppo ed in pa by he Na ional Science Founda ion g an DMS-0200880 and by he Max Planck Ins i u e, Leipzig. JC acknowledges an FPDI g an om ‘La Jun a de Andaluc´ıa’ and pa ial suppo unde he Eu opean Commission RTN p ojec LOCNET, HPRN-CT-1999-00163 and he MECD/FEDER p ojec FIS2004-01183. PGK g a e ully acknowledges suppo om NSF-DMS-0204585, he Eppley Founda ion o Resea ch and om an NSF- CAREER awa d. We a e hank ul o Se gej Flach o b inging Re . [18] o ou a en ion. We a e also indeb ed o an anonymous e e ee o b inging o ou a en ion a ecen pape [14], as well as se e al ea lie ele an wo ks. [1] S. Aub y, Physica 103D, 201 (1997); S. Flach and C.R. 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