Domain wall dynamics in expanding spaces

Domain wall dynamics in expanding spaces F ancisco J. Cao,1,2,*Elías Zamo a-Sille o,3,† and Niu ka R. Quin e o3,‡ 1Depa amen o de Física A ómica, Molecula y Nuclea , Uni e sidad Complu ense de Mad id, A enida Complu ense s/n, E-28040 Mad id, Spain 2LERMA, Obse a oi e de Pa is, Labo a oi e Associé au CNRS UMR 8112, 61 A enue de l’Obse a oi e, 75014 Pa is, F ance 3Depa amen o de Física Aplicada I, E.U.P., Uni e sidad de Se illa, Vi gen de Á ica 7, 41011 Se illa, Spain 共Recei ed 1 Decembe 2005; published 12 May 2006兲 We s udy he e ec s on he dynamics o kinks due o expansions and con ac ions o he space. We show ha he p opaga ion eloci y o he kink can be adiaba ically uned h ough slow expansions and/o con ac ions, while i s wid h is gi en as a unc ion o he eloci y. We also analyze he case o as expansions and/o con ac ions, whe e we a e no longe on he adiaba ic egime. In his case he kink mo es mo e slowly a e an expansion-con ac ion cycle as a consequence o he loss o ene gy h ough adia ion. All hese e ec s a e nume ically s udied in he nonlinea Klein-Go don equa ions 共bo h o he sine-Go don and o he ␾ 4po en- ial兲, and hey a e also s udied wi hin he amewo k o he collec i e coo dina e e olu ion equa ions o he wid h and he cen e o mass o he kink. These collec i e coo dina e e olu ion equa ions a e ob ained wi h a p ocedu e ha allows us o conside e en he case o la ge expansions and/o con ac ions. DOI: 10.1103/PhysRe E.73.056603 PACS numbe 共s兲: 05.45.Y , 45.20.Jj, 02.30.J I. INTRODUCTION Domain walls a e he in e ace be ween wo egions whe e he sys em is close o wo di e en s able o me a- s able s a es. Thei o ma ion and dynamics a e ele an o many domains o physics as solid s a e physics 共magne ic ma e ials兲关1,2兴, liquid c ys als science 关3兴, plasma physics 关4,5兴, and cosmology 共phase ansi ions in he ea ly uni e se兲 关6,7兴. Spa ially ex ended sys ems wi h mo e han one s able o me as able s a e can p esen domain walls, as is he case o sys ems ha ha e unde gone a i s o de phase ansi ion. An expansion o con ac ion o hese sys ems can be ealized h ough an ex e nal ac ion on he sys em 共magne ic ma e i- als, liquid c ys als, plasmas兲, o hey a e al eady p esen in he sys em dynamics 共phase ansi ions in he ea ly uni e se兲. The kink solu ions o he one-dimensional nonlinea Klein-Go don equa ion o po en ials wi h wo o mo e s able s a es 共as sine-Go don o ␸ 4兲p o ide models o he p opaga ion o plane domain walls in he di ec ion pe pen- dicula o he domain wall. These models ha e been success- ully used in se e al con ex s as in he desc ip ion o he dynamics o domain walls in e odis o i e ma e ials 关2兴, and he magne ohyd odynamic mode ajec o y in e e sed- ield pinch expe imen s 关4,5兴. On he o he hand, expansions and con ac ions can be ealized in magne ic ma e ials h ough he mechanical ac ion on he ma e ial 关8兴, and in plasmas h ough con inemen 关9兴; while in he case o phase ansi ions in he ea ly uni e se, he expansion is al eady p esen in he sys em dynamics. These expansions and/o con ac ions can be pa ame ized wi h a scale ac o a共 兲 when hey a e homogeneous in space 关10兴. The cha ac e is ic imes o he ime a ia ion o a共 兲can be sho e o longe han he cha ac e is ic imes o he kink dynamic. Wi h hese ac s in mind, in his pape we conside kinks o he nonlinea Klein-Go don equa ions o he sine-Go don and he ␾ 4po en ials 共as a pa adigma ic examples o in e- g able and nonin eg able nonlinea Klein-Go don equa ions, espec i ely兲, and we s udy hei dynamics in slow and as expanding and/o con ac ing spaces. In o de o achie e his pu pose, ou pape is o ganized as ollows: In Sec. II we in oduce he nonlinea Klein-Go don equa ions in expanding and/o con ac ing spaces, and de i e he con inui y equa ions o he ene gy and momen um den- si ies. In he nex sec ion, Sec. III, we use he Rice Ansa z 关11兴in hese con inui y equa ions, and we educe he p ob- lem wi h an in ini e numbe o deg ees o eedom o an app oxima e desc ip ion in e ms o wo collec i e coo di- na es 共CC兲, he cen e and he wid h o he kink. In Sec. IV, we s udy he kink dynamic in bo h slow and as expanding and/o con ac ing spaces using he collec i e coo dina e e olu ion equa ions, and in eg a ing nume ically he com- ple e nonlinea Klein-Go don equa ions. We i s conside he case o he slow expansion and/o con ac ion wi hin he adiaba ic app oxima ion, and la e we discuss he as expan- sion and/o con ac ion case and he depa u es om he adiaba ici y ha eme ge. Finally, in he conclusions, we sum- ma ize and discuss he esul s. The p e ious sec ions a e complemen ed wi h wo Appen- dixes whe e some echnical de ails ela ed o expanding and/o con ac ing spaces a e e iewed, and he equi alence be ween se e al collec i e coo dina es app oaches 关12–14兴 o his p oblem is shown. II. KLEIN-GORDON EQUATION IN EXPANDING SPACES The Klein-Go don equa ion in nonexpanding spaces can be ob ained by ex emizing he ac ion de ined by he La- g angian densi y, S= 冕 d dx L= 冕 d dx 冋 1 2 ␾ 2−1 2 ␾ x 2−U共 ␾ 兲 册 .共1兲 *Elec onic add ess: [email p o ec ed] †Elec onic add ess: elias@eule .us.es ‡Elec onic add ess: niu ka@eule .us.es PHYSICAL REVIEW E 73, 056603 共2006兲 1539-3755/2006/73共5兲/056603共8兲©2006 The Ame ican Physical Socie y056603-1 In an expanding 共o con ac ing兲space wi h one spa ial dimension he space- ime dis ance be ween e en s is gi en by ds2=d 2−a2共 兲dx2,共2兲 whe e a共 兲is he scale ac o ha gi es he dila ion o he physical spa ial dis ances 共see Appendix A兲. Thus, he physi- cal spa ial dis ance be ween wo poin s wi h coo dina es x1 and x2is d12共 兲=a共 兲兩x2−x1兩.共3兲 The Klein-Go don equa ion in his expanding space can be ob ained no ing ha a e he spa ial dila ion dx→a共 兲dx he ac ion o a Klein-Go don ield is S= 冕 d dxa共 兲L= 冕 d dxa共 兲 冋 1 2 ␾ 2−1 2 ␾ x 2 a2共 兲−U共 ␾ 兲 册 , 共4兲 whe e he subsc ip s and xindica e he ime and space pa ial de i a i es, espec i ely, and U共 ␾ 兲is he nonlinea Klein-Go don po en ial. The Eule -Lag ange equa ion o he ield ␾ , ⳵ ⳵ ␦ 关a共 兲L兴 ␦ ␾ + ⳵ ⳵ x ␦ 关a共 兲L兴 ␦ ␾ x − ␦ 关a共 兲L兴 ␦ ␾ =0, 共5兲 gi es he equa ion o mo ion ␾ +H共 兲 ␾ − ␾ xx a2共 兲+dU d ␾ =0, 共6兲 wi h H=a /a. We ha e o keep in mind ha he physical dis ance is a共 兲兩x2−x1兩and no 兩x2−x1兩共 he la e is usually called he como ing dis ance兲. Thus, when a共 兲g ows he space expe- iences an elonga ion and when a共 兲dec eases he space ex- pe iences a con ac ion. We de ine he ene gy densi y ␳ E共 ,x兲, he momen um den- si y ␳ P共 ,x兲, and he momen um cu en jP共 ,x兲, as he ime- ime, space- ime, and space-space con a a ian componen s, espec i ely, o he ene gy-momen um enso . We ob ain ␳ E=1 2 ␾ 2+1 2 ␾ x 2 a2+U共 ␾ 兲,共7兲 ␳ P=− 1 a2 ␾ ␾ x,共8兲 jP=1 a2 冉 1 2 ␾ 2+1 2 ␾ x 2 a2−U共 ␾ 兲 冊 .共9兲 The ene gy-momen um conse a ion 共in he co a ian sense兲 gi es he ollowing ela ions: ⳵␳ E ⳵ +H共 ␳ E+a2jP兲+ ⳵␳ P ⳵ x=0, 共10兲 ⳵␳ P ⳵ +3H ␳ P+ ⳵ jP ⳵ x=0, 共11兲 ha also can be ob ained using he equa ion o mo ion 共6兲. This also implies ela ions be ween he o al ene gy E =兰adx ␳ E, he o al momen um P=兰adx ␳ P, and he o al momen um cu en JP=兰adxj P. They a e gi en by dE d +Ha2JP=0 共12兲 关p o ided ␳ P共 ,⬁兲= ␳ P共 ,−⬁兲兴, and dP d +2HP =0 共13兲 关p o ided jP共 ,⬁兲=jP共 ,−⬁兲兴. No e ha o a space ha is nei he expanding no con ac ing 关i.e., a共 兲=1兴Eand Pa e conse ed quan i ies 关1兴. III. COLLECTIVE COORDINATES ANALYSIS FOR THE PERTURBED KLEIN-GORDON EQUATION In he p e ious sec ion we ha e shown ha he momen- um and he ene gy a e no longe conse ed quan i ies in expanding and/o con ac ing spaces, ins ead hey sa is y he con inui y equa ions 共12兲and 共13兲, espec i ely. We a e in- e es ed in he e olu ion o he kink which we desc ibe h ough wo collec i e coo dina es, i s wid h and i s cen e o mass posi ion. The e o e, we neglec he possible ene gy ans e o o he deg ees o eedom. In o de o ob ain he equa ions o mo ion o hese wo CCs we use he Rice Ansa z 关11兴in he con inui y equa ions 共12兲and 共13兲关wi h he ene gy and momen um gi en by Eqs. 共7兲–共9兲兴, and we s udy wo pa icula nonlinea po en ials, he sine-Go don 共SG兲po en ial U共 ␾ 兲=1−cos共 ␾ 兲, and he ␾ 4 po en ial U共 ␾ 兲=1 4共1− ␾ 2兲2. The domain walls in hese sys- ems a e ep esen ed by kinklike solu ions ha desc ibe he ansi ion be ween wo egions in di e en minima o he po en ial. Fo he sine-Go don po en ial he Rice Ansa z as- sumes he unc ion ␾ 共x, 兲= 4 a c an 再 exp 冋 x−X共 兲 l共 兲 册 冎 ,共14兲 whe eas o he ␾ 4, he Rice Ansa z eads ␾ 共x, 兲= anh 冋 x−X共 兲 l共 兲 册 ,共15兲 whe e X共 兲and l共 兲 ep esen in bo h cases he cen e and he wid h o he kink, espec i ely, in he co-mo ing ame. No e ha o he unpe u bed sys em 关 ake, in Eq. 共6兲,a共 兲=1兴 he kinklike solu ions a e ep esen ed by Eqs. 共14兲and 共15兲wi h X共 兲= and l共 兲=ls=l0冑1− 2, being and ls he cons an eloci y and wid h o he kink, espec i ely, and l0=1 共l0 =冑2兲 he wid h o he soli on a es o SG 共 ␸ 4兲. The ime dependence o he scale ac o a共 兲共expansions and/o con- ac ions兲implies changes in he wid h o he kink, as we show la e , jus i ying he choice o he Rice Ansa z. Inse ing he Ansa z, Eq. 共14兲o 共15兲, in he exp essions CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲 056603-2 o ␳ P, ␳ E, and jPand in eg a ing o e x, we ob ain P共 兲=M0l0X ˙ a共 兲l,共16兲 E共 兲=M0l0 2 aX ˙2 l+ ␣ M0l0 2 al ˙2 l+M0 2 冉 l0 al +al l0 冊 ,共17兲 JP共 兲=M0l0 2 X ˙2 al + ␣ M0l0 2 l ˙2 al +M0 2a2 冉 l0 al −al l0 冊 ,共18兲 espec i ely; whe e ␣ = ␲ 2/12 and M0=8 o he sine- Go don and ␣ =共 ␲ 2−6兲/12 and M0=2冑2/3 o ␸ 4. These exp essions a e eplaced in he con inui y equa ions 共12兲and 共13兲, gi ing he ollowing sys ems o o dina y di e en ial equa ions 共ODE兲 o X共 兲and l共 兲: X ˙=a共 兲P共 兲l共 兲 M0l0 ,共19兲 dP d =−2HP,共20兲 ␣ 关l ˙2−2Hll ˙−2ll ¨兴=l2 l0 2 冉 1+a2共 兲P2 M0 2 冊 −1 a2,共21兲 whe e he do s deno e he de i a i e wi h espec o .I he ini ial condi ions a e hose o a kink wi h an ini ial posi- ion X共0兲and an ini ial eloci y X ˙共0兲, his implies l共0兲=l0冑1−X ˙2共0兲and l ˙共0兲=0, and P共0兲is de e mined by Eq. 共19兲a =0. The same collec i e coo dina es e olu- ion equa ions a e ob ained using he Lag angian me hod 关14,15兴o he gene alized a elling wa e Ansa z 共GTWA兲 关12,16兴共based on p ojec ion echniques兲, as i is shown in Appendix B. This se o e olu ion equa ions, Eqs. 共19兲–共21兲, in ol es he a iables X共 兲and l共 兲 h ough he momen um P共 兲de- ined in Sec. II. No e ha he equa ion o he momen um is linea , and he e o e i can be sol ed exac ly yielding P共 兲=P共0兲 a2共 兲,共22兲 wi h P共0兲=X ˙共0兲M0l0/冑1−X ˙2共0兲. The o he wo equa ions a e nonlinea and coupled, and hey a e analyzed in de ail in he nex sec ion. IV. KINK DYNAMICS IN EXPANDING AND/OR CONTRACTING SPACES Fi s o all, le us ema k ha he main physical a iables associa ed wi h he kink p opaga ion a e he physical cen e o he kink, Xphys共 兲=a共 兲X共 兲, i s physical wid h lphys共 兲 =a共 兲l共 兲, and i s physical eloci y Vphys共 兲=HXphys共 兲+a共 兲dX共 兲 d .共23兲 The i s e m on he hs is he con ibu ion o he kink e- loci y due o he expansion o con ac ion o he space 共i.e., i u ns o ze o when he space inishes i s elonga ion o con- ac ion H=0兲. On he o he hand, he second e m is he peculia eloci y o he kink wi h espec o he p opaga ing space, Vpec共 兲=a共 兲dX共 兲 d .共24兲 Fo he sake o cla i y, we analyze sepa a ely he cases o slow and as expansion and/o con ac ion. A. Slow expansion and/o con ac ion (adiaba ic egime) No e om Eqs. 共19兲–共21兲 ha he adiaba ic app oxima ion implies ␣ 关l ˙2−2Hll ˙−2ll ¨兴Ⰶ1 a2.共25兲 Hence, om Eq. 共21兲we ob ain he ela ion lphys共 兲=l0冑1−Vpec 2共 兲,共26兲 ha links he physical wid h and he peculia eloci y o he kink. Fu he mo e, om Eqs. 共19兲and 共22兲–共26兲we ob ain Vpec共 兲=P共0兲 冑M0 2a2共 兲+P2共0兲.共27兲 F om he p e ious Eqs. 共26兲and 共27兲we see ha an expan- sion decele a es he peculia mo ion o he kink and makes i wide , asymp o ically lim a共 兲→⬁ Vpec共 兲= 0, lim a共 兲→⬁ lphys共 兲=l0.共28兲 On he o he hand, a con ac ion accele a es he peculia mo- ion o he kink and makes i sha pe , asymp o ically lim a共 兲→0 Vpec共 兲= 1, lim a共 兲→0 lphys共 兲=0. 共29兲 The adiaba ici y condi ions can be ob ained om Eq. 共25兲 and equi e only slow expansions and/o con ac ions 兩H兩Ⰶ⍀R 1−Vpec 2,共30兲 兩H ˙兩Ⰶ⍀R 2 共1−Vpec 2兲2,共31兲 whe e ⍀R=1/共冑 ␣ l0兲is he so-called Rice equency o ze o eloci y, ⍀R=冑12/ ␲ =1.10..., o SG and ⍀R=冑6/共 ␲ 2−6兲 =1.24..., o ␸ 4. We ha e s udied he dynamics o di e en expansion and con ac ion a es, bo h o he sine-Go don and he ␾ 4po en- ials. In pa icula , Fig. 1 共and also Fig. 2兲shows he esul s DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲 056603-3 o an expansion ollowed by a con ac ion pa ame ized by he unc ion a共 兲=1+⌬a 2 anh 冉 − 0 ⌬ 冊 −⌬a 2 anh 冉 − 1 ⌬ 冊 ,共32兲 wi h ⌬a, 0, 1, and ⌬ cons an s. 共⌬a ep esen s he changes in a; 0and 1, he imes when he expansion and con ac ion ake place, espec i ely, and ⌬ is he cha ac e is ic ime in- e al whe e he changes in a ake place.兲 In Fig. 1 we show he e ec s on a kink o a slow expan- sion ollowed by a slow con ac ion ha e i y he adiaba ic condi ions 关H⬃⌬a/⌬ =0.1 and H ˙⬃⌬a/共⌬ 兲2=0.01兴. In his case, when he adiaba ic condi ions a e e i ied, bo h he CCs equa ions and he adiaba ic app oxima ions a e in good ag eemen wi h he exac esul s, especially o he cen e o he kink Xphys共 兲and i s peculia eloci y Vpec共 兲共see Fig. 1兲. I is impo an o no e ha he ag eemen is good e en when he a ia ions o aa e la ge. This implies ha he main e ec s o a slow expansion and/o con ac ion a e he change o he wid h and he speed o he p opaga ion o he kink, ollowing he adiaba ic ela- ions 共26兲and 共27兲. The e o e, hese esul s show ha he speed o a kink can be uned by slowly expanding and/o con ac ing he space. No e ha he CCs e olu ion equa ions go u he and a e able o p edic he small oscilla ions o he kink wid h 关23兴 p oduced by he slow expansion and/o con ac ion 共see Fig. 1兲. The e o e, hey can compu e de ia ions om adiaba ici y excluding he adia i e e ec s 共because hey in ol e he ans e o ene gy o o he deg ees o eedom兲. I is impo an o s ess ha he collec i e coo dina e e o- lu ion equa ions ob ained wi h an Ansa z wi h ixed wid h and a a iable cen e o he kink does no p edic any a ia- ion o he kink speed due o expansions and/o con ac ions. The e o e, an Ansa z ha allows he wid h o he kink o e ol e, like he Rice Ansa z, is an essen ial ing edien in o de o ob ain he co ec a ia ion o he speed o he kink unde expansions and/o con ac ions. I is in e es ing o e- ma k ha , in he ␾ 4model, he oscilla ions o he wid h a e ela ed o exci a ions o he in e nal mode 关11兴, while in he sine-Go don equa ion i has been shown ha he exci a ion o ce ain phonons can imply oscilla ions in he shape o he kink 关17,18兴. FIG. 1. Sine-Go don kink dy- namics in a slow expanding and la e con ac ing space. Top-le panel: scale ac o a共 兲, Eq. 共32兲 wi h ⌬a=1, 0=50, 1=150, and ⌬ =10. Top- igh panel: spa ial p o ile o he kink ␾ 共x, ix兲a imes ix=0 共solid line兲, ix=100 共long-dashed line兲, and ix=200 共dashed line兲. Th ee ollowing panels: posi ion o he cen e o he kink Xphys共 兲, peculia kink e- loci y Vpec共 兲, and wid h o he kink lphys共 兲as unc ions o ime gi en by he nume ical simula- ions o he pa ial di e en ial equa ion 关PDE兴共solid line兲, he nume ical solu ions o he CCs equa ion 共long-dashed line兲, and he adiaba ic app oxima ion 共dashed line兲. Bo om- igh panel: momen um o he kink P共 兲exac solu ion. Ini ial condi ions: a kink wi h X共0兲=10 and X ˙共0兲=0.1. CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲 056603-4 On he o he hand, in gene al, he applicabili y o he pe u ba i e app oaches is es ic ed o slow and small ex- pansions and/o con ac ions. The pe u ba i e app oaches ew i e Eq. 共6兲in he o m ␾ − ␾ xx+dU/d ␾ = ⑀ ⬅−H共 兲 ␾ −关1/a2共 兲−1兴 ␾ xx, whe e ⑀ is ea ed as a small pe u ba ion. The e o e, hei ange o applicabili y is limi ed no only o small Hbu also o small 共1/a2−1兲, i.e., slow and small expansions and/o con ac ions. B. Fas expansion and/o con ac ion (nonadiaba ic egime) When he expansion and/o con ac ion is as e he adia- ba ic app oxima ion b eaks, and nonadiaba ic e ec s appea as adia ion 共compa e he kink p o iles in Figs. 1 and 2兲 and as a change in he inal wid h and eloci y o he kink 关 ha a e no longe hose p edic ed by Eqs. 共26兲and 共27兲兴. 共See Fig. 2, ha co esponds o he nonadiaba ic egime H⬃⌬a/⌬ =2 and H ˙⬃⌬a/共⌬ 兲2=4.兲The CCs e olu ion equa ions p edic s ong oscilla ions in he eloci y and he wid h o he kink. In he nume ical in eg a ion o he com- ple e e olu ion equa ions 关Eq. 共6兲兴 he oscilla ions a e indeed p esen in he wid h. Howe e , he e a e o he deg ees o eedom o which he ene gy can be ans e ed. This esul s in he damping o he oscilla ions h ough adia ion emis- sion. The loss o ene gy can also be easily shown no ing ha a e an expansion-con ac ion cycle he kink mo es slowe 共see Fig. 2兲. We would like o s ess ha all he p e ious commen s abou slow and as expansions and/o con ac ions apply bo h o he sine-Go don and he ␾ 4po en ial. We ha e pe - o med nume ical simula ions and compa isons o he adia- ba ic app oxima ion, he CCs e olu ion equa ions, and he ull e olu ion equa ion o bo h po en ials ob aining analo- gous esul s. V. CONCLUSIONS We ha e s udied he e ec s on a kink o slow and as expansions and/o con ac ions o he media, bo h o he sine-Go don and o he ␾ 4equa ions. We ha e used he Rice Ansa z 关11兴in he con inui y equa ions o he momen um and he ene gy, in o de o ob ain he e olu ion equa ions o he wid h and he cen e o a kink in an expanding and/o con ac ing space. In gene al, his se o ODEs canno be FIG. 2. Sine-Go don kink dy- namics in a as expanding and la e con ac ing space. Top-le panel: scale ac o a共 兲, Eq. 共32兲 wi h ⌬a=1, 0=40, 1=100 and ⌬ =0.5. Top- igh panel: spa ial p o ile o he kink ␾ 共x, ix兲a imes ix=37.5 共solid line兲, ix =75 共long-dashed line兲, and ix =150 共dashed line兲. The h ee ol- lowing panels: posi ion o he cen- e o he kink Xphys共 兲, peculia kink eloci y Vpec共 兲, and wid h o he kink lphys共 兲as unc ions o ime gi en by he nume ical simu- la ions o he PDE 共solid line兲, he nume ical solu ions o he CCs equa ion 共long-dashed line兲, and he adiaba ic app oxima ion 共dashed line兲. Bo om- igh panel: momen um o he kink P共 兲exac solu ion. Ini ial condi ions: a kink wi h X共0兲=10 and X ˙共0兲=0.1. DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲 056603-5 sol ed analy ically and he solu ion mus be ob ained nu- me ically. Howe e , his e ec i e desc ip ion in e ms o he collec i e coo dina es usually gi es ele an in o ma ion and insigh o he e olu ion and he phenomena in ol ed 关16,19–22兴. In his case he CCs app oxima ion is able o desc ibe e y accu a ely he dynamics o he kink when he p ocess is slow, e en when he a ia ions o aa e la ge. We ha e ound ha he main e ec s o a slow expansion and/o con ac ion a e he change o he wid h and he eloci y o he kink, ollowing he adiaba ic ela ions 共26兲and 共27兲.In addi ion, he collec i e coo dina e e olu ion equa ions o he wid h and he cen e o he kink a e able o ake in o accoun he small de ia ions om adiaba ici y 共excluding ex- plici ly he adia i e e ec s, because hey a e ela ed o ans- e ence o ene gy o o he deg ees o eedom兲. An impo an consequence o he adiaba ic ela ions is ha he speed o a kink can be uned by a slowly expansion and/o con ac ion o space. Thus, his esul p o ides a mechanism o con ol he speed o a kink whene e we can expand and con ac he media whe e i p opaga es. On he o he hand, as expansions and/o con ac ions b eak he adiaba ic app oxima ion, gi ing ise o nonadia- ba ic e ec s, as, o example, adia ion, ha imply a change in he inal wid h and eloci y o he kink 共i.e., hey a e no longe hose p edic ed by he adiaba ic ela ions兲. The collec- i e coo dina e e olu ion equa ions p edic s ong oscilla- ions in he wid h o he kink. In he ull p oblem, hese oscilla ions a e also p esen . Howe e , he e a e o he de- g ees o eedom o which he ene gy o hese oscilla ions can be ans e ed. This esul s in emission o adia ion and in he damping o hese oscilla ions. In his sense, he as expansion and/o con ac ion o space p o ides a me hod o implemen a as change in he speed o he kink, and also o gene a e phonons. ACKNOWLEDGMENTS We acknowledge inancial suppo om he Minis e io de Ciencia y Tecnología o Spain unde G an s No. BFM2003- 02547/FISI 共F.J.C.兲, No. NAN2004-09087-C0303 共F.J.C.兲, and No. FIS2005-973 共E.Z.S., N.R.Q.兲, and om he Jun a de Andalucía h ough he p ojec s FQM-0207 and 00481 共E.Z.S., N.R.Q.兲. In addi ion, E.Z.S. hanks he Uni e si y o Se illa o i s inancial suppo . APPENDIX A: EXPANDING SPACES This appendix includes some echnical de ails ela ed o expanding spaces. 1. Me ic The me ic in an homogeneous and iso opous la space is gi en by ds2=d 2−a2共 兲dx2=g ␮ ␯ dx ␮ dx ␯ ,共A1兲 whe e a共 兲is he scale ac o o he expansion and/o con- ac ion, he me ic enso is g ␮ ␯ = 冉 10 0−a2共 兲 冊 .共A2兲 I s in e se, g ␮ ␯ , de ined by g ␮ ␯ g ␯ ␭=g ␮ ␭= ␦ ␮ ␭共wi h ␦ ␮ ␭a K o- necke del a兲is g ␮ ␯ = 冢 10 0−1 a2共 兲 冣 .共A3兲 The con ac ion wi h g ␮ ␯ lowe s indices, and he con ac ion wi h g ␮ ␯ aises indices. 2. In a ian elemen o space- ime olume De ining g⬅− De g ␮ ␯ ,共A4兲 i can be shown 共关10兴pages 98 and 99兲 ha 冑gd dx 共A5兲 is an in a ian olume elemen unde gene al coo dina e ans o ma ions. 共d dx ans o ms wi h he Jacobian, while 冑g ans o ms wi h he in e se o he Jacobian.兲 The e o e he app op ia e ela ion be ween he ac ion S, and he Lag angian densi y Lis S= 冕 d dx冑gL.共A6兲 This de ini ion implies ha i he ac ion is in a ian unde gene al coo dina e ans o ma ions he Lag angian densi y is also in a ian . In ou case 冑g=a共 兲. 3. Ene gy-momen um enso The ene gy-momen um enso is T ␮ ␯ = ⳵ ␮ ␾ ⳵ ␯ ␾ −g ␮ ␯ L.共A7兲 Some use ul ela ions o calcula e i s componen s a e ⳵ x ␾ =gx ␴ ⳵ ␴ ␾ =gxx ⳵ x ␾ =− ␾ x a2,共A8兲 ⳵ ␾ = ⳵ ␾ = ␾ .共A9兲 The ime- ime componen o he ene gy-momen um enso o ene gy densi y is ␳ E⬅T = ⳵ ␾ ⳵ ␾ −g 冋 1 2 ␾ 2−1 2 ␾ x 2 a2−U共 ␾ 兲 册 =1 2 ␾ 2+1 2 ␾ x 2 a2+U共 ␾ 兲,共A10兲 he space- ime componen o momen um densi y is ␳ P⬅Tx =T x = ⳵ ␾ ⳵ x ␾ −g xL=− 1 a2 ␾ ␾ x,共A11兲 and he space-space componen o momen um cu en is CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲 056603-6 jP⬅Txx =1 a2 冉 1 2 ␾ 2+1 2 ␾ x 2 a2−U共 ␾ 兲 冊 .共A12兲 APPENDIX B: EQUIVALENCE AMONG DIFFERENT CC APPROACHES In his appendix we show ha he e olu ion equa ions o he collec i e coo dina es ob ained wi h he de i a ion o he momen um and he ene gy 关13兴共 ha we used in Sec. III兲a e he same as hose ob ained wi h he Lag angian me hod 关14,15兴and wi h he so-called GTWA 关12,16兴共based on p o- jec ion echniques兲. In all cases, we use he Rice Ansa z as an app oxima ed kinklike solu ion o he ollowing pe u bed nonlinea Klein-Go don equa ion: ␾ − ␾ xx a2共 兲=−dU d ␾ −关H共 兲+ ␤ 共 兲兴 ␾ +z共x, , ␾ 兲,共B1兲 whe e ␤ 共 兲is a ime-dependen damping coe icien and z共x, , ␾ 兲 ep esen s a gene ic pe u ba ion on he sys em. No e ha o ␤ =0 and z共x, , ␾ 兲=0 we eco e he sys em 共6兲, in oduced in Sec. II. 1. Gene alized Lag angian o malism In oducing a new ime a iable ␶ ⬅ 冕 0 d ⬘ c共 ⬘兲,共B2兲 whe e c共 兲is gi en by c共 兲⬅exp 再 冕 0 关H共 ⬘兲+ ␤ 共 ⬘兲兴d ⬘ 冎 =a共 兲 a共0兲exp 再 冕 0 ␤ 共 ⬘兲d ⬘ 冎 , 共B3兲 Eq. 共B1兲becomes he dissipa ionless equa ion, ␾ ␶␶ C2共 ␶ 兲− ␾ xx A2共 ␶ 兲=−dU d ␾ +Z共x, ␶ , ␾ 兲,共B4兲 wi h C共 ␶ 兲=c关 共 ␶ 兲兴,A共 ␶ 兲=a关 共 ␶ 兲兴, and Z共x, ␶ , ␾ 兲=z关x, 共 ␶ 兲, ␾ 兴. The e olu ion equa ion 共B4兲can be ob ained om he ac ion S= 冕 d ␶ dx L= 冕 d ␶ dx 再 1 2 ␾ ␶ 2−1 2 C2共 ␶ 兲 A2共 ␶ 兲 ␾ x 2 −C2共 ␶ 兲关U共 ␾ 兲+M共x, ␶ , ␾ , ␾ x兲兴 冎 共B5兲 Lbeing he Lag angian densi y, and M共x, ␶ , ␾ , ␾ x兲an “Eule -Lag ange in eg al” o Z共x, ␶ , ␾ 兲. Some o he possible in eg a ions a e 共i兲 he ␾ unc ional in eg al M共x, ␶ , ␾ 兲=− 冕 D ␾ Z共x, ␶ , ␾ 兲,共B6兲 and 共ii兲 he ollowing in eg a ion o he pa icula case whe e Z共x, ␶ , ␾ 兲=F共x, ␶ 兲G共 ␾ 兲关15兴: M共x, ␶ , ␾ , ␾ x兲= ␾ xG共 ␾ 兲 冕 x0 x dx⬘F共x⬘, ␶ 兲.共B7兲 Once we de ine he Lag angian L= 冕 –⬁ +⬁ dx L,共B8兲 he Eule -Lag ange e olu ion equa ions o he collec i e a iables a e ob ained inse ing a gi en Ansa z in his La- g angian and using ei he exp ession 共B6兲o 共B7兲. This p o- cedu e is he gene aliza ion o he Lag angian o malism de- eloped in 关14兴. Using he Rice Ansa z de ined by 共14兲and 共15兲共 o he SG and ␾ 4po en ials, espec i ely兲and a e some s aigh - o wa d calcula ions, we ob ain he Lag angian as a unc ion o ou collec i es a iables and hei de i a i es wi h espec o ␶ L共X,X⬘,l,l⬘兲=M0l0 2l共X⬘兲2+ ␣ M0l0 2l共l⬘兲2 −1 2M0C2共 ␶ 兲 冉 l l0 +l0 A2共 ␶ 兲l 冊 −C2共 ␶ 兲 冕 −⬁ +⬁ dx M共x, ␶ , ␾ , ␾ x兲.共B9兲 Replacing L共X,X⬘,l,l⬘兲in o he Eule -Lag ange equa ions, d d ␶ 冉 ⳵ L ⳵ X⬘ 冊 = ⳵ L ⳵ X,共B10兲 d d ␶ 冉 ⳵ L ⳵ l⬘ 冊 = ⳵ L ⳵ l,共B11兲 and ew i ing he equa ions o mo ion in he ime a iable , we ge X ˙=a共 兲P共 兲l共 兲 M0l0 ,共B12兲 dP d =−共 ␤ +2H兲P−1 a共 兲 再 冕 −⬁ +⬁ d ␪ z共X+ ␪ l, , ␾ 兲 ␾ ␪ 冎 , 共B13兲 ␣ 关l ˙2−2共 ␤ +H兲ll ˙−2ll ¨兴=l2 l0 2 冉 1+a2共 兲P2 M0 2 冊 −1 a2 +2l2 M0l0 冕 −⬁ +⬁ d ␪ z共X+ ␪ l, , ␾ 兲 ␪␾ ␪ , 共B14兲 whe e ␪ =关x−X共 兲兴/land he do s deno e he de i a i e wi h espec o . These a e he e olu ion equa ions o he collec- DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲 056603-7 i e a iables ob ained o ei he o he wo p e ious exp es- sions o M共x, , ␾ , ␾ x兲Eq. 共B6兲o Eq. 共B7兲共in hei espec- i e egime o alidi y兲. Taking in hese equa ions ␤ =0 and z=0, we ob ain he Eqs. 共19兲–共21兲de i ed in Sec. III. 2. GTWA In o de o apply he GTWA we ew i e Eq. 共B1兲as ␾ ˙= ␺ , ␺ ˙= ␾ xx a2共 兲−dU d ␾ −关H共 兲+ ␤ 共 兲兴 ␺ +z共x, , ␾ 兲.共B15兲 The p ocedu e o ob ain he CCs equa ions wi h he GTWA consis s o inse ing ou speci ic unc ional o m o ␾ , Eq. 共14兲 o sine-Go don o 共15兲 o ␾ 4po en ial, in o Eq. 共B15兲, mul iplying he i s equa ion by ⳵ ␺ / ⳵ Xand he second one by ⳵ ␾ / ⳵ X, aking hei di e ence and in eg a ing o e x, equa ing he esul wi h ze o; and epea ing he same p oce- du e wi h ⳵ ␺ / ⳵ land ⳵ ␾ / ⳵ l. This gi es us he same o dina y di e en ial equa ions o X共 兲,P共 兲, and l共 兲as hose ob ained in he p e ious subsec ion 关see Eqs. 共B12兲–共B14兲兴. 关1兴A. C. Sco , Nonlinea Science 共Ox o d Uni e si y, Ox o d, 1999兲. 关2兴M. A. Collins, A. Blumen, J. F. Cu ie, and J. Ross, Phys. Re . B19, 3630 共1979兲;19, 3645 共1979兲. 关3兴Kalman B. Migle and Robe B. Meye , Phys. Re . Le . 66, 1485 共1991兲. 关4兴H. K. Eb aheem, J. L. Shohe , and A. C. Sco , Phys. Re . Le . 88, 235003 共2002兲. 关5兴J. L. Shohe , B. R. Ba mish, H. K. Eb aheen, and A. C. Sco , Phys. Plasmas 11, 3877 共2004兲. 关6兴A. Linde, Pa icle Physics and In la iona y Cosmology 共Ha - wood Academic Publishe s, Chu , Swi ze land, 1990兲. 关7兴P. Anninos, S. Oli ei a, and R. A. Ma zne , Phys. Re . D 44, 1147 共1991兲. 关8兴D.-H. Han, J.-G. Zhu, J. H. Judy, and J. M. Si e senb, Appl. Phys. Le . 70, 664 共1997兲; L. Lano e, G. Ausanio, V. Ianno i, G. Pepe, G. Ca o enu o, P. Ne i, and L. Nicolous, Phys. Re . B63, 054438 共2001兲. 关9兴D. J. G o e and D. M. Meade, Nucl. Fusion 25, 1167 共1985兲. 关10兴S. Weinbe g, G a i a ion and Cosmology 共Wiley, New Yo k, 1972兲. 关11兴M. J. Rice and E. J. Mele, Solid S a e Commun. 35, 487 共1980兲. 关12兴G. M. Wysin, F. G. Me ens, A. R. Völkel, and A. R. Bishop, in Nonlinea Cohe en S uc u es in Physics and Biology, ed- i ed by K. H. Spa schek and F. G. Me ens, NATO Ad anced S udies Ins i u e, Se ies B: Physics 共Plenum, New Yo k, 1994兲, Vol. 329; F. G. Me ens, H. J. Schni ze , and A. R. Bishop, Phys. Re . B 56, 2510 共1997兲. 关13兴N. R. Quin e o, A. Sánchez, and F. G. Me ens, Phys. Re . E 62, 5695 共2000兲. 关14兴N. R. Quin e o and E. Zamo a-Sille o, Physica D 197,63 共2004兲. 关15兴Sa a Cuenda and Angel Sánchez, Chaos 15, 023502 共2005兲. 关16兴N. R. Quin e o, A. Sánchez, and F. G. Me ens, Phys. Re . Le . 84, 871 共2000兲. 关17兴Da id K. Campbell, Jona han F. Shon eld, and Cha les A. Winga e, Physica D 9,1共1983兲. 关18兴N. R. Quin e o and Panayo is G. Ke ekidis, Phys. Re . E 64, 056608 共2001兲. 关19兴F. G. Bass, Yu. S. Ki sha , V. V. Kono op, and Yu. A. Sini syn, Phys. Rep. 157,63共1988兲; Yu i S. Ki sha and Bo is A. Mal- omed, Re . Mod. Phys. 61, 763 共1989兲. 关20兴D. W. McLaughlin and A. C. Sco , Phys. Re . A 18, 1652 共1978兲. 关21兴A. Sánchez and A. R. Bishop, SIAM Re . 40, 579 共1998兲. 关22兴G. Cos an ini, F. Ma chesoni, and M. Bo omeo, Phys. Re . E 65, 051103 共2002兲; M. Sale no and Y. Zolo a yuk, ibid. 65, 056603 共2002兲; L. Mo ales-Molina, N. R. Quin e o, F. G. Me ens, and A. Sánchez, Phys. Re . Le . 91, 234102 共2003兲. 关23兴The wid h and he cen e o he kink posi ion o he PDE esul s a e ob ained i ing he Rice Ansa z, Eq. 共14兲o Eq. 共15兲, o he kink p o ile gi en by he PDE e olu ion 共6兲. CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲 056603-8