P u b l i s h e d f o r SISSA b y S p r i n g e r
Received: January 20, 2017 Accepted: March 13, 2017 Published: May 19, 2017
Boundary scattering in the ϕ4 model
Patrick Dorey,a Aliaksei Halavanau,b,c Jam es M ercer,a,d Tomasz Romanczukiewicze and Yasha Shnirb f’g
aD ep a rtm en t o f M athem atical Sciences, D urham U niversity, D urham DH1 3LE, U.K.
bD ep a rtm en t o f Theoretical P hysics and A strophysics, B SU , M in sk Independence A ven u e 4, B elarus cF erm i N a tio n a l Laboratory,
P in e St. and K irk Rd., Z IP 60511, M ail S ta tio n 221, B atavia, Illinois, U .S.A . dD eloitte M C S Lim ited,
H ill House, 1 L ittle N ew Street, London, E C
4
A 3 T R , U.K.eIn stitu te o f P hysics, Jagiellonian U niversity, Lojasiew icza 11, 30-348 Krakow, P oland f B L T P , J IN R ,
141980 Dubna, R ussia
g In stitu te o f P hysics, Oldenburg U niversity, P ostfach 2503 D-26111 Oldenburg, G erm any
E-mail: p .e .d o re y @ d u rh a m .a c .u k , a lia k s e i.h a la v a n a u @ g m a il.c o m , j m e r c e r @ d e l o i t t e .c o .u k , t r o m @ t h . i f . u j . e d u . p l , s h n ir @ m a th s .tc d .ie
A b s t r a c t : We stu d y b o u n d ary scatterin g in th e m odel on a half-line w ith a one- p a ra m ete r fam ily of N eum an n-ty p e b o u n d ary conditions. A rich variety of phenom ena is observed, which extends previously-studied behaviour on th e full line to include regimes of near-elastic scattering , th e resto ratio n of a m issing scatterin g window, and th e creatio n of a kink or oscillon th ro u g h th e collision-induced decay of a m etastab le b o u n d ary sta te . We also stu d y th e decay of th e v ib ration al bo u n d ary m ode, and explore different scenarios for its relaxation and for th e creation of kinks.
K e y w o r d s : Field Theories in Lower Dim ensions, N o n p e rtu rb a tiv e Effects, Solitons M onopoles and In sta n to n s
A r X i v e P r i n t : 1508.02329
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C o n te n ts
1 I n tr o d u c tio n 1
2 T h e m o d e l 2
3 N u m e r ic a l r e s u lts 3
4 K in k -b o u n d a r y fo r c e s a n d t h e lo c a tio n o f t h e lo w -v e lo c ity w in d o w 5
5 T h e b o u n d a r y m o d e 7
6 T h e r e so n a n c e m e c h a n is m in b o u n d a r y s c a tt e r in g 9
7 R a d ia tiv e d e c a y o f t h e b o u n d a r y m o d e 13
8 H ig h e r -o r d e r n o n lin e a r e ffe c ts a n d a m p litu d e -d e p e n d e n t d e c a y r a te s 16 9 C r e a tio n o f k in k s fro m a n e x c it e d b o u n d a r y 17
10 C o n c lu s io n s 18
A N u m e r ic a l m e t h o d s 20
B S u p p le m e n ta r y m a te r ia l 21
1 In tr o d u c tio n
System s w ith boundaries, defects and im purities have been intensively studied in statistical physics and field theory, b o th a t th e classical and th e q u a n tu m levels. O ften th e key physics of th e m odel can be cap tu red , possibly after dim ensional reduction, by a simple 1+1 dim ensional field th eo ry on a half line. E xam ples include th e K ondo problem [1], fluxon p ro p ag ation in long Josephson ju n ctio n s [2], th e XXZ m odel w ith b o u n d ary m agnetic field [3], an im p urity in an in teracting electron gas [4], th e sine-G ordon [5] and T oda [6] m odels, m onopole catalysis [7], th e L u ttin g e r liquid [8], and a toy m odel m otivated by M -theory [9].
Especially since th e work of G hoshal and Zam olodchikov [5], th ere has been great interest in b o u n d ary conditions com patible w ith bulk integrability, and m any such m odels tu rn o ut to be of direct physical interest. However less a tte n tio n has been paid to th e equally if not m ore physically-relevant cases of non-integrable bo u n d ary system s, even at th e classical level. T his is p erhaps a sham e, as it is now known th a t non-integrable classical
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field theories, even in 1+ 1 dim ensions, can exhibit rem arkably rich p a tte rn s of behaviour not seen in th eir integrable co u n te rp arts [10- 14].
In this p ap er we exam ine th e 04 th eory in 1+1 dim ensions, restricted to a half line by a sim ple N eum ann-type ‘m agnetic field’ b o u n d ary condition. (T he sine-G ordon m odel w ith a non-integrable b o u n d ary was recently investigated in [15].) T he 04 th eo ry on a full line is sim ilar to th e sine-G ordon m odel in th a t b o th su p p o rt topological kinks and antikinks; th e 0 4 th eo ry also has an intriguing and still not fully-understood co u n te rp art of th e sine-G ordon breath er, th e oscillon [16]. We chose th e m agnetic field b o u n d ary condition in p a rt because of its simplicity, and in p a rt because th e scatterin g of kinks against such a bo u n d ary provides a n a tu ra l deform ation of th e full-line scatterin g problem s which are already known to exhibit in tricate p a tte rn s of resonant scatterin g [10- 13]. In some regimes our results do indeed resem ble th e p a tte rn of scatterin g windows observed in kink-antikink collisions on th e full line, while in others we find novel phenom ena including a new ty p e of
‘sh arp-edged’ scatterin g window. Even th o u g h th e th eo ry is not integrable, it tu rn s o ut to be possible to give an accu rate an aly tical description of some aspects of th is behaviour. We com plem ent these studies w ith an investigation of th e decay of th e v ib ration al b o un d ary m ode th ro u g h nonlinear couplings to scatterin g states, and of th e creation of kinks by an excited boundary. An interesting feature of th e b o u n d ary m ode decay, discussed in section 8, is th a t w ith suitable initial conditions a period of relatively slow decay can be followed by a sudden b u rst of rad iatio n from th e b o u n d ary as a new decay channel opens.
W hile this pap er is self-contained, we have also m ade a num ber of short movies to illu strate aspects of th e discussion, which can be found as supplem entary m aterial. A fter a brief exp lanatio n of th e num erical m ethods used to o b tain our plots in appen d ix A , these are listed in app en d ix B .
2 T h e m o d el
We consider a rescaled 04 th eo ry w ith vacua 0 v € {—1, + 1} on th e left half-line —to <
x < 0. T he bulk energy and L agrangian densities are E = T + V and L = T - V respectively, where
T = 1 0 ? and V = 2 0X + 2 (^ 2 - 1)2 . (2.1) T he s ta tic full-line kink and antikink, 0 K (x) = ta n h (x - x0) and 0 k ( x ) = —0k (x), have rest m ass M = 4 /3 and in terp o late betw een th e two vacua. Including a b o u n d ary energy - H 0 o, w here 00 = 0(0, t) and H can be in terp reted as a b o u n d ary m agnetic field, yields th e N eum ann-typ e b o u n d ary condition 0 X(0 ,t) = H at x = 0 .
For 0 < H < 1 th ere are four sta tic solutions to th e equations of m otion, shown in figure 1. Two of them , 0 i(x ) = ta n h (x - X0) and 02(x) = ta n h (x + X0) w ith X0 = cosh-1(1/ y l H j ), are restriction s of regular full-line kinks to th e half-line, while th e o th er two, 03(x) = - c o th (x - X1) and 04(x) = - c o th (x + X1) w ith X 1 = sin h-1( 1 / ^ / |H |) are irregular on th e full line. On th e half line, 0 3 is non-singular and corresponds to th e absolute m inim um of th e energy, while 0 1 is m etastab le, and 0 2 is th e unstab le saddle-point betw een 0 3 and 01. T h eir energies can be found by rew riting E [0] = V d x - H 0 0 in
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x
Figure 1. Static solutions for H = 1/2.
Bogom olnyi form as
(2.2)
Since 0 i and 02 satisfy 0 x = 1 — 02 we have 0 i(0 ) = — \J 1 - H , 02(0) = a/ 1—H ; while (03)x = 03 — 1 and so 03(0) = \ / 1 + H . Taking th e u p p er and lower signs in (2.2) as a p p ro priate,
(2.3) As H increases th ro u g h 1, 0i merges w ith 02 and d isappears, leaving 03 as th e only s ta tic solution for H > 1. For H < 0 th e sto ry is th e same, w ith 0 and H negated th ro u g h o u t, so th e physically-relevant solutions are 0i(x) := — 0i(x), i = 1... 3.
3 N u m e r ic a l r esu lts
We to o k initial conditions corresponding to an antikink a t x0 = —10 travelling tow ards th e b o u n d ary w ith velocity vi > 0. (We found th e setu p w ith an incident antik ink easier to visualise, b u t our results apply equally to kink-boundary collisions on n egating 0 and H.) T hus th e initial profile was 0(x, 0) = 01(x) — ta n h (q(x — x0)) + 1 for H > 0 and 0(x, 0) = 03(x) — ta n h (q(x — x0)) + 1 for H < 0, w here y = 1 / 1 — v2. O ur real interest was in th e problem w ith th e initial antik ink infinitely far from th e boundary, b u t th e rapid decay of th e an tik in k -b o u n d ary force (4.1) , calculated below, m eant th a t erro r in tak ing x0 = —10 was small.
To solve th e system num erically, we restricted it to an interval of length L, w ith th e N eum ann b o u n d ary condition im posed at x = 0 and a D irichlet condition a t x = —L. (Since we to o k run tim es such th a t rad iatio n did not have tim e to reflect from th e e x tra
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E [0i] = 3 - 2 ( 1 - H) 3/2 , E [02] = 2 + 2 ( 1 - H ) 3 /2 , E [03] = 2 - 2 ( 1 + H )3 /2 .
1 f0 2 r i i0
E [0] = 2 (0 X ± (0 2- 1) )2dx T - 0 3- 0 - H 0o .
2 J —co L3 J —^
F ig u re 2. Final antikink velocities as functions of initial velocities. The dashed line indicates the result for a purely elastic collision. In the fifth plot, a kink can also be produced: its velocity is shown in red. The horizontal dotted lines in plots a and b show the relevant values of vcr ( H), as given by equation (4.2) below.
b o u n d ary and retu rn , th e b o u n d ary condition a t x = - L was anyway irrelevant.) We used a 4th order finite-difference m ethod, explained in m ore detail in ap pen dix A , on a grid of N = 1024 nodes w ith L = 100, so th e sp atial step was 5x « 0.1, and a 6th-order sym plectic in te g rato r for th e tim e stepping function, w ith tim e step St = 0.04. Selected runs were rep eated w ith o th er values of x0, L, N and St to check th e stab ility of our results.
O ur sim ulations revealed a rich picture, aspects of which are sum m arised in figures 2 and 3 . For all ( H , vi) pairs w ith H < H c « 0.6, th e an tikin k eith er reflects off th e b o u n d ary w ith some velocity Vf, or becomes stuck to it — corresponding to Vf = 0 — to form a ‘b o u n d ary oscillon’. T his la tte r configuration oscillates w ith a large (of order one) am plitu de, and a below -bulk-threshold basic frequency. J u s t like th e bulk oscillon (which it becomes in th e lim it H ^ 0), it th e n decays very slowly into rad iatio n. At H = 0 (figure 2(c)) th e plot of |vf| as a function of vi reproduces th e well-known stru c tu re of resonant scatterin g windows in K K collisions on a full line [10-12]. For negative values of H (figures 2(a) and (b)) new features emerge. For vi small, th e antik ink is reflected elastically from th e b o u n d ary w ith very little rad iatio n . As vi increases above an H -d ep en d en t critical value vcr, th e an tikink is tra p p e d by th e boundary, leaving only rad iatio n in th e final state.
Increasing vi fu rth er, scatterin g windows begin to open, until vi exceeds an u p p er critical
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Figure 3 . A ‘phase diagram’ of antikink-boundary collisions. The plot shows the value of the field at x = 0 a time tf = |x0|/vj + 100 after the start of the simulation, as function of the boundary magnetic field H and the initial velocity v*.
value and th e antikin k again always escapes. If th e an tik in k does escape, its speed |Vf | is always larger th a n some m inim al value very slightly lower th a n vcr, so (in co n trast to th e full-line situatio n) Vf is a discontinuous function of vp giving th e windows th e sharp edges m entioned in th e in tro du ctio n. For sm all positive values of H (figure 2 (d)), Vf is instead a continuous function of Vj, th e sequence of windows for H = 0 shifting tow ards lower velocities while preserving its general stru ctu re. Finally, for H > H c (figure 2 (e)) o th er new phenom ena arise which have no c o u n te rp arts in th e full-line theory; these will b e discussed fu rth e r in la te r sections.
4 K in k -b o u n d a ry fo rces and th e lo c a tio n o f th e lo w -v e lo c ity w in d o w
To u n d e rsta n d th e novel window of near-elastic scatterin g at low initial velocities w hen H is negative, seen in figures 2(a) and 2(b), we s ta rt by evaluating th e s ta tic force betw een a single an tikin k and th e boundary. P lacing th e antikink a t x = xo < 0, we add a possibly- singular ‘im age’ kink a t x i > 0 in such a way th a t th e com bined configuration satisfies th e bo u n d ary condition a t x = 0. From th e sta n d a rd full-line result, th e force on th e antikink from th e im age kink is equal to 32e-2(xi-Xo), or m inus th is if th e image kink is singular. For
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Figure 4 . Example collisions for H = -0 .4 (left column) and H = 0.9 (right column), illustrating various scattering scenarios. At negative H : (a) elastic recoil for low impact velocity; (b) saddle point production at the critical velocity, the antikink finishing on the top of the potential barrier;
(c) single bounce with the antikink escaping back over the barrier. At positive H : (d) single bounce with the excitation of the H > 0 boundary mode; (e) kink production via collision-induced boundary decay; (f) bulk oscillon production. See also movies M01-M07 and M11 of appendix B .
|H | ^ 1 and |x 0| ^ 1 we find th a t th e bo u n d ary condition requires e 2x1 = 4H + e2x° , so
F = 3 2 Q H + e2x°^ e2x0. (4.1)
For H < 0 th e force is repulsive far from th e boundary, only becom ing a ttra c tiv e nearer in.
W hen x 0 = 2 log(—1 H ), x \ = to and th e force vanishes, th e antikink-kink configuration reducing to th e u n stab le s ta tic solution ( 2.
Now consider, again for H < 0, an an tikin k moving tow ards th e boundary. If its velocity vi is small, th e n it w on’t have sufficient energy to overcome th e initially-repulsive force, and it will be reflected w ith ou t ever com ing close to x = 0, and w ith ou t significantly exciting any o th er modes; th is behaviour is illu strated in figure 4 (a). Increasing vi , a t some critical value vcr th e energy will be ju s t enough reach th e to p of th e p o ten tial barrier and create th e s ta tic saddle-point configuration ( 2, as shown in figure 4(b). T he value of vcr can be deduced on energetic grounds: th e initial energy is | (1 — v^ ,) -1 /2 + E [ ( 3], while th e final energy is E [ ( 2] = 3 + 3( 1 + H ) 3/2. E q u a tin g th e two,
vcr(H ) = ^ 1 — 4 ( ( 1 + H) 3/2 + (1—H) 3/2) - 2 . (4.2)
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If vi is ju s t larger th a n vcr, th e antikin k can overcome th e p o ten tial barrier and ap proach th e boundary; energy is th e n lost to o th er m odes and so it is unable to retu rn , and is tra p p e d a t th e boundary. T hus vcr ( H ) m arks th e u p p er lim it of th e windows of alm ost-perfectly- elastic scatterin g seen in figures 2(a) and 2(b), and th e lower edge of th e ‘fractal to n g u e ’ occupying th e left half of figure 3 . T he curve vi = vcr( H ) is included in figure 3 ; it m atches our num erical results rem arkably well. Indeed, it can be seen from figure 7 below th a t th e m axim um erro r is of th e order of 0.5%, which is ra th e r small given th a t rad iatio n was ignored in th e derivation. Sim ilar argum ents show th a t, w ithin th is approxim ation, vcr is th e sm allest possible speed for any escaping antikink, explaining th e sharp (discontinuous) edges of all windows w hen H < 0.
5 T h e b o u n d a r y m o d e
N ext we consider th e p e rtu rb a tiv e sector of th e model, th a t is solutions of th e form 0 (x , t) = 0s ( x )+ n (x , t) w here 0s (x) is a sta tic solution to th e equations of m otion and n(x, t) is small.
T he full-line th eo ry has a continuum of small linear p e rtu rb a tio n s ab o u t each vacuum w ith m ass m = 2, while a sta tic kink 0 K (x) = ta n h (x — X o) has two discrete norm alizable m odes — th e tra n sla tio n a l m ode, and a vib ratio n al m ode w ith frequency w1 = a/3 — and a continuum of above-threshold states n(x, t) = eiwtnk(x) w here w2 = 4 + k2 and [18]
Vk(x) = e -ik(x-X o) ( —1 — k2 + 3ik ta n h (x — X o) + 3 ta n h 2(x — X 0)) . (5.1) T urning now to th e half-line theory, we can regard 0 K (x) instead as th e sta tic half-line solution 01(x) to th e b o u n d ary th eory w ith 0 < H < 1 and 0 ( —ro) = —1. Its linear p e rtu rb a tio n s m ust now satisfy dxn(x) = 0 a t x = 0. S etting k = in th is yields
k3 — 30oK2 + (602 — 4)k — 6^0 + 60o = 0 (5.2) w here 0 o = 01(0) = —a/ 1 — H and th e frequency wB of th e corresponding b o u n d ary m ode satisfies w2B = 4 — k2. T he solutions of (5.2) for b o th negative and positive values of 0 o are shown on th e left-hand plot of figure 5; note th a t only solutions w ith k > 0 can give rise to localised modes, and of these, k m ust be less th a n 2 for wB to be real and th e m ode stable.
We will denote th e corresponding norm alised profile function as (x) := niK( x ) /n iK (0), w here niK(x) is given by (5.1) w ith k = iK.
For 0 < H < 1, we have —1 < 0 o < 0 and (5.2) has ju s t one positive solution k, which satisfies k < 2: this is th e single v ib rational m ode, localised near to th e boundary. T he linear p e rtu rb a tio n s of 02(x), th e saddle-point solution, are also described by ( 5.2) , b ut now w ith 0 o = 0 2(0) = + a /1 — H . For these cases (5.2) has two positive solutions b u t one is larger th a n 2: this is th e u n stab le m ode of 02(x). Finally, for H < 0, th e contin u atio n of (5.2) to 0 o < —1 governs th e spectrum of fluctuations ab o u t 03(x), th e H < 0 vacuum in th e 0 ( —rc>) = —1 sector. T here are no positive solutions in th is regim e and hence no v ib ratio n al m odes of th e b o u n d ary for H < 0. T he rig ht-hand plot of figure 5 sum m arises th e situ atio n, p lo ttin g th e images of th e positive-K p a rts of th e curves shown on th e left u nd er th e m apping (0 o, k) ^ (H , w2B) = (1—0oi 4 —K2). T he grey dashed p a rts of th e curve
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00 H
Figure 5 . Linearised boundary mode analysis: on the left, the solutions of equation (5.2) as a function of 0o = 0(0); on the right, the frequencies of localised boundary modes as a function of H .
Frequency w
Figure 6 . Power spectra at the boundary after a collision with v = 0.5, for H = -0 .1 (upper) and H = 0.3 (lower).
visible for H < 0 are included for com pleteness b u t do not describe v ib ration al m odes of physical solutions — th ey correspond to ‘p e rtu rb a tio n s ’ of th e singular solution 0 4(x).
T hese findings are confirm ed by our num erical results. Figure 6 shows th e Fourier transfo rm s of 0(0, t) for 30 < t < 3030, for antik in k -b o u n d ary collisions w ith initial velocity vi = 0.5, and H = - 0 .1 and 0.3. T h e final velocity Vf of th e reflected an tikin k is -0.3 8 259 6 for H = - 0 .1 and -0 .4 5 4 0 1 4 for H = 0.3, so in b o th cases tra n sla tio n a l energy is lost to o th er m odes durin g th e collision.
For H = - 0 .1 , th e b o u n d ary does not have an intern al m ode, and only radiativ e m odes w ith frequencies near to 2, th e m ass threshold, rem ain n ear to th e boundary. T he internal
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m ode of th e reflected an tik ink has frequency wp b u t th is m ode can no t be observed a t th e b o u n d ary since it is exponentially suppressed there. However nonlinear couplings w ith o th er excitations create waves w ith frequencies a t above-threshold m ultiples of w1 [19], which can pro p ag ate back to th e boundary. T he u p p er plot of figure 6 shows peaks at Q1 = 2 and Q2 = ^ (2 w 1), w here Q(w) = 7(w + k ( w ) v f ) is th e D oppler-shifted frequency of rad iatio n e m itte d from th e moving kink m easured on th e boundary. H igher harm onics a t Q3 = Q(3w1) and Q4 = Q(4w1) are also visible, along w ith com binations of th e internal m ode of th e antikin k and th e lowest continuum m ode such as = Q(2 + w1) and Q6 = 2 + Q (2w1).
M any of these m odes are also present in th e H = 0.3 sp ectrum shown in th e lower plot of figure 6, alb eit a t shifted locations because of th e different final an tikink velocity.
However th e plot is dom inated by th e internal b o u n d ary m ode w ith frequency Q10 = wB = 1.888459. T he higher harm onics Q11 = 2wB and Q12 = 3wB are also visible, while interactions betw een rad iatio n from th e outgoing antikink and th e b o u n d ary m ode lead to peaks at Q13 = wB + Q (2w1) and Q14 = wB — Q(2w1).
6 T h e r eso n a n ce m ech a n ism in b o u n d a r y s c a tte r in g
For small nonzero values of |H |, th e resonant energy exchange m echanism governing sc at
terin g in th e bulk 0 4 m odel is changed in two ways in th e b o u n d ary theory: (i) th e a ttra c tiv e force acting on th e antik in k n ear to th e bo u n d ary is m odified, in p a rticu la r becom ing re
pulsive a t g reater distances when H is negative; (ii) after th e initial im pact, energy can be stored not only in th e in ternal m ode of th e antikink, b u t also, for positive values of H , in th e bo u n d ary m ode. T hese factors change th e resonance condition for energy to be retu rn ed to th e tra n sla tio n a l m ode of th e antikink on a subsequent im pact after some integer num ber of oscillations of th e a n tik in k ’s in ternal m ode, shifting (and, for negative H , sharpening) th e windows seen in figures 2 (a )-(d ). This re tu rn can hap p en after two, th re e or m ore bounces from th e boundary, leading to a hierarchy of m ultibounce windows as in th e full-line situ atio n . O ur num erical results suggest th a t for small positive values of H th e con trib u tio n of th e bo u n d ary m ode in th e resonant energy tran sfer is not significant.
For larger values of |H | o th er new features app ear. For H < 0 th e first is th e resurrec
tio n of a tw o-bounce window th a t was observed to be m issing from th e full-line scatterin g process by C am pbell et al. in [10]. Figure 2 (a) includes a scatterin g window centred at vj ~ 0.245 which is n o t th e con tin u ation of any of th e windows seen in figures 2 (b )-(d ); th e sam e window can be seen in figure 3 running from (H , Vj) = ( —0.135, 0.202) to (H , Vj) = ( —0.489, 0.417), and th e to p plot of figure 7, running from (H , vj — vcr) = ( —0.135, 0.084) to (H , vj) = ( —0.489,0.023). T he em ergence of th is window as H decreases below H « —0.136 is shown in m ore detail in th e left-hand set of plots of figure 8. T he n a tu re of th e new window is m ade clear by th e plots in figure 9, which shows 0(0, t) for vj inside th e first th re e tw o-bounce windows for H = —0.2, and also for H = 0, which is equivalent to th e full-line case. T he ‘w obbles’ betw een th e large dips in such plots count th e oscillations of th e intern al antik ink m ode betw een bounces [10]. As can be seen from th e figure, th e m inim um num ber of oscillations su p p o rtin g an tikink escape is one sm aller for H = —0.2
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Figure 7 . Zoomed-in views of the region near the tip of the fractal tongue of figure 3. Note that the vertical axes show multiples of v* — vcr, where vcr = vcr (H ) is the theoretical upper limit of the near-elastic scattering window given by the formula (4.2).
th a n it was for H = 0, giving rise to th e e x tra window. A com plem entary process of
‘window d e stru c tio n ’ occuring for H > 0 can be seen on a close exam ination of figure 3, and on th e righ t-h an d set of plots of figure 8. D ecreasing H fu rth er, we also observed interestin g stru c tu re s at th e tip of th e fractal tongue, near to H = - 1 , w ith resonance windows m erging to give rise to a p a tte rn of half-rings on th e phase diagram . These are shown in th e lower two plots in figure 7.
For H > 0, th e scatterin g can induce th e m etastab le 01 bo u n d ary to decay to 03, th e tru e ground sta te , w ith th e creatio n of an e x tra kink. T his process, which has no analogue in th e full-line theory, is visible in th e e x tra red ‘kin k ’ line in figure 2 (e). T he principal region of b o u n d ary decay occupies th e solid red area on th e right edge of figure 3 , and is exam ined in m ore d etail in figure 10. If th e bo u n d ary m ode is sufficiently strongly excited by th e initial an tikin k im pact, it behaves as an in term ed iate s ta te prior to th e escape of a kink from th e boundary, analogous to th e in term ediate oscillon s ta te in th e process of
J H E P 0 5 ( 2 0 1 7 ) 1 0 7
F ig u re 8. Scans of ^>(0, t) for various values of H , showing the emergence (on the left) and destruction (on the right) of a two-bounce window as H moves away from zero.
Figure 9 . Plots of 0(0, t) for Vj inside the first three two-bounce windows for H = -0 .2 (left) and for H = 0 (right).
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J H E P 0 5 ( 2 0 1 7 ) 1 0 7
F ig u re 10. Antikink-boundary scattering at large H . (a): a zoomed-in view of figure 3 showing the value of the field at x = 0 a time tf = |x0|/vj + 100 after the start of the simulation; (b): the measured final velocity of the reflected antikink; (c): the measured final velocity of the emitted kink, if present, white being plotted otherwise; (d): the difference between these two velocities.
K K p air p ro d uctio n on th e full line [17, 19, 20]. D epending on th e ir relative velocities, th e reflected antikink and th e subsequently-em itted kink m ay a p p e ar sep arately in th e final sta te , or recom bine to form a bulk oscillon. Such collisions lead them selves to a fractal-like stru c tu re w ith windows w here th e antik ink and kink sep arate interspersed w ith regions of oscillon production, ju s t as in th e full-line th eory (though w ith added com plications due to interference w ith rad iatio n from th e b o u n dary). Some of th is s tru c tu re can be seen in figure 10(d), w here th e blue regions inside th e zone of b o u n d ary decay show windows of antikink and kink separation, while th e yellow regions correspond to th e p roduction of a bulk oscillon, and also in th e movies M11 and M12. Spacetim e plots of some of th e relevant processes, for H = 0.90, are shown in th e right panels of figure 4: scatterin g of th e an tikin k w ith excitatio n of th e b o u n d ary m ode, b u t no kink p ro d uctio n (d); produ ctio n of a separated K K pair, w ith th e b o u n d ary decaying to th e tru e ground s ta te (e); and recom bination of th e K K p air to form a bulk oscillon (f). A fu rth e r intriguing featu re of th e region of b o u n d ary decay, clearly visible in figures 3 and 10, is th e cusp-like nick, term in a tin g at (H , vj) « (1,0.365), which splits it into two disconnected p arts. T his appears to be associated w ith a velocity-dependent vanishing of th e effective coupling betw een th e incident antik ink and th e b o u n d ary m ode. It would be very interesting to find an an alytical un d erstan d in g of th is phenom enon, b u t we will leave th is for fu tu re work.
J H E P 0 5 ( 2 0 1 7 ) 1 0 7
7 R a d ia tiv e d eca y o f th e b o u n d a r y m o d e
A significant featu re of th e 04 m odel is th a t its sp ectru m of p e rtu rb a tiv e oscillations around th e s ta tic kink or an tik in k solutions contains an in tern al v ib ratio nal m ode. If th e am pli
tu d e of th e excitatio n is small enough and nonlinear corrections can be neglected, this m ode oscillates w ith alm o st-co n stan t am plitude A and frequency wd = a/3. For larger am plitud es nonlinearities s ta rt to play an im p o rta n t role. It has been shown [19] th a t th e first anharm onic correction to th e in ternal m ode oscillation results in th e app earan ce of an outgoing wave w ith frequency 2wd, which is above th e m ass th reshold. T he corresponding ra te of radiativ e energy loss is d E / d t A4, causing the mode to decay. The resulting tim e dependence of th e am plitu d e of th e internal m ode follows th e law d A /d t ~ A 3, where th e explicit value of th e pro p o rtio n ality co n stan t can be found using a G reen’s function technique [19].
For our b o u n d ary theory, we have observed a sim ilar p a tte rn in th e decay of small- a m p litu de excitations of th e b o u n d ary m ode, b u t w ith a num ber of interestin g new features.
For small positive values of H , th e frequency wB of th e linearised b o u n d ary m ode, as predicted by (5.2) , satisfies 2wB > 2, and so th e second harm onic of th is m ode is able to p ro p ag ate in th e b u lk.1 B u t as th e b o u n d ary m agnetic field H increases, th e frequency of th e b o u n d ary m ode decreases, and w hen H > H2 « 0.925, 2wB dips below 2 and th e situ a tio n changes. T he second harm onic can no longer p ro p ag ate into th e bulk, and th is channel of radiative energy loss from th e b o u n d ary is term in a te d . Only th e next harm onic, which app ears in th e th ird order of th e p e rtu rb a tio n series, can be seen in th e power spectrum . T he rad iatio n loss ra te becomes d E / d t ~ A6 and th e decay ra te is reduced to d A /d t ~ A 5.
T he situ a tio n changes again as H increases beyond H = H3 « 0.982, when 3wB falls below 2 and th e th ird harm onic joins th e second, tra p p e d below th e m ass threshold.
T heoretically, as H ^ 1 and wb ^ 0 th is p a tte rn will rep eat an infinite num ber of tim es, so th a t whenever n + i < wB < n , th e am plitu de of th e decaying m ode should satisfy, to leading order, th e equatio n d A /d t A 2ra-i.
F igure 11 shows th e behaviour of th e field on th e bo u n d ary and at x = —50, in th e far field zone, w ith initial conditions 0 (x , 0) = 01(x) + 0.05 nB (x), 0 t (x, 0) = 0 and H = 0.90 < H 2. T he power spectrum of th e field on th e bo u n d ary is dom in ated by bo u n d ary m ode oscillating w ith th e theoretically predicted frequency wB = 1.08509. T here are also two peaks a t 2wB and 3wB . Since wB < m, this lowest m ode cann o t p ropagate and indeed, th ere is no tra c e of it in th e far field zone. T he m ode w ith th e frequency 2wb is already in th e scatterin g spectru m , so this m ode does p ropagate, causing th e energy loss from th e bo u n d ary m ode, as seen in figure 11a .
T he picture is different w hen H = 0.94 > H2 (see figure 12) . T he m ode w ith frequency wB = 0.93643 still dom inates th e power sp ectru m of th e b o u n d ary excitations, b u t its decay is m uch slower, reflecting th e fact th a t th e m ode 2wB is now below th e m ass threshold and can no t p ro p ag ate into th e bulk. As can be seen from th e power sp ectrum in th e far field zone p lo tted in figure 12(d), th e rad iatio n is m uch less th a n in th e previous case. T here are
1R ecall th a t m = 2 is th e m ass threshold for th e bulk theory.
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J H E P 0 5 ( 2 0 1 7 ) 1 0 7
Figure 11. Evolution of the boundary mode for H = 0.90 < H 2, with initial conditions ^>(x, 0) =
^i(x) + 0 .05 (x), ^t(x, 0) = 0: a) the amplitude of the boundary mode as a function of time;
b) its power spectrum, found by taking the Fourier transform of ^>(0, t) for 0 < t < 1600; c) the values of n (x,t) := ^ (x ,t) — ^ i(x ) at x = -5 0; d) its power spectrum, taken from n ( - 50, t) for 200 < t < 1600. In plots a) and c), solid non-transparent lines join local extrema of the measured field at the given position.
Figure 12. The same sequence of plots as in figure 11, but now for H = 0.94 > H2 .
J H E P 0 5 ( 2 0 1 7 ) 1 0 7
H2 H3
H
F ig u re 13. Maximal radiation amplitude |nmax(x)|, where n (x,t) = ( ( x ,t) — (i( x ) is the deviation of the field from its static value, at x = —50 for the kicked initial conditions ((x , 0) = ( 1 (x), ( t (x, 0) = Aqwb Vb (x) as a function of H , for three different values of Aq.
Impetus Aq
F ig u re 14. A log-log plot of the maximal radiation amplitude at x = —50 for the kicked initial conditions ((x , 0) = (1(x), ( t (x, 0) = Aqwb nB(x) as a function of Aq. Note th at for small Aq and H < H2 the power law decay rate is universal.
two dom inant frequencies, w = 2 and w = 3wB . T he presence of th e peak a t 3wB is n a tu ra l, since th is is th e first harm onic above th e m ass threshold. T he peak at w = 2 originates from near-threshold bulk modes, excited by th e initial conditions via th e nonlinearities, which disperse only slowly away from th e bo u n d ary [21].
A n o ther te s t of th e scenario is to consider th e rad iatio n from a “kicked” b o u n d ary initial condition ( ( x , 0) = (1(x), ( t (x, 0) = A0wB (x) in th e far field zone. O ur num erical results for th is case are presented in figures 13 and 14.
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J H E P 0 5 ( 2 0 1 7 ) 1 0 7
F igure 13 shows th e H -dep en d ence of m axim al am p litud e of th e field m easured at x = —50, far away from th e boundary, for th re e small values of th e initial im p etus A0 given to th e bo u n d ary m ode. N ote th a t th e rad iatio n am p litu de drops sharply w hen H crosses H2 and H 3, as predicted by our general considerations.
F igure 14 shows a log-log plot of th e dependence of th e m axim al am p litude at x = —50 on A0. For small values of A0 and H < H 2, all curves have th e sam e slope, fittin g th e expected ~ A2 dependence. T he curve for H = 0.95 shows a significant redu ction in th e rad iatio n am plitude, reflecting th e loss of a decay channel as H passes H 2. However its slope for small values of A0 app ears to be relatively unchanged from th a t of th e previous curves, even th o u g h our previous considerations based on th e p ro pagatio n of th e th ird harm onic would suggest an ~ Aj] dependence. It m ay be th a t slow (near-threshold) bulk m odes, visible in figure 12(d) in th e peak at w = 2, are obscuring th e effect we are looking for. It is possible th a t this could be teste d by w aiting significantly longer before m easuring th e rad iatio n , to allow th e slow m odes to die away, b u t a m ore-detailed stu d y would be needed to draw a clear conclusion.
A n o ther interesting feature visible on each curve is th a t as A0 reaches some (curve- dep end ent) critical value, th e rad iatio n flux suddenly dips. As will be discussed in th e next section, th is effect is associated w ith th e nonlinear effect of th e reduction in th e frequency of th e b o u n d ary m ode w ith increasing am plitude.
Finally, for even larger values of th e intial im p etus we can see a large increase of th e a m plitud e of th e field in th e far zone. T his is a sign ature of a n o n -p ertu rb ativ e effect, th e ex citatio n a t th e b o u n d ary becom ing stro ng enough to destabilise it completely, w ith th e em ission of a kink into th e bulk flipping th e field th ere into th e o th er vacuum . Some fu rth e r observations concerning th is phenom enon are rep o rted in section 9 below.
8 H ig h er-o rd er n o n lin ea r effects and a m p litu d e -d e p e n d e n t d eca y ra tes In th e last section we principally considered b o u n d ary m ode decay in th e sm all-am plitude regim e w here th e b o u n d ary m ode itself could be tre a te d linearly. For larger am plitudes th e frequency of th e m o d e’s oscillation is lowered, ju s t as in th e case of an anharm onic oscillator or th e sim ple pendulum . N um erical sim ulations of th e oscillations of a full-line kink [19] also exhibit th is behaviour, which is typical for m any nonlinear system s.
In th e evaluation of th e critical values H n above, we im plicitly assum ed th a t th e am pli
tu d e of th e ex citatio n was small, so th a t its frequency was th a t predicted by th e linearised equations. However for larger am plitudes, given th e am p litu de-d ep enden t frequency re
du ctio n ju s t described, it is possible th a t even for H < H 2, th e a ctu al frequency of th e b o u n d ary m ode, wB , will be lower th a n m /2 . T h en th e decay ra te will be slower th a n th a t observed for sm aller am plitudes, since th e second harm onic will not couple directly w ith any p ro p ag ating bulk modes. However, th e am p litude of th e bo u n d ary m ode will decrease w ith tim e due to th e outgoing radiation, causing its frequency to grow. Provided H < H 2, once th e am plitu d e has decreased far enough, th e second harm onic will en ter th e scatterin g spectrum . In such a case we can expect to observe an intriguing phenom enon:
while initially th e rad iatio n flux from th e b o u n d ary is relatively small and th e decay rate
J H E P 0 5 ( 2 0 1 7 ) 1 0 7
F ig u re 15. Evolution of the boundary mode for H = 0.8393 < H2 and large initial amplitude A 0 = 0.3. Plots a) and c) show the time evolution of the amplitudes of the boundary mode and radiation field respectively, from the values of the field at x = 0 and x = -5 0 . Plots b) and d) show power spectra at the positions x = 0 and x = -5 0 . The black lines show the power spectra from
^(0,t) for times 0 < t < 1600 (plot b)) and ^( —50, t) for times 200 < t < 1600 (plot d)). The purple and green lines and filled areas show the power spectra for times before and after the transition (0 < t < 750 and 750 < t < 1600 for x = 0; 200 < t < 800 and 800 < t < 1600 for x = —50).
ra th e r slow, after some tim e th ere will be a sudden increase of th e rad iatio n flux and a sw itch to a m uch faster decay rate.
N um erical work confirms th a t th is effect really exists, as can be seen in figure 15 and movie M13, which show th e decay of th e am plitude of th e b o u n d ary m ode. For ab o u t th e first 750 u n its of tim e th e am p litu d e changes very slowly, albeit w ith a small m odulation, after which th ere is a sudden tra n sitio n to a m uch m ore rapid decay. In th e far field zone th is effect can be observed as a sudden ju m p of th e rad iatio n flux, by ab o u t one order of m agnitude.
In th e power sp ectrum p lo tted in figure 15d one can clearly see a large peak ju s t below w = 1, which is th e initial frequency of th e m ode. W hile th e am p litud e slowly decreases th e frequency grows un til it crosses 1, after which point th e decay runs m uch faster. We can also see a d rift of th e frequency up to = 1.24666.
T his slow -then-fast behaviour is rem iniscent of higher-dim ensional oscillon decay.
In [22- 24] it was observed th a t oscillons in two and th re e sp atial dim ensions lose th eir energy very slowly for ten s of th o usands of oscillations until th ey reach some critical fre
quency, above which th ey quickly decay to th e vacuum . 9 C r ea tio n o f k in ks from an e x c ite d b o u n d a r y
T he final phenom enon we investigated was th e creation of kinks from th e m etastab le b o u n d ary. We previously observed th a t th is could be induced in certain scatterin g processes at
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J H E P 0 5 ( 2 0 1 7 ) 1 0 7
large H . To view it in isolation, we instead excited th e b o u n d ary m ode directly, taking initial conditions of th e two types ( “stre tc h e d ” and “kicked” ) used earlier. F irst, we used initial condition
0 (x , 0) = 0 1(x) + AonB (x), 0t (x, 0) = 0 (9.1) w ith nB(x) th e b o u n d ary profile for th e linearised prolem , as an ap proxim ation to the b o u n d ary m ode a t its largest deviation from equilibrium ; and second, we took
0 (x , 0) = ^1 (x), 0 t(x , 0) = AoWbnB(x) (9.2) representing th e “kicked” boundary. As before, we norm alized th e profile of th e b o u nd ary m ode in such a way th a t nB (0) = 1.
In b o th cases, if A0 is tak en to be sufficiently small, th e b o u n d ary oscillates w ith fre
quency wB and th e am p litu d e A0. However, as A0 becomes larger, th e nonlinear processes discussed above s ta rt to play a significant role and fu rth er, as th e initial energy of th e excited m ode becomes sufficient, outgoing kinks can be observed in th e far zone, as seen in figures 16 and 17.
N ote th a t for large H ^ 1 th e b o u n d ary m ode profile resembles th e difference betw een th e unstable bo u n d ary solution (a saddle point of th e energy) and th e stab le b o u ndary solution:
, ta n h (x + Xo) - ta n h (x - Xo)
nB(x) * --- 2 ta n h (X o )--- (9'3) Therefore th e b o u n d ary m ode, w ith ap p ro p riate am plitude, being added to th e static b o u n d ary solution yields th e u n stab le boundary. W hen th e solution crosses th e saddle point of energy it decays into a n o th e r sta tic solution w ith an additional kink is em itted from th e boundary.
Therefore th e critical value of th e am plitude of th e b o u n d ary m ode for th e production of th e kinks is:
Acrit = 02 — 01 = 2V1 — H . (9.4) T his critical am p litud e is in very good agreem ent w ith th e first ty p e of th e initial conditions for positive values of A0. Only for very small values of A0 is th ere a sym m etry A0 ^ — A0. For larger A0 > 0 th e ex citatio n have less energy th a n th e excitation for — A0. Therefore th e critical line for kink creation, from th e left side of th e plot, is m uch closer to th e centre (A0 = 0).
For initial conditions of th e second type, th e energy for A0 and —A0 is exactly th e sam e and therefore th e plots look m uch m ore sym m etric. For H ^ 1 th e critical am p litud e is alm ost exactly Acrit = \/1 — H , h alf as big as in th e first case.
10 C o n c lu sio n s
O ur investigations of th e b o u n d ary 0 4 th eo ry have shown th a t it offers a considerably richer variety of resonance phenom ena th a n th e bulk theory, w ithin a settin g w here analytical progress can be m ade. K ey features include th e m odification of th e force leading to th e sharpen in g of window boundaries and th e new critical velocity vcr, th e resurrection of th e
J H E P 0 5 ( 2 0 1 7 ) 1 0 7
F ig u re 16. The field value on the boundary at time t f = 50 for initial conditions ((x , 0) = ( B (x) + AQnB (x), ( t (x, 0) = 0. The blue colour represents the region without kink creation. The dashed line corresponds to the function H = 1 — A2/4.
F ig u re 17. The field value on the boundary at time tf = 50 for initial conditions ((x , 0) = ( B (x), ( t (x, 0) = AqwbnB(x). The blue colour represents the region without kink creation. The dashed line corresponds to the function H = 1 — A2.
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J H E P 0 5 ( 2 0 1 7 ) 1 0 7
first ‘m issing’ scatterin g window, th e observation of th e b o u n d ary oscillon, and th e collision- induced decay of th e m etastab le b o u n d ary vacuum for H n ear to 1. M uch of our work has been num erical and m any issues rem ain for fu rth e r study, th e m ost pressing being th e developm ent of a reliable m oduli space approxim ation in corpo ratin g th e bo u n d ary degrees of freedom (see [9] for some earlier work on this issue). This m odel is sufficiently simple th a t it should offer th e ideal playground for th e developm ent of b e tte r an aly tical techniques for th e un d erstan d in g of m ore general nonintegrable field theories.
A c k n o w le d g m e n ts
We would like to th a n k P io tr Bizoń, R o b ert P arin i and W ojtek Zakrzewski for discussions.
T he work of P E D was su p p o rted by an ST FC C onsolidated G ran t, S T /L 0 00 40 7/1 , and by th e GATIS M arie C urie F P 7 netw ork (gatis.desy.eu) u nd er R E A G ran t A greem ent No 317089. AH th an k s th e BSU S tu den t G ran t P ro g ram for su p p o rt, and YS th an k s th e R ussian F o und atio n for Basic R esearch (G ran t No. 16-52 -12012), D FG (G ran t LE 838/12-2) and JIN R Bogoljubov-Infeld P rogram m e.
A N u m e r ic a l m e th o d s
In th is app en d ix we describe some details of th e num erical m eth ods used in our sim ulation.
In our num erical code we used th e following discretization:
u n = 0 ( —n h ), n = 0 . . . N . (A .1) To calculate sp atial derivatives we used a fo u rth-order central difference scheme
D 2Un = 12h2 ( _ Un-2 + 16u n-1 — 30un + 16u n+1 — Un+2) (A .2) for all points far enough from th e boundary, n < 2. T his scheme can be derived using Lagrange polynom ial approxim ation:
n+m n+m . 7
u (x ) = ^ u ^ ( x ) , ^ ( x ) = ^ ih _ j ' h . (A.3)
i= n-m j= n-m
j=i
However for th e two points closest to th e b o u n d ary we have to use a different basis 3
u(x ) = H lo (x ) + ^ u iiiri (x), (A.4)
i=0 where
2 3
M ( x ) = ( ih)2 n i h - f . (A -5)
j= i
« * ) = ( 1 + 1 1 * ) n x _ j h h . (a .6 )
J H E P 0 5 ( 2 0 1 7 ) 1 0 7
« « ( * )= x n (A.7)
j= 1 j
N ote th a t for points w ithin th e in terpo lation intervals
4 ( x j ) = / j ( x j ) = , ^(xi) = 0, Ę(0) = 0, / ( 0 ) = 1. (A.8) T he above relations prove th a t form ula (A .4) really in terpo lates th e function w ith appro- p riete bo u n d ary condition
From th is ap proxim ation it is straightforw ard to calculate th e second derivative for th e first two points:
B S u p p le m e n ta r y m a teria l
We have p repared a num ber of short movies, labelled M01 . . . M13, to illu strate aspects of our findings, which are listed in th is appendix. T he movies them selves can be found as sup plem entary m aterial.
T he first six movies show th e processes depicted in figure 4 (a )-(f) : M 01_B ndryScattering_H m inus040_v020.m ov: H = —0.4, v = 0.20 < vcr( H )
A lm ost-perfect reflection of th e incident antikink, which has insufficient energy to get over th e saddle-point p o ten tial barrier.
M 02_B ndryScattering_H m inus040_v0333.m ov: H = —0.4, v = 0.333 = vcr( H )
A ntikink incident a t th e critical velocity, leading to th e creation of th e saddle-point configuration. N ote, this movie (and th e associated figure 4(b)) is som ew hat idealised, as in practice it is im possible to tu n e th e initial velocity finely enough to hit th e tru e critical velocity precisely. In stead, we patched to g eth e r an an im atio n up to t = 40 w ith th e sta tic solution th ereafter.
M 03_B ndryScattering_H m inus040_v040.m ov: H = —0.4, v = 0.40 > vcr( H )
Single bounce, w ith subsequent escape of th e antikink. N ote th a t th e acceleration of th e antikink after it surpasses th e p o ten tial b a rrier is clearly visible.
M 0 4_ B n dryS cattering_H plus090_v035.m ov: H = 0.9, v = 0.35
Single bounce, w ith ex citatio n of b o th th e H > 0 b o u n d ary m ode and th e internal m ode of th e antikink.
M 0 5_ B n dryS cattering_H plus090_v037.m ov: H = 0.9, v = 0.37
Single bounce exciting th e b o u n d ary m ode strongly enough to induce decay of th e m etastab le bo u n d ary sta te , creating an ad ditio nal kink in th e bulk.
u (n h ) = u n , n < 3 and u ;(0) = H . (A.9)
(A.10) (A.11)
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J H E P 0 5 ( 2 0 1 7 ) 1 0 7
D 2u 0 = — ^ 1 2 (85u0 + 6 6 h H — 108u1 + 27u2 — 4u 3) , D 2u 1 =
-^2
(29u0 + 6h H — 54u1 — 2u3 + 27u2) .M 06 _B ndryS cattering_H plus090_v039.m ov: H = 0.9, v = 0.39
A sim ilar process to M5, b u t here th e relative velocities of th e em itted kink and an tik ink are such th a t a bulk oscillon is form ed in stead of a sep arated kink-antikink pair.
T he next six movies scan th ro u g h a range of velocities a t co n stan t H . Movies M07-M10 show
‘tra d itio n a l’ window form ation as in th e full-line case (equivalent to M09). T he sharpened edges of th e windows for H < 0, caused by th e presence th ere of a p o ten tial barrier, are visible on careful com parision of M7 and M8 (for H < 0) w ith M9 and M10 (for H > 0). For movies M11 and M12, H is in th e region w here collision-induced b o u n d ary decay is possible.
M 07_V elocityScan_H m inus040.m ov: H = - 0 .4 M 08_V elocityScan_H m inus020.m ov: H = - 0 .2 M 09_V elocityScan_H 000.m ov: H = 0
M 10_V elocityScan_H plus020.m ov: H = 0.2 M 11_V elocityScan_H plus090.m ov: H = 0.9 M 12_V elocityScan_H plus095.m ov: H = 0.95
Finally, movie M13 shows th e slow -then-fast relaxation of th e b o u n d ary m ode. Four plots are shown: on th e left, th e field values next to, and slightly fu rth e r from, th e b o u n dary;
on th e right, th e am plitu d e of th e field a t th e boundary, and an e stim ate w (0 ,t) := 2 n /T of its in stan tan eo u s frequency, w here T is th e tim e betw een successive m inim a of 0(0, t).
M 13_R elaxation_H plus08393_A 030.m ov: H = 0.8393, A0 = 0.3
O p e n A c c e s s . This article is d istrib u ted under th e term s of th e C reative Com m ons A ttrib u tio n License (CC -B Y 4.0) , which perm its any use, d istrib u tio n and reprodu ction in any m edium , provided th e original au th o r(s) and source are credited.
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