Nonlinear response of the sine-Gordon breather to an ac driver

PHYSD2598Physica D 2598 (2000) 1–12**Nonlinear** **response** **of** **the** **sine**-**Gordon** brea**the**r **to** **an** a.c. **driver**Abstr**ac**tK. Forinash a,∗ , C.R. Willis ba Division **of** Natural Sciences, Indi**an**a University Sou**the**ast, 4201 Gr**an**t Line Road, New Alb**an**y, IN 47150-6405, USAb Department **of** Physics, Bos**to**n University, 590 Commonwealth Avenue, Bos**to**n, MA 02215, USAReceived 10 J**an**uary 2000; received in revised form 28 September 2000; **ac**cepted 29 September 2000Communicated by I. Gabi**to**vWe investigate **the** nonlinear **response** **of** **the** continuum **sine**-**Gordon** (SG) brea**the**r **to** **an** a.c. **driver**. We use **an** **an**satz byMatsuda which is **an** ex**ac**t collective variable (CV) solution for **the** unperturbed SG brea**the**r **an**d uses only a single CV, r(t),which is **the** separation between **the** center **of** masses **of** **the** kink **an**d **an**tikink that make up **the** brea**the**r. We show that in**the** presence **of** a **driver** with **an** amplitude below **the** breakup threshold **of** **the** brea**the**r in**to** kink **an**d **an**tikink, **the** a.c.-drivenSG is quite **ac**curately described by **the** r(t), which is a solution **of** **an** ordinary differential equation for a one-dimensionalpoint particle in a potential V(r)driven by **an** a.c. **driver** **an**d with **an** r-dependent mass, M(r). That is below **the** thresholdfor breakup, **the** solution for a driven r(t) **an**d **the** use **of** **the** Matsuda identity gives a solution for **the** a.c.-driven SG, whichis very close **to** **the** ex**ac**t simulation **of** **the** a.c.-driven SG. We use a wavelet tr**an**sform **to** **an**alyze **the** frequency dependence**of** **the** time-dependent nonlinear **response** **of** **the** SG brea**the**r **to** **the** a.c. **driver**. We find **the** wavelet tr**an**sforms **of** **the** CVsolution **an**d **of** **the** simulation **of** **the** a.c.-driven SG are qualitatively very similar **to** e**ac**h o**the**r **an**d **of**ten agree quite wellqu**an**titatively. In cases **of** breakup **of** **the** brea**the**r in**to** K **an**d A, where **the**re is no appreciable radiation **of** phonons, we find**the** CV solution is very close **to** **the** ex**ac**t simulation result. © 2000 Elsevier Science B.V. All rights reserved.PACS: 05.40.+j; 03.40.-t; 74.50.+rKeywords: Sine-**Gordon**; Collective variable solution; Kink-**an**tikink inter**ac**tions1. IntroductionIn this paper, we use a collective variable (CV) appro**ac**h**to** investigate **the** nonlinear **response** **of** a continuum**sine**-**Gordon** (SG) brea**the**r **to** **an** a.c. **driver**which satisfies **the** differential equationφ ,tt − φ ,xx + V ,φ = 0,V(φ)≡ Γ0 2 (1 − cos φ)εf (t)φ (1.1)∗ Corresponding author.E-mail address: kforinas@ius.edu (K. Forinash).0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S0167-2789(00)00194-9where f(t)is **the** a.c. **driver**, ε **the** strength **of** **the** **driver****an**d Γ0 2 dimensionless. The CV, r(t), is **the** separationbetween **the** center **of** mass **of** **the** kink (K) **an**d **the** **an**tikink(A) that make up **the** brea**the**r. The equation **of**motion for **the** one-dimesional “particle” r(t) is givenby New**to**n’s law for a particle whose mass, M(r),depends on r moving in a potential U(r) derived in[1] **an**d driven by **an** a.c. **driver**, see Eq. (2.4). In **the**absence **of** **the** a.c. **driver** **the** solution, r(t), when insertedin**to** **the** Matsuda identity [2] gives **the** rigoroussolution **of** **the** SG brea**the**r. When we add **the** a.c.**driver** **to** **the** r(t) equation **of** motion, solve for r(t),UNCORRECTED PROOF

PHYSD25982 K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12**an**d substitute in**to** **the** Matsuda identity we obtain **an**approximate solution for **the** SG brea**the**r in **the** presence**of** **the** a.c. **driver**. The solution in **the** presence **of****the** **driver** is only approximate because **the** identity isonly ex**ac**t for **the** undriven brea**the**r. However, as weshow in Section 3, **the** approximate CV solution for**the** brea**the**r is very close **to** **the** ex**ac**t simulation resultfor **the** a.c.-driven SG for **driver** amplitudes below **the**amplitude for **the** breakup **of** **the** brea**the**r in**to** K **an**dA. In some cases even at breakup **of** **the** brea**the**r in**to**K **an**d A **the** CV solution is very close **to** **the** ex**ac**tsimulation result. Consequently we are able **to** obtainquite-detailed knowledge **of** **the** nonlinear **response** **of****the** SG solution **to** **an** a.c. **driver** by **an**alyzing **the** solution**of** a one-dimensional ordinary differential equationfor r(t) which yields **the** ex**ac**t brea**the**r solutionwhen **the** a.c. **driver** is not present.The frequencies that appear in r(t) due **to** **the** nonlinearparticle potential, U(r), agree very well qualitatively**an**d qu**an**titatively with **the** simulation **of** **the**full a.c.-driven SG equation. Also **the** amplitude **of** **the****driver** required for **the** breakup **of** **the** brea**the**r in**to** K**an**d A agree qualitatively **an**d fairly well qu**an**titatively.For **the** r**an**ge **of** **driver** amplitudes **an**d frequenciesused in this paper we find **the** intensity **of** **the** phononsradiated **to** be small even in **the** case **of** breakup **of** **the**brea**the**r in**to** K–A. The main reason that phonon radiationis small is that r(t) has **to** be driven nonlinearlyaway from its (nonlinear) motion in **the** undriven casebefore **the**re c**an** be radiation **of** phonons. Below **the**threshold for breakup **the** radiation intensity, I = ε 2m ,where m>1 **an**d ε is **the** dimensionless **driver** amplitudewhich we take in this paper **to** be in **the** r**an**ge0.04 ≤ ε ≤ 0.2. Near breakup only a linear deviationfrom **the** undriven r(t) is needed **to** cause **the** radiation**of** **the** phonons observed in **the** simulations.We compare **the** frequency spectrum **of** **the** CV approximationwith **the** ex**ac**t simulation **of** **the** drivenSG by using a wavelet description which enables us**to** see **the** frequencies that appear in both **the** CV approximation**an**d **the** simulation **of** **the** full a.c.-drivencontinuum SG equation at different times in **the**ir evolution.For example we find that below **the** thresholdfor breakup both **the** CV approximation **an**d **the** ex**ac**tsimulation show a nonlinear frequency “beating” between**the** **driver** **an**d **the** brea**the**r which shows up in**the** phonon b**an**d. We also find that in all cases when**the** ex**ac**t simulation undergoes breakup in**to** aK**an**dA **the** CV approximation also breaks up in**to** aK**an**dA.The nonlinear **response** **of** **the** undamped a.c.-drivenSG brea**the**r in this paper is time dependent **an**d containsa spectrum **of** frequencies with variable amplitudeswhich depend sensitively on **the** amplitude **an**dfrequency **of** **the** **driver**. The nonlinear **response** contrastswith **the** damped a.c.-driven SG [3] where **the**main **response** is a const**an**t shift (modulation) **of** **the**undriven brea**the**r frequency which depends on **the****driver** amplitudes, **driver** frequency **an**d on **the** fixedvalue **of** **the** energy **of** **the** kink. The fixed energy is determinedby bal**an**cing **the** energy loss due **to** dampingwith **the** energy gain from **the** **driver**. In Section 2, wederive **the** equations **of** motion for r(t).We compare**the** results for **the** CV approximation with **the** simulation**of** **the** a.c.-driven SG in Section 3. In Section 4,we summarize **an**d conclude **the** paper.2. CV equations **of** motionWe first briefly review **the** results for **the** rigoroussingle CV **the**ory **of** **the** brea**the**r **an**d **the** K–A system.The **an**alytic solution for **the** brea**the**r is[ ]φ b (x, t) = 4 t**an** −1 Γb cos(ωt − θ), (2.1)ω b cosh(Γ x)where **the** center **of** mass **of** **the** brea**the**r is located atzero, θ determines **the** initial phase, ω b is **the** brea**the**rfrequency **an**d Γ b < 1 obeys **the** dispersion law ωb 2 +Γb 2 = 1. From Matsuda’s identity for **the** brea**the**r [2],we define r(t) byr(t) ≡ 2 [ ]sinh −1 Γbcos(ωt − θ) . (2.2)Γ b ω bWhen we substitute Eq. (2.2) in **the** **an**alytic solution,(2.1), **an**d use **the** following identity t**an** −1 y −t**an** −1 z = t**an** −1 [y − z/(1 + yz)], we obtainφ(x,t)= 4 t**an** −1 e Γ(x−r(t)/2)−4 t**an** −1 e Γ(x+r(t)/2) , (2.3)UNCORRECTED PROOF

PHYSD2598where here **an**d in **the** remainder **of** **the** paper we drop**the** subscript b on Γ , φ **an**d ω. Eq. (2.3) is also **the**ex**ac**t solution for K–A when we repl**ac**e Γ by γ ≡(1 − 4ṙ2 1 ) −1/2 **an**d r(t) is given by[ ] 2r(t) = 2γ −1 sinh −1 γ ṙtsinh .ṙ 2For details, see [1]. The close relationship between **the**ex**ac**t brea**the**r **an**d **the** K–A solution follows from **the**f**ac**t that **the** brea**the**r is a bound state **an**d **the** K–Ais **an** unbound state **of** **the** “same” potential U(Γ,r),where Γ>1 for K–A **an**d Γ 1 is substitutedin Eq. (2.1) we get **the** ex**ac**t K–A solution. Nextwe add **the** a.c. **driver** whose potential is V ext (r) ≡4εf (t) ∫ ∞−∞ dx{ t**an**−1 Γ(x−2 1 r)− t**an**−1 Γ(x+2 1 r)},where f(t)= ε sin Ωt.Note that dV ext /dr =−16εΓf (t) **an**d dV ext /dΓ =0. Thus **the** a.c. **driver** drives r(t) directly. However,even if Γ was treated as a dynamical variable **the** a.c.**driver** would not drive Γ directly, i.e., Γ(t) wouldexperience **the** **driver** only through **the** **driver**’s effec**to**n r(t).K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12 33. Comparison **of** CV approximation withsimulationIn this section, we **an**alyze some typical cases **of** **the**a.c.-driven SG for **the** full SG partial differential equation(S) **an**d for **the** CV description determined by **the**solution r(t) **of** **the** ordinary differential equation (2.4)substituted in Eq. (2.3). Our purpose is **to** first examine**the** time-dependent nonlinear **response** **of** **the** SGbrea**the**r **to** **the** a.c. **driver** **an**d second **to** show how **the**much simpler CV description which is completely determinedby **the** solution r(t) **of** Eq. (2.4), provides uswith information that appreciably increases our underst**an**ding**of** **the** nonlinear **response** **of** **the** SG brea**the**r**to** **an** a.c. **driver**. In order **to** compare **the** CV calculationwith **the** full simulation equations (S) **of** **the**SG brea**the**r **the**y were both iterated with a st**an**dardfifth-order Runge–Kutta method [4] **an**d **the**ir outputcompared. Typically in **the** absence **of** a **driver** with atime step **of** 0.01 **the** energy is conserved **to** **an** **ac**cur**ac**y**of** 0.001% over a whole simulation which lastedon **the** order **of** 300 time units. With a **driver** **the** energyis not expected **to** be conserved ex**ac**tly, however,in all cases **the** energy fluctuated around a const**an**t averageenergy which did not increase or decrease signific**an**tlyover **the** length **of** **the** simulations. Differentinitial phase conditions were examined **an**d found **to**affect mainly **the** time needed until breakup (drivingin reson**an**ce was extremely sensitive). To avoid questionsrelated **to** phase **the** **driver** was turned on veryslowly (over 10–20 time units) in most cases exceptwhere noted.We find that a very useful technique **to** present frequency**an**d time-dependent information in our simulationsis **to** use a wavelet tr**an**sform [5]. The underlyingidea **of** **the** wavelet tr**an**sform used in this paper issimilar **to** that **of** **the** “windowed” Fourier tr**an**sformdefined byTf(t, ξ) =∫ ∞−∞f(x)g(x− t)e −i2πξx dx (3.1)which is **the** Fourier tr**an**sform with a window functiong inserted, which limits **the** calculation **of** frequency **to****an** interval **of** time around time t. (The window functiong c**an** be chosen **to** be a Gaussi**an**, which results inUNCORRECTED PROOF

PHYSD25984 K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12a Gabor tr**an**sform.) The wavelet tr**an**sform we use differsfrom this tr**an**sform in that **the** “window” ch**an**geswidth as **the** frequency ξ is ch**an**ged. For high frequencies,**the** window is narrow (which will catch rapidch**an**ges in frequency better); for low frequencies, **the**window is wider (which localizes frequency better).The continuous wavelet tr**an**sform is well suited forlocalizing frequency **an**d for localizing time whereas**the** windowed Fourier tr**an**sform c**an** **of**ten localize onedomain well but not **the** o**the**r.An inherent limitation **of** **the** tr**an**sform restricts howwell time **an**d frequency c**an** simult**an**eously be localizedwith this method. In a fashion ma**the**maticallysimilar **to** **the** uncertainty principle, both time **an**d frequencydomains c**an**not be made arbitrarily small. Thislimitation m**an**ifests itself in our graphs as a trade-**of**fbetween how “blurry” **the** image is vertically (i.e., localization**of** frequency), **an**d how blurry **the** image ishorizontally (i.e., localization **of** time).Wavelet tr**an**sforms were performed on **the** amplitudeoscillations **of** **the** central site **of** **the** brea**the**r as**the** brea**the**r evolved in time. The continuous wavelettr**an**sform **of** Grossm**an** **an**d Morlet was used which,for a function f on **the** line, is defined by∫ ∞Wf(t, ξ) = f(x)ω(ξ(x− t))ξ dx,−∞( ) 1ω(x)=σ √ e i2πx e −x2 /2σ 2 . (3.2)2πThe tr**an**sform Wf(t, ξ) shows information about **the**strength **of** **the** signal f at time t **an**d frequency ξ. Intensityplots (**of**ten called spectrograms) **of** frequencyversus time are rendered as graphs showing **the** intensity**of** **the** frequencies present as a function **of** time.The graphs are arbitrarily scaled **to** fit **the** r**an**ge **of** intensityvalues.We start with a comparison **of** **the** full SG, Eq. (1.1),with **the** CV approximation using **the** solution **of**Eq. (2.4) for r(t) for some typical values **of** **the** parameters.Although we examined a wide r**an**ge **of** brea**the**r**an**d **driver** frequencies in this paper we present only**the** ω = 0.9 **an**d 0.7 results because as **the** brea**the**rfrequency is decreased **the** discreteness **of** **the** simulationsstarts **to** become import**an**t **an**d **the** radiation**of** phonons due **to** **the** Peirels–Nabarro potential **the**ninterfere with **the** radiation **of** phonons caused by **the**nonlinearity in r(t) produced by **the** **driver**.In Fig. 1 we compare **the** shape **of** **the** SG brea**the**rin **the** simulation **of** **the** a.c.-driven SG brea**the**r (solidline) **an**d for **the** CV approximation **of** **the** a.c.-drivenSG (dotted line). Note that **the** CV approximationagrees with S in **the** diagram very closely everywherebut in **the** wings **of** **the** line. We found this agreement ismaintained for long periods **of** time in all cases belowthreshold for breakup. In Fig. 1 **the** **to**tal number **of**brea**the**r oscillations at this point has been 42 oscillationsfor **the** simulation **an**d 43 oscillations for **the** CVcalculation. The reason for **the** oscillation in **the** wings**of** **the** simulation S but not in **the** CV is because **the**CV approximation does not possess phonon degrees**of** freedom but **the** simulation S does. The oscillationhas a frequency **of** approximately 1.08 (where **the**phonon b**an**d edge has a frequency **of** 1 in our units)which is very close **to** **the** SG brea**the**r quasimode (see[6]). The quasimode is caused by **the** oscillation **of** **the**shape **of** a kink or a brea**the**r **an**d since **the** frequencyis in **the** phonon b**an**d near **the** b**an**d edge it is a quasimode**an**d not a pure mode. The corresponding modein **the** φ 4 equation is in **the** gap **an**d is a pure mode.We find that **the** quasimode is produced at t = 0inS because when **the** a.c. **driver** is applied at t = 0it causes **an** inst**an**t**an**eous ch**an**ge in **the** shape **of** **the**brea**the**r which **the**n oscillates **an**d radiates phonons **of**frequency **of** 1.08. Thus we find that CV **an**d S areremarkably close **to**ge**the**r **an**d differ at long times because**of** **the** quasimode radiation **of** phonons. If **the** a.c.**driver** is turned on sufficiently slowly **the** quasimodewill not be excited **an**d S **an**d CV will remain close **to**e**ac**h o**the**r as long as **the** a.c. **driver** is well below **the**threshold for breakup. In some cases (see Fig. 3a **an**db) we find S radiates phonons due **to** **the** inter**ac**tion**of** **the** a.c. **driver** with **the** nonlinear brea**the**r oscillationswhich are not related **to** slope oscillations. Nearbreakup **of** **the** brea**the**r in**to** K **an**d A for appreciable**driver** amplitudes **the**re c**an** be instability, chaotic behavior**an**d sometimes copious radiation **of** phononswhich for **the** full simulation may lead **to** a breakupin**to** multiple brea**the**rs ra**the**r th**an** K–A. In most **of****the** cases we investigated at breakup **the** phonon radiationis weak **an**d **the**re is a strong similarity betweenUNCORRECTED PROOF

PHYSD2598K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12 5Fig. 1. Simulation **an**d CV calculation **of** **the** brea**the**r shape captured at maximum amplitude shortly before 300 time units. The brea**the**rfrequency is 0.9, **the** **driver** frequency is 0.7 **an**d **the** **driver** amplitude is 0.02 in dimensionless units.S **an**d CV breakdown when **the** a.c. **driver** has a smallamplitude.The amplitude **of** **the** **driver** required for breakupin**to** K **an**d A as a function **of** **the** **driver** frequency ispresented in Fig. 2. Except at very low driving frequencyΩ = 0.1 **an**d reson**an**ce Ω = 0.9 **the** thresholdamplitude for breakup for S **an**d CV are close **to**e**ac**h o**the**r. When **the** driving frequency is reson**an**twith **the** brea**the**r frequency **the** agreement with S **an**dCV breaks down because at reson**an**ce a large number**of** phonons are radiated. It is somewhat surprisingthat **the** agreement between S **an**d CV at breakup awayfrom **the** reson**an**ce is so good. Different initial phaseswere tried **an**d found only **to** affect **the** length **of** timeuntil breakup. Fig. 2 is for in-phase initial conditions.In Fig. 3a **an**d b we show **the** breakup **of** **the** brea**the**rin**to** K–A for S **an**d for CV for Ω = 0.4, ω = 0.7 **an**ddriving amplitude **of** 0.1. In **the** time before breakupS **an**d CV agree very well **an**d **the** times **of** breakupdiffer by only seven time steps. After breakup K **an**dA recede from e**ac**h o**the**r at **the** same speed **an**d K**an**d A e**ac**h oscillate at **the** driving frequency. However,it is clear that **the**re is a difference in **the** speed**of** **the** K (**an**d A) in S which is 0.17 **an**d **the** speed **of****the** K (**an**d A) in CV which is 0.70. The reason for**the** difference speeds in S **an**d CV is that at breakupS radiates phonons while **the** CV solution c**an**not radiatephonons. The **to**tal energy is approximately **the**same in both cases **an**d **the** **to**tal rest energy is **the** samein both cases. Thus **the** energy radiated by phononsin S shows up as higher kinetic **of** K **an**d A in **the**CV calculation. An examination **of** **the** amplitude atsmaller scales shows phonons present in **the** simulationcase. Most breakups were in**to** K–A pairs but oc-UNCORRECTED PROOF

PHYSD25986 K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12Fig. 2. Driving amplitude needed **to** cause breakup within 300 time units with various **driver** frequencies for a brea**the**r with a frequency**of** 0.9, **driver** on from **the** beginning. Simulation values are denoted by ×, CV calculation by +.casionally **the** simulation broke up in**to** two smallerbrea**the**rs.In Fig. 4a **an**d b we show **the** wavelet tr**an**sform **of** abrea**the**r with ω = 0.9, Ω = 0.6 **an**d amplitude = 0.1which is well below breakup. The **driver** is turned ongradually over 10 time units starting at t = 50. We seethat **the**re are frequencies above **the** b**an**d edge around1.15 in both S **an**d CV. These frequencies arise from**the** nonlinear brea**the**r oscillation driven by **the** a.c.**driver**. In **the** simulation **the**se frequencies lead **to** radiation**of** phonons. A beating between ω **an**d Ω c**an**be seen in both case with a time duration **of** about 10time units. (The small variations in intensity **of** approximatelyfive time units are **an** artif**ac**t **of** **the** calculation**an**d are not physical.) There is a faint intermediatefrequency between **an**d just below **the** frequency0.8 in both S **an**d CV. Also **the**re is a faint frequency at**the** b**an**d edge in both S **an**d CV. The CV calculationshows **the** brea**the**r frequency shifts downward slightlywhile **the** simulation does not because **the** simulationis radiating phonons which causes **the** frequency **to**rise thus c**an**celing **the** drops in frequency.In Fig. 5a **an**d b we show **the** wavelet tr**an**sformwith ω = 0.9, Ω = 0.8 **an**d amplitude = 0.1 which isjust above breakup for S **an**d just below breakup forCV with **the** **driver** turned on gradually over 10 timeunits starting at t = 50. The frequency shifts down **to****the** **driver** frequency in both S **an**d CV. At t = 150 **the**breakup **of** S in**to** K **an**d A is complete. In **the** timebetween t = 50 **an**d 150 **the** agreement between S**an**d CV for frequencies at 0.9 **an**d below is very goodwith b**an**ds around 0.8, between 0.6 **an**d 0.7 **an**d be-UNCORRECTED PROOF

PHYSD2598K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12 7Fig. 3. Brea**the**r breakup in**to** kink–**an**tikink pair (simulation (a), CV calculation (b)). Driving frequency is 0.4, brea**the**r frequency is 0.7**an**d driving amplitude is 0.1. Speed **of** **the** kink (or **an**tikink) after breakup is approximately 0.17 for **the** simulation **an**d 0.70 for **the** CVcalculation.UNCORRECTED PROOF

PHYSD25988 K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12Fig. 4. Wavelet tr**an**sform **of** **the** central location **of** **the** brea**the**r with initial frequency 0.9, **driver** frequency 0.6 **an**d amplitude 0.1 (wellbelow breakup). The wavelet tr**an**sform **of** **the** simulation is shown in (a), **the** CV calculation in (b). The small variations in intensity **of**approximately five time units are **an** artif**ac**t **of** **the** calculation **an**d are not physical.tween 0.4 **an**d 0.5. Since **the** CV solution c**an**not radiateor have shape mode oscillations we c**an** reasonablyconclude that **the** frequencies in **the** phonon b**an**din **the** CV solution arise from **the** inter**ac**tion between**the** **driver** frequency **an**d **the** frequencies which arisefrom **the** difference between **the** undriven **an**d drivenr(t). The frequency spectra **of** **the** simulation **an**d **the**CV calculation are nearly identical (as was **the** casefor all wavelet tr**an**sforms for cases below breakup).As a final example **of** **the** use **of** **the** CV, r(t),**to**increase our underst**an**ding **of** **the** a.c.-driven SG weconsider **the** cases with **the** same driving frequencyΩ = 0.7, brea**the**r frequency ω = 0.9 **an**d threedifferent but close driving amplitudes **of** 0.119765,0.119768 **an**d 0.119770 shown in Fig. 6. In all threecases **the** SG brea**the**r breaks up in**to** K **an**d A. In mostcases we examined **the** breakup occurs smoothly asin Fig. 6a. In Fig. 6b (which differs by only a rela-UNCORRECTED PROOF

PHYSD2598K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12 9Fig. 5. Wavelet tr**an**sform **of** **the** central location **of** **the** brea**the**r with initial frequency 0.9, **driver** frequency 0.8 **an**d amplitude 0.1 (right atbreakup for simulation (a), just below breakup for CV calculation (b)). The small variations in intensity **of** approximately five time unitsare **an** artif**ac**t **of** **the** calculation **an**d are not physical.tive percentage **of** **the** **driver** amplitude from Fig. 6a**of** 0.0025%), we see irregular behavior for **an** appreciabletime before breakup. In Fig. 6c (which differsonly by a relative percentage ch**an**ge **of** **the** **driver** amplitudefrom Fig. 6b **of** 0.0017%), we see **the** development**of** a multifrequency quasiperiodic modulation **of****the** brea**the**r frequency which might be signaling **the**onset **of** chaotic behavior. The three very different resultsfor three nearly equal **driver** amplitudes demonstratevery effectively **the** effic**ac**y **of** using **the** CV,r(t), **to** underst**an**d **the** a.c.-driven SG in **the** breakupregion.The cases we discussed in this section are typical**an**d are a representative selection from **the** m**an**y caseswe simulated. In **the** region away from breakup **the**agreement is very good. The only signific**an**t qualitativedifference is that for S phonons c**an** be radiatedwhich only has **an** appreciable effect for cases nearbreakup. For cases **of** breakup where phonon radiationis low **the**re is good agreement between S **an**d CV.UNCORRECTED PROOF

PHYSD259810 K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12Fig. 6. Plot **of** r(t) showing brea**the**r breakup in**to** K–A pair for **the** driving frequency 0.7, brea**the**r frequency 0.9 **an**d three differentamplitudes. (a) Driver amplitude = 0.119765; (b) **driver** amplitude = 0.119768; (c) **driver** amplitude = 0.119770. In **the**se cases **the** **driver**was turned on gradually over 20 time units starting at t = 10.UNCORRECTED PROOF

PHYSD25984. Discussion **an**d conclusionsPreviously **the** authors **of** [7] investigated **the**breakup **of** **the** SG brea**the**r in**to** K **an**d A by atime-independent const**an**t force **an**d obtained goodagreement between **the**ir **the**ory **an**d **the**ir numerical results.In a comprehensive review paper on “Soli**to**ns innearly integrable systems” [8] **the** authors carried outextensive investigations **of** several problems includingkink–**an**tikink inter**ac**tions **an**d brea**the**r dynamics usinggeneral perturbation induced evolution equationsfor **the** parameters where **the** time dependence **of** **the**parameters arises solely from **the** perturbation. Theyobtained results for a r**an**ge **of** problems from breakup**of** **the** brea**the**r in**to** kink **an**d **an**tikink **an**d for caseswith external damping **an**d driving terms, including**the** results **of** [7]. The most fundamental difference **of**our appro**ac**h from **the**irs is that **the** CV, r(t), has timedependence even without a perturbation that gives **the**rigorous **an**alytic solution, using **the** Matsuda identity,for **the** brea**the**r solution **to** **the** SG partial differentialequation. Any external perturbation only produces **ac**h**an**ge in **the** time dependence **of** **the** solution **of** **the**r(t) equation **of** motion which in turn causes a ch**an**gein **the** brea**the**r time dependence. Consequently **the**time-dependent ch**an**ge in **the** spectra **of** r(t) due **to****the** perturbation directly, using **the** Matsuda identityleads **to** **the** ch**an**ge **of** **the** time-dependent spectra **of****the** SG brea**the**r.The good agreement between S **an**d CV belowbreakup **of** **the** brea**the**r in**to** K **an**d A for **the** parameterr**an**ge we use in this paper indicates **the** dominateeffect on **the** brea**the**r by **the** a.c. **driver** is **to** ch**an**ge**the** dist**an**ce, r(t), between **the** center **of** masses **of** **the**bound K **an**d A that constitutes **the** brea**the**r from **the**value **of** r(t) in **the** undriven brea**the**r. Consequently**the** ch**an**ge in slope, Γ(t), is small or zero in **the**cases we consider. Note **the** spatially independent a.c.**driver** is very different th**an** cases where **the** brea**the**rexperiences a spatially dependent **driver**. In thosecases **the** **driver** **of**ten causes **the** slope **to** oscillate.The reason **the** slope Γ is const**an**t or nearly const**an**tfor **the** a.c. **driver** (away from breakup) as opposed **to****the** spatially dependent **driver** is that if you introduceΓ(t) as a CV **an**d work out **the** rigorous equationsK. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12 11**of** motion for Γ(t) you find that **the** a.c. **driver** doesnot appear in **the** Γ equation **of** motion. The reasonis that **the** brea**the**r shape mode is orthogonal **to** **the**a.c. **driver** (see [5]). Thus **the** only way that Γ(t)c**an**develop a time dependence is through **the** inter**ac**tionwith r(t) which is directly driven by **the** a.c. **driver**.For **the** parameter r**an**ge we use in this paper **the** onlypl**ac**e where **the** Γ slope mode could start **to** oscillateis at **the** breakup **of** **the** brea**the**r in**to** K **an**d A where**the** slope is altered at breakup. Paren**the**tically **the**time dependence **of** **the** slope in Fig. 1 is not directlydue **to** **the** a.c. **driver** but is a consequence **of** **the** initialcondition at t = 0 **of** **the** a.c.-driven SG equation.In [3] **the** authors expressed **the**ir results for **the**damped a.c.-driven SG in terms **of** a const**an**t frequencymodulation. We c**an** also express our undampedresults as a frequency modulation by solvingEq. (2.2) for θ(t) in terms **of** r(t) **an**d **the**n taking **the**time derivative **of** θ(t) which is **the** time-dependentfrequency modulation (which depends only on r(t)**an**d ṙ(t)). In **the** present paper, we are only considering**the** undamped case so **the** frequency modulationis **an** oscilla**to**ry function **of** time with a rich spectra **of**frequencies which appear in our wavelet tr**an**sforms.From **the** wavelet tr**an**sform we see that below breakup**the** frequencies that appear in **the** phonon b**an**d aredue **to** **the** nonlinearity in **the** difference between **the**driven **an**d undriven r(t) caused by **the** a.c. **driver**.The results on **the** a.c.-driven SG suggest a set **of**problems worth pursuing. When damping is added **to****the** SG in **the** underdamped limit **the**re will be a r**an**ge**of** phenomena such as those that appear in **the** treatments**of** **the** underdamped pendulum which would apply**to** **the** r(t) equation **of** motion. Then with **the** use**of** **the** Matsuda identity we could find **the** new brea**the**rbehavior **of** **the** SG that results from **the** correspondingr(t) behavior. A second area **of** study would be **the**effect **of** moderately stronger **driver**s even though **the**agreement would not be as good as **the** weaker **driver**squ**an**titatively but **the** qualitative agreement would stillbe good. The case **of** damped driven SG has beentreated in [9–11] **an**d it would be interesting **to** comparethose results with **the** present appro**ac**h using CV.A fourth region worth studying fur**the**r is driving atfrequencies in reson**an**ce with **the** brea**the**r. Finally **the**UNCORRECTED PROOF

PHYSD259812 K. Forinash, C.R. Willis / Physica D 2598 (2000) 1–12behavior **of** r(t) might be **an** interesting **to**ol for investigating**the** chaotic nature **of** **the** breakup.References[1] C.D. Ferguson, C.R. Willis, Physica D 119 (1998) 283.[2] T. Matsuda, Lett. Nuovo Cimen**to** 24 (1979) 207.[3] P.S. Lomdahl, M.R. Samuelsen, Phys. Rev. A 34 (1986)664.[4] W.H. Press, B.P. Fl**an**nery, S.A. Teukolsky, W.T. Vetterling,Numerical Recipes, Cambridge University Press, Cambridge,1989.[5] W.C. L**an**g, K. Forinash, Am. J. Phys. 66 (1998) 794.[6] R. Boesch, C.R. Willis, Phys. Rev. B 42 (1990) 2290.[7] P.S. Lomdahl, O.H. Olsen, M.R. Samuelsen, Phys. Rev. A29 (1984) 350.[8] Y. Kivshar, B. Malomed, Rev. Mod. Phys. 61 (1989) 763.[9] J.D. Kaup, A. Newell, Phys. Rev. B 18 (1978) 5162.[10] R. Grauer, Y. Kivshar, Phys. Rev. E 48 (1993) 4791.[11] B. Birnir, R. Grauer, Commun. Math. Phys. 162 (1994) 539.UNCORRECTED PROOF