Domain wall dynamics in expanding spaces

Domain wall dynamics in expanding spaces Francisco J. Cao,1,2,*Elías Zamora-Sillero,3,† and Niurka R. Quintero3,‡ 1Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, Avenida Complutense s/n, E-28040 Madrid, Spain 2LERMA, Observatoire de Paris, Laboratoire Associé au CNRS UMR 8112, 61 Avenue de l’Observatoire, 75014 Paris, France 3Departamento de Física Aplicada I, E.U.P., Universidad de Sevilla, Virgen de África 7, 41011 Sevilla, Spain 共Received 1 December 2005; published 12 May 2006兲 We study the effects on the dynamics of kinks due to expansions and contractions of the space. We show that the propagation velocity of the kink can be adiabatically tuned through slow expansions and/or contractions, while its width is given as a function of the velocity. We also analyze the case of fast expansions and/or contractions, where we are no longer on the adiabatic regime. In this case the kink moves more slowly after an expansion-contraction cycle as a consequence of the loss of energy through radiation. All these effects are numerically studied in the nonlinear Klein-Gordon equations 共both for the sine-Gordon and for the ␾ 4poten- tial兲, and they are also studied within the framework of the collective coordinate evolution equations for the width and the center of mass of the kink. These collective coordinate evolution equations are obtained with a procedure that allows us to consider even the case of large expansions and/or contractions. DOI: 10.1103/PhysRevE.73.056603 PACS number共s兲: 05.45.Yv, 45.20.Jj, 02.30.Jr I. INTRODUCTION Domain walls are the interface between two regions where the system is close to two different stable or metastable states. Their formation and dynamics are relevant for many domains of physics as solid state physics 共magnetic materials兲关1,2兴, liquid crystals science 关3兴, plasma physics 关4,5兴, and cosmology 共phase transitions in the early universe兲 关6,7兴. Spatially extended systems with more than one stable or metastable state can present domain walls, as is the case of systems that have undergone a first order phase transition. An expansion or contraction of these systems can be realized through an external action on the system 共magnetic materials, liquid crystals, plasmas兲, or they are already present in the system dynamics 共phase transitions in the early universe兲. The kink solutions of the one-dimensional nonlinear Klein-Gordon equation for potentials with two or more stable states 共as sine-Gordon or ␸ 4兲provide models for the propagation of plane domain walls in the direction perpendicular to the domain wall. These models have been successfully used in several contexts as in the description of the dynamics of domain walls in ferrodistortive materials 关2兴, and the magnetohydrodynamic mode trajectory in reversedfield pinch experiments 关4,5兴. On the other hand, expansions and contractions can be realized in magnetic materials through the mechanical action on the material 关8兴, and in plasmas through confinement 关9兴; while in the case of phase transitions in the early universe, the expansion is already present in the system dynamics. These expansions and/or contractions can be parametrized with a scale factor a共t兲 when they are homogeneous in space 关10兴. The characteristic times for the time variation of a共t兲can be shorter or longer than the characteristic times of the kink dynamic. With these facts in mind, in this paper we consider kinks of the nonlinear Klein-Gordon equations for the sine-Gordon and the ␾ 4potentials 共as a paradigmatic examples of integrable and nonintegrable nonlinear Klein-Gordon equations, respectively兲, and we study their dynamics in slow and fast expanding and/or contracting spaces. In order to achieve this purpose, our paper is organized as follows: In Sec. II we introduce the nonlinear Klein-Gordon equations in expanding and/or contracting spaces, and derive the continuity equations for the energy and momentum densities. In the next section, Sec. III, we use the Rice Ansatz 关11兴in these continuity equations, and we reduce the problem with an infinite number of degrees of freedom to an approximate description in terms of two collective coordinates 共CC兲, the center and the width of the kink. In Sec. IV, we study the kink dynamic in both slow and fast expanding and/or contracting spaces using the collective coordinate evolution equations, and integrating numerically the complete nonlinear Klein-Gordon equations. We first consider the case of the slow expansion and/or contraction within the adiabatic approximation, and later we discuss the fast expansion and/or contraction case and the departures from the adiabaticity that emerge. Finally, in the conclusions, we summarize and discuss the results. The previous sections are complemented with two Appendixes where some technical details related to expanding and/or contracting spaces are reviewed, and the equivalence between several collective coordinates approaches 关12–14兴to this problem is shown. II. KLEIN-GORDON EQUATION IN EXPANDING SPACES The Klein-Gordon equation in nonexpanding spaces can be obtained by extremizing the action defined by the Lagrangian density, S= 冕 dt dx L= 冕 dt dx 冋 1 2 ␾ t 2−1 2 ␾ x 2−U共 ␾ 兲 册 .共1兲 *Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] PHYSICAL REVIEW E 73, 056603 共2006兲 1539-3755/2006/73共5兲/056603共8兲©2006 The American Physical Society056603-1 In an expanding 共or contracting兲space with one spatial dimension the space-time distance between events is given by ds2=dt2−a2共t兲dx2,共2兲 where a共t兲is the scale factor that gives the dilation of the physical spatial distances 共see Appendix A兲. Thus, the physical spatial distance between two points with coordinates x1 and x2is d12共t兲=a共t兲兩x2−x1兩.共3兲 The Klein-Gordon equation in this expanding space can be obtained noting that after the spatial dilation dx→a共t兲dx the action for a Klein-Gordon field is S= 冕 dt dxa共t兲L= 冕 dt dxa共t兲 冋 1 2 ␾ t 2−1 2 ␾ x 2 a2共t兲−U共 ␾ 兲 册 , 共4兲 where the subscripts tand xindicate the time and space partial derivatives, respectively, and U共 ␾ 兲is the nonlinear Klein-Gordon potential. The Euler-Lagrange equation for the field ␾ , ⳵ ⳵ t ␦ 关a共t兲L兴 ␦ ␾ t + ⳵ ⳵ x ␦ 关a共t兲L兴 ␦ ␾ x − ␦ 关a共t兲L兴 ␦ ␾ =0, 共5兲 gives the equation of motion ␾ tt +H共t兲 ␾ t− ␾ xx a2共t兲+dU d ␾ =0, 共6兲 with H=at/a. We have to keep in mind that the physical distance is a共t兲兩x2−x1兩and not 兩x2−x1兩共the latter is usually called the comoving distance兲. Thus, when a共t兲grows the space experiences an elongation and when a共t兲decreases the space experiences a contraction. We define the energy density ␳ E共t,x兲, the momentum density ␳ P共t,x兲, and the momentum current jP共t,x兲, as the timetime, space-time, and space-space contravariant components, respectively, of the energy-momentum tensor. We obtain ␳ E=1 2 ␾ t 2+1 2 ␾ x 2 a2+U共 ␾ 兲,共7兲 ␳ P=− 1 a2 ␾ t ␾ x,共8兲 jP=1 a2 冉 1 2 ␾ t 2+1 2 ␾ x 2 a2−U共 ␾ 兲 冊 .共9兲 The energy-momentum conservation 共in the covariant sense兲 gives the following relations: ⳵␳ E ⳵ t+H共 ␳ E+a2jP兲+ ⳵␳ P ⳵ x=0, 共10兲 ⳵␳ P ⳵ t+3H ␳ P+ ⳵ jP ⳵ x=0, 共11兲 that also can be obtained using the equation of motion 共6兲. This also implies relations between the total energy E =兰adx ␳ E, the total momentum P=兰adx ␳ P, and the total momentum current JP=兰adxj P. They are given by dE dt +Ha2JP=0 共12兲 关provided ␳ P共t,⬁兲= ␳ P共t,−⬁兲兴, and dP dt +2HP =0 共13兲 关provided jP共t,⬁兲=jP共t,−⬁兲兴. Note that for a space that is neither expanding nor contracting 关i.e., a共t兲=1兴Eand Pare conserved quantities 关1兴. III. COLLECTIVE COORDINATES ANALYSIS FOR THE PERTURBED KLEIN-GORDON EQUATION In the previous section we have shown that the momentum and the energy are no longer conserved quantities in expanding and/or contracting spaces, instead they satisfy the continuity equations 共12兲and 共13兲, respectively. We are interested in the evolution of the kink which we describe through two collective coordinates, its width and its center of mass position. Therefore, we neglect the possible energy transfer to other degrees of freedom. In order to obtain the equations of motion for these two CCs we use the Rice Ansatz 关11兴in the continuity equations 共12兲and 共13兲关with the energy and momentum given by Eqs. 共7兲–共9兲兴, and we study two particular nonlinear potentials, the sine-Gordon 共SG兲potential U共 ␾ 兲=1−cos共 ␾ 兲, and the ␾ 4 potential U共 ␾ 兲=1 4共1− ␾ 2兲2. The domain walls in these systems are represented by kinklike solutions that describe the transition between two regions in different minima of the potential. For the sine-Gordon potential the Rice Ansatz assumes the function ␾ 共x,t兲= 4 arctan 再 exp 冋 x−X共t兲 l共t兲 册 冎 ,共14兲 whereas for the ␾ 4, the Rice Ansatz reads ␾ 共x,t兲= tanh 冋 x−X共t兲 l共t兲 册 ,共15兲 where X共t兲and l共t兲represent in both cases the center and the width of the kink, respectively, in the co-moving frame. Note that for the unperturbed system 关take, in Eq. 共6兲,a共t兲=1兴the kinklike solutions are represented by Eqs. 共14兲and 共15兲with X共t兲=vtand l共t兲=ls=l0冑1−v2, being vand lsthe constant velocity and width of the kink, respectively, and l0=1 共l0 =冑2兲the width of the soliton at rest for SG 共 ␸ 4兲. The time dependence of the scale factor a共t兲共expansions and/or contractions兲implies changes in the width of the kink, as we show later, justifying the choice of the Rice Ansatz. Inserting the Ansatz, Eq. 共14兲or 共15兲, in the expressions CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲 056603-2 for ␳ P, ␳ E, and jPand integrating over x, we obtain P共t兲=M0l0X ˙ a共t兲l,共16兲 E共t兲=M0l0 2 aX ˙2 l+ ␣ M0l0 2 al ˙2 l+M0 2 冉 l0 al +al l0 冊 ,共17兲 JP共t兲=M0l0 2 X ˙2 al + ␣ M0l0 2 l ˙2 al +M0 2a2 冉 l0 al −al l0 冊 ,共18兲 respectively; where ␣ = ␲ 2/12 and M0=8 for the sine- Gordon and ␣ =共 ␲ 2−6兲/12 and M0=2冑2/3 for ␸ 4. These expressions are replaced in the continuity equations 共12兲and 共13兲, giving the following systems of ordinary differential equations 共ODE兲for X共t兲and l共t兲: X ˙=a共t兲P共t兲l共t兲 M0l0 ,共19兲 dP dt =−2HP,共20兲 ␣ 关l ˙2−2Hll ˙−2ll ¨兴=l2 l0 2 冉 1+a2共t兲P2 M0 2 冊 −1 a2,共21兲 where the dots denote the derivative with respect to t.If the initial conditions are those of a kink with an initial position X共0兲and an initial velocity X ˙共0兲, this 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 determined by Eq. 共19兲at t=0. The same collective coordinates evolution equations are obtained using the Lagrangian method 关14,15兴or the generalized travelling wave Ansatz 共GTWA兲 关12,16兴共based on projection techniques兲, as it is shown in Appendix B. This set of evolution equations, Eqs. 共19兲–共21兲, involves the variables X共t兲and l共t兲through the momentum P共t兲defined in Sec. II. Note that the equation for the momentum is linear, and therefore it can be solved exactly yielding P共t兲=P共0兲 a2共t兲,共22兲 with P共0兲=X ˙共0兲M0l0/冑1−X ˙2共0兲. The other two equations are nonlinear and coupled, and they are analyzed in detail in the next section. IV. KINK DYNAMICS IN EXPANDING AND/OR CONTRACTING SPACES First of all, let us remark that the main physical variables associated with the kink propagation are the physical center of the kink, Xphys共t兲=a共t兲X共t兲, its physical width lphys共t兲 =a共t兲l共t兲, and its physical velocity Vphys共t兲=HXphys共t兲+a共t兲dX共t兲 dt .共23兲 The first term on the rhs is the contribution to the kink velocity due to the expansion or contraction of the space 共i.e., it turns to zero when the space finishes its elongation or contraction H=0兲. On the other hand, the second term is the peculiar velocity of the kink with respect to the propagating space, Vpec共t兲=a共t兲dX共t兲 dt .共24兲 For the sake of clarity, we analyze separately the cases of slow and fast expansion and/or contraction. A. Slow expansion and/or contraction (adiabatic regime) Note from Eqs. 共19兲–共21兲that the adiabatic approximation implies ␣ 关l ˙2−2Hll ˙−2ll ¨兴Ⰶ1 a2.共25兲 Hence, from Eq. 共21兲we obtain the relation lphys共t兲=l0冑1−Vpec 2共t兲,共26兲 that links the physical width and the peculiar velocity of the kink. Furthermore, from Eqs. 共19兲and 共22兲–共26兲we obtain Vpec共t兲=P共0兲 冑M0 2a2共t兲+P2共0兲.共27兲 From the previous Eqs. 共26兲and 共27兲we see that an expansion decelerates the peculiar motion of the kink and makes it wider, asymptotically lim a共t兲→⬁ Vpec共t兲= 0, lim a共t兲→⬁ lphys共t兲=l0.共28兲 On the other hand, a contraction accelerates the peculiar motion of the kink and makes it sharper, asymptotically lim a共t兲→0 Vpec共t兲= 1, lim a共t兲→0 lphys共t兲=0. 共29兲 The adiabaticity conditions can be obtained from Eq. 共25兲 and require only slow expansions and/or contractions 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 where ⍀R=1/共冑 ␣ l0兲is the so-called Rice frequency for zero velocity, ⍀R=冑12/ ␲ =1.10..., for SG and ⍀R=冑6/共 ␲ 2−6兲 =1.24..., for ␸ 4. We have studied the dynamics for different expansion and contraction rates, both for the sine-Gordon and the ␾ 4poten- tials. In particular, Fig. 1 共and also Fig. 2兲shows the results DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲 056603-3 for an expansion followed by a contraction parametrized by the function a共t兲=1+⌬a 2tanh 冉 t−t0 ⌬t 冊 −⌬a 2tanh 冉 t−t1 ⌬t 冊 ,共32兲 with ⌬a,t0,t1, and ⌬tconstants. 共⌬arepresents the changes in a;t0and t1, the times when the expansion and contraction take place, respectively, and ⌬tis the characteristic time interval where the changes in atake place.兲 In Fig. 1 we show the effects on a kink of a slow expansion followed by a slow contraction that verify the adiabatic conditions 关H⬃⌬a/⌬t=0.1 and H ˙⬃⌬a/共⌬t兲2=0.01兴. In this case, when the adiabatic conditions are verified, both the CCs equations and the adiabatic approximations are in good agreement with the exact results, especially for the center of the kink Xphys共t兲and its peculiar velocity Vpec共t兲共see Fig. 1兲. It is important to note that the agreement is good even when the variations of aare large. This implies that the main effects of a slow expansion and/or contraction are the change of the width and the speed of the propagation of the kink, following the adiabatic relations 共26兲and 共27兲. Therefore, these results show that the speed of a kink can be tuned by slowly expanding and/or contracting the space. Note that the CCs evolution equations go further and are able to predict the small oscillations of the kink width 关23兴 produced by the slow expansion and/or contraction 共see Fig. 1兲. Therefore, they can compute deviations from adiabaticity excluding the radiative effects 共because they involve the transfer of energy to other degrees of freedom兲. It is important to stress that the collective coordinate evolution equations obtained with an Ansatz with fixed width and a variable center of the kink does not predict any variation of the kink speed due to expansions and/or contractions. Therefore, an Ansatz that allows the width of the kink to evolve, like the Rice Ansatz, is an essential ingredient in order to obtain the correct variation of the speed of the kink under expansions and/or contractions. It is interesting to remark that, in the ␾ 4model, the oscillations of the width are related to excitations of the internal mode 关11兴, while in the sine-Gordon equation it has been shown that the excitation of certain phonons can imply oscillations in the shape of the kink 关17,18兴. FIG. 1. Sine-Gordon kink dynamics in a slow expanding and later contracting space. Top-left panel: scale factor a共t兲, Eq. 共32兲 with ⌬a=1, t0=50, t1=150, and ⌬t=10. Top-right panel: spatial profile of the kink ␾ 共x,tfix兲at times tfix=0 共solid line兲,tfix=100 共long-dashed line兲, and tfix=200 共dashed line兲. Three following panels: position of the center of the kink Xphys共t兲, peculiar kink velocity Vpec共t兲, and width of the kink lphys共t兲as functions of time given by the numerical simulations of the partial differential equation 关PDE兴共solid line兲, the numerical solutions of the CCs equation 共long-dashed line兲, and the adiabatic approximation 共dashed line兲. Bottom-right panel: momentum of the kink P共t兲exact solution. Initial conditions: a kink with 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 the other hand, in general, the applicability of the perturbative approaches is restricted to slow and small expansions and/or contractions. The perturbative approaches rewrite Eq. 共6兲in the form ␾ tt− ␾ xx+dU/d ␾ = ⑀ f⬅−H共t兲 ␾ t −关1/a2共t兲−1兴 ␾ xx, where ⑀ fis treated as a small perturbation. Therefore, their range of applicability is limited not only to small Hbut also to small 共1/a2−1兲, i.e., slow and small expansions and/or contractions. B. Fast expansion and/or contraction (nonadiabatic regime) When the expansion and/or contraction is faster the adiabatic approximation breaks, and nonadiabatic effects appear as radiation 共compare the kink profiles in Figs. 1 and 2兲 and as a change in the final width and velocity of the kink 关that are no longer those predicted by Eqs. 共26兲and 共27兲兴. 共See Fig. 2, that corresponds to the nonadiabatic regime H⬃⌬a/⌬t=2 and H ˙⬃⌬a/共⌬t兲2=4.兲The CCs evolution equations predict strong oscillations in the velocity and the width of the kink. In the numerical integration of the complete evolution equations 关Eq. 共6兲兴 the oscillations are indeed present in the width. However, there are other degrees of freedom to which the energy can be transferred. This results in the damping of the oscillations through radiation emission. The loss of energy can also be easily shown noting that after an expansion-contraction cycle the kink moves slower 共see Fig. 2兲. We would like to stress that all the previous comments about slow and fast expansions and/or contractions apply both for the sine-Gordon and the ␾ 4potential. We have performed numerical simulations and comparisons of the adiabatic approximation, the CCs evolution equations, and the full evolution equation for both potentials obtaining analogous results. V. CONCLUSIONS We have studied the effects on a kink of slow and fast expansions and/or contractions of the media, both for the sine-Gordon and for the ␾ 4equations. We have used the Rice Ansatz 关11兴in the continuity equations for the momentum and the energy, in order to obtain the evolution equations for the width and the center of a kink in an expanding and/or contracting space. In general, this set of ODEs cannot be FIG. 2. Sine-Gordon kink dynamics in a fast expanding and later contracting space. Top-left panel: scale factor a共t兲, Eq. 共32兲 with ⌬a=1, t0=40, t1=100 and ⌬t=0.5. Top-right panel: spatial profile of the kink ␾ 共x,tfix兲at times tfix=37.5 共solid line兲,tfix =75 共long-dashed line兲, and tfix =150 共dashed line兲. The three following panels: position of the center of the kink Xphys共t兲, peculiar kink velocity Vpec共t兲, and width of the kink lphys共t兲as functions of time given by the numerical simulations of the PDE 共solid line兲, the numerical solutions of the CCs equation 共long-dashed line兲, and the adiabatic approximation 共dashed line兲. Bottom-right panel: momentum of the kink P共t兲exact solution. Initial conditions: a kink with 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 solved analytically and the solution must be obtained numerically. However, this effective description in terms of the collective coordinates usually gives relevant information and insight of the evolution and the phenomena involved 关16,19–22兴. In this case the CCs approximation is able to describe very accurately the dynamics of the kink when the process is slow, even when the variations of aare large. We have found that the main effects of a slow expansion and/or contraction are the change of the width and the velocity of the kink, following the adiabatic relations 共26兲and 共27兲.In addition, the collective coordinate evolution equations for the width and the center of the kink are able to take into account the small deviations from adiabaticity 共excluding explicitly the radiative effects, because they are related to transference of energy to other degrees of freedom兲. An important consequence of the adiabatic relations is that the speed of a kink can be tuned by a slowly expansion and/or contraction of space. Thus, this result provides a mechanism to control the speed of a kink whenever we can expand and contract the media where it propagates. On the other hand, fast expansions and/or contractions break the adiabatic approximation, giving rise to nonadiabatic effects, as, for example, radiation, that imply a change in the final width and velocity of the kink 共i.e., they are no longer those predicted by the adiabatic relations兲. The collective coordinate evolution equations predict strong oscillations in the width of the kink. In the full problem, these oscillations are also present. However, there are other degrees of freedom to which the energy of these oscillations can be transferred. This results in emission of radiation and in the damping of these oscillations. In this sense, the fast expansion and/or contraction of space provides a method to implement a fast change in the speed of the kink, and also to generate phonons. ACKNOWLEDGMENTS We acknowledge financial support from the Ministerio de Ciencia y Tecnología of Spain under Grants 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 from the Junta de Andalucía through the projects FQM-0207 and 00481 共E.Z.S., N.R.Q.兲. In addition, E.Z.S. thanks the University of Sevilla for its financial support. APPENDIX A: EXPANDING SPACES This appendix includes some technical details related to expanding spaces. 1. Metric The metric in an homogeneous and isotropous flat space is given by ds2=dt2−a2共t兲dx2=g ␮ ␯ dx ␮ dx ␯ ,共A1兲 where a共t兲is the scale factor for the expansion and/or contraction, the metric tensor is g ␮ ␯ = 冉 10 0−a2共t兲 冊 .共A2兲 Its inverse, g ␮ ␯ , defined by g ␮ ␯ g ␯ ␭=g ␮ ␭= ␦ ␮ ␭共with ␦ ␮ ␭a Kronecker delta兲is g ␮ ␯ = 冢 10 0−1 a2共t兲 冣 .共A3兲 The contraction with g ␮ ␯ lowers indices, and the contraction with g ␮ ␯ raises indices. 2. Invariant element of space-time volume Defining g⬅− Det g ␮ ␯ ,共A4兲 it can be shown 共关10兴pages 98 and 99兲that 冑gdtdx 共A5兲 is an invariant volume element under general coordinate transformations. 共dt dx transforms with the Jacobian, while 冑gtransforms with the inverse of the Jacobian.兲 Therefore the appropriate relation between the action S, and the Lagrangian density Lis S= 冕 dt dx冑gL.共A6兲 This definition implies that if the action is invariant under general coordinate transformations the Lagrangian density is also invariant. In our case 冑g=a共t兲. 3. Energy-momentum tensor The energy-momentum tensor is T ␮ ␯ = ⳵ ␮ ␾ ⳵ ␯ ␾ −g ␮ ␯ L.共A7兲 Some useful relations to calculate its components are ⳵ x ␾ =gx ␴ ⳵ ␴ ␾ =gxx ⳵ x ␾ =− ␾ x a2,共A8兲 ⳵ t ␾ = ⳵ t ␾ = ␾ t.共A9兲 The time-time component of the energy-momentum tensor or energy density is ␳ E⬅Ttt = ⳵ t ␾ ⳵ t ␾ −gtt 冋 1 2 ␾ t 2−1 2 ␾ x 2 a2−U共 ␾ 兲 册 =1 2 ␾ t 2+1 2 ␾ x 2 a2+U共 ␾ 兲,共A10兲 the space-time component or momentum density is ␳ P⬅Txt =Ttx = ⳵ t ␾ ⳵ x ␾ −gtxL=− 1 a2 ␾ t ␾ x,共A11兲 and the space-space component or momentum current is CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲 056603-6 jP⬅Txx =1 a2 冉 1 2 ␾ t 2+1 2 ␾ x 2 a2−U共 ␾ 兲 冊 .共A12兲 APPENDIX B: EQUIVALENCE AMONG DIFFERENT CC APPROACHES In this appendix we show that the evolution equations for the collective coordinates obtained with the derivation of the momentum and the energy 关13兴共that we used in Sec. III兲are the same as those obtained with the Lagrangian method 关14,15兴and with the so-called GTWA 关12,16兴共based on projection techniques兲. In all cases, we use the Rice Ansatz as an approximated kinklike solution of the following perturbed nonlinear Klein-Gordon equation: ␾ tt − ␾ xx a2共t兲=−dU d ␾ −关H共t兲+ ␤ 共t兲兴 ␾ t+z共x,t, ␾ 兲,共B1兲 where ␤ 共t兲is a time-dependent damping coefficient and z共x,t, ␾ 兲represents a generic perturbation on the system. Note that for ␤ =0 and z共x,t, ␾ 兲=0 we recover the system 共6兲, introduced in Sec. II. 1. Generalized Lagrangian formalism Introducing a new time variable ␶ ⬅ 冕 0 tdt⬘ c共t⬘兲,共B2兲 where c共t兲is given by c共t兲⬅exp 再 冕 0 t 关H共t⬘兲+ ␤ 共t⬘兲兴dt⬘ 冎 =a共t兲 a共0兲exp 再 冕 0 t ␤ 共t⬘兲dt⬘ 冎 , 共B3兲 Eq. 共B1兲becomes the dissipationless equation, ␾ ␶␶ C2共 ␶ 兲− ␾ xx A2共 ␶ 兲=−dU d ␾ +Z共x, ␶ , ␾ 兲,共B4兲 with C共 ␶ 兲=c关t共 ␶ 兲兴,A共 ␶ 兲=a关t共 ␶ 兲兴, and Z共x, ␶ , ␾ 兲=z关x,t共 ␶ 兲, ␾ 兴. The evolution equation 共B4兲can be obtained from the action 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 the Lagrangian density, and M共x, ␶ , ␾ , ␾ x兲an “Euler-Lagrange integral” of Z共x, ␶ , ␾ 兲. Some of the possible integrations are 共i兲the ␾ functional integral M共x, ␶ , ␾ 兲=− 冕 D ␾ Z共x, ␶ , ␾ 兲,共B6兲 and 共ii兲the following integration for the particular case where 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 define the Lagrangian L= 冕 –⬁ +⬁ dx L,共B8兲 the Euler-Lagrange evolution equations for the collective variables are obtained inserting a given Ansatz in this Lagrangian and using either expression 共B6兲or 共B7兲. This procedure is the generalization of the Lagrangian formalism developed in 关14兴. Using the Rice Ansatz defined by 共14兲and 共15兲共for the SG and ␾ 4potentials, respectively兲and after some straightforward calculations, we obtain the Lagrangian as a function of our collectives variables and their derivatives with respect to ␶ 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into the Euler-Lagrange equations, d d ␶ 冉 ⳵ L ⳵ X⬘ 冊 = ⳵ L ⳵ X,共B10兲 d d ␶ 冉 ⳵ L ⳵ l⬘ 冊 = ⳵ L ⳵ l,共B11兲 and rewriting the equations of motion in the time variable t, we get X ˙=a共t兲P共t兲l共t兲 M0l0 ,共B12兲 dP dt =−共 ␤ +2H兲P−1 a共t兲 再 冕 −⬁ +⬁ d ␪ z共X+ ␪ l,t, ␾ 兲 ␾ ␪ 冎 , 共B13兲 ␣ 关l ˙2−2共 ␤ +H兲ll ˙−2ll ¨兴=l2 l0 2 冉 1+a2共t兲P2 M0 2 冊 −1 a2 +2l2 M0l0 冕 −⬁ +⬁ d ␪ z共X+ ␪ l,t, ␾ 兲 ␪␾ ␪ , 共B14兲 where ␪ =关x−X共t兲兴/land the dots denote the derivative with respect to t. These are the evolution equations for the collec- DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲 056603-7 tive variables obtained for either of the two previous expressions of M共x,t, ␾ , ␾ x兲Eq. 共B6兲or Eq. 共B7兲共in their respective regime of validity兲. Taking in these equations ␤ =0 and z=0, we obtain the Eqs. 共19兲–共21兲derived in Sec. III. 2. GTWA In order to apply the GTWA we rewrite Eq. 共B1兲as ␾ ˙= ␺ , ␺ ˙= ␾ xx a2共t兲−dU d ␾ −关H共t兲+ ␤ 共t兲兴 ␺ +z共x,t, ␾ 兲.共B15兲 The procedure to obtain the CCs equations with the GTWA consists of inserting our specific functional form for ␾ , Eq. 共14兲for sine-Gordon or 共15兲for ␾ 4potential, into Eq. 共B15兲, multiplying the first equation by ⳵ ␺ / ⳵ Xand the second one by ⳵ ␾ / ⳵ X, taking their difference and integrating over x, equating the result with zero; and repeating the same procedure with ⳵ ␺ / ⳵ land ⳵ ␾ / ⳵ l. This gives us the same ordinary differential equations for X共t兲,P共t兲, and l共t兲as those obtained in the previous subsection 关see Eqs. 共B12兲–共B14兲兴. 关1兴A. C. Scott, Nonlinear Science 共Oxford University, Oxford, 1999兲. 关2兴M. A. Collins, A. Blumen, J. F. Currie, and J. Ross, Phys. Rev. B19, 3630 共1979兲;19, 3645 共1979兲. 关3兴Kalman B. Migler and Robert B. Meyer, Phys. Rev. Lett. 66, 1485 共1991兲. 关4兴H. K. Ebraheem, J. L. Shohet, and A. C. Scott, Phys. Rev. Lett. 88, 235003 共2002兲. 关5兴J. L. Shohet, B. R. Barmish, H. K. Ebraheen, and A. C. Scott, Phys. Plasmas 11, 3877 共2004兲. 关6兴A. Linde, Particle Physics and Inflationary Cosmology 共Harwood Academic Publishers, Chur, Switzerland, 1990兲. 关7兴P. Anninos, S. Oliveira, and R. A. Matzner, Phys. Rev. D 44, 1147 共1991兲. 关8兴D.-H. Han, J.-G. Zhu, J. H. Judy, and J. M. Sivertsenb, Appl. Phys. Lett. 70, 664 共1997兲; L. Lanotte, G. Ausanio, V. Iannotti, G. Pepe, G. Carotenuto, P. Netti, and L. Nicolous, Phys. Rev. B63, 054438 共2001兲. 关9兴D. J. Grove and D. M. Meade, Nucl. Fusion 25, 1167 共1985兲. 关10兴S. Weinberg, Gravitation and Cosmology 共Wiley, New York, 1972兲. 关11兴M. J. Rice and E. J. Mele, Solid State Commun. 35, 487 共1980兲. 关12兴G. M. Wysin, F. G. Mertens, A. R. Völkel, and A. R. Bishop, in Nonlinear Coherent Structures in Physics and Biology, edited by K. H. Spatschek and F. G. Mertens, NATO Advanced Studies Institute, Series B: Physics 共Plenum, New York, 1994兲, Vol. 329; F. G. Mertens, H. J. Schnitzer, and A. R. Bishop, Phys. Rev. B 56, 2510 共1997兲. 关13兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. E 62, 5695 共2000兲. 关14兴N. R. Quintero and E. Zamora-Sillero, Physica D 197,63 共2004兲. 关15兴Sara Cuenda and Angel Sánchez, Chaos 15, 023502 共2005兲. 关16兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. Lett. 84, 871 共2000兲. 关17兴David K. Campbell, Jonathan F. Shonfeld, and Charles A. Wingate, Physica D 9,1共1983兲. 关18兴N. R. Quintero and Panayotis G. Kevrekidis, Phys. Rev. E 64, 056608 共2001兲. 关19兴F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157,63共1988兲; Yuri S. Kivshar and Boris A. Malomed, Rev. Mod. Phys. 61, 763 共1989兲. 关20兴D. W. McLaughlin and A. C. Scott, Phys. Rev. A 18, 1652 共1978兲. 关21兴A. Sánchez and A. R. Bishop, SIAM Rev. 40, 579 共1998兲. 关22兴G. Costantini, F. Marchesoni, and M. Borromeo, Phys. Rev. E 65, 051103 共2002兲; M. Salerno and Y. Zolotaryuk, ibid. 65, 056603 共2002兲; L. Morales-Molina, N. R. Quintero, F. G. Mertens, and A. Sánchez, Phys. Rev. Lett. 91, 234102 共2003兲. 关23兴The width and the center of the kink position for the PDE results are obtained fitting the Rice Ansatz, Eq. 共14兲or Eq. 共15兲, to the kink profile given by the PDE evolution 共6兲. CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲 056603-8