Crossover dark soliton dynamics in ultracold one-dimensional Bose gases D.J. Frantzeskakis1, P.G. Kevrekidis2 and N.P. Proukakis3 1 Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece 2 Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, U.S.A. 3 School of Mathematics and Statistics, University of Newcastle, Merz Court, Newcastle NE1 7RU, United Kingdom 7 Ultracold confined one-dimensional atomic gases are predicted to support dark soliton solutions 0 arising from a nonlinear Schr¨odinger equation of suitable nonlinearity. In weakly-interacting (high 0 density)gases,thenonlinearityiscubic,whereasanapproximatemodelfordescribingthebehaviour 2 of strongly-interacting (low density) gases is one characterized by a quintic nonlinearity. We use n an approximate analytical expression for the form of the nonlinearity in the intermediate regimes a to show that, near the crossover between the two different regimes, the soliton is predicted and J numericallyconfirmedtooscillateatafrequencyofp2/3Ω,whereΩistheharmonictrapfrequency. 3 ] I. INTRODUCTION Regarding the dynamical features of the moving DSs r e intrapped1DBosegases,theaboveworksrevealedthat, h in the absence of other dissipative mechanisms, the soli- Dark solitons (DSs), the most fundamental nonlin- t ton oscillates in the trap with a frequency which differs o ear excitations of the one-dimensional defocusing non- . between the weakly and strongly interacting regimes. In t linear Schr¨odinger (NLS) equation, have been studied in a particular,inthepresenceofaharmonicconfiningpoten- a broad range of physical systems. Apart from the the- m tial of frequency Ω, the study of the cubic (quintic) NLS oretical work, experimental studies on DSs include their predicts such an oscillation frequency to be αΩ, where - observationeitherastemporalpulsesinopticalfibers[1], d α = 1/√2 (α = 1) for the weakly [10, 11, 12, 13, 14] n orasspatialstructuresin bulk mediaandwaveguides[2] (strongly[14,16])interactingBosegas. Thelatterresult o (see also [3] for a review), the excitation of a nonpropa- for the strongly interacting case is identical to the corre- c gating kink in a parametrically-drivenshallow liquid [4], [ sponding one obtained by a full many-body calculation standing DSs in a discrete mechanical system [5], high- [22]. 1 frequency DSs in thin magnetic films [6], and so on. The transition between the weakly and strongly inter- v Recently, DSs have attracted much attention in the 0 actingregimesisusuallycharacterizedbyasingleparam- physics of atomic Bose-Einsteincondensates (BECs) [7], 5 eter, denoted by γ, quantifying the ratio of the average wheredarkmatter-wavesolitonshavealsobeenobserved 0 interaction energy to the kinetic energy calculated with 1 experimentally [8]. Dark solitons in BECs are known meanfieldtheory[19]. Thisparametervariessmoothlyas 0 to be more robust in one-dimensional (1D) geometries the interatomic coupling is increased from values γ 1 7 and at very low temperatures (a regime which is cur- ≪ (weakly interacting regime), to γ 1 (strongly inter- 0 rently experimentally accessible [9]). For that reason, ≫ acting regime); thus, an approximate“crossoverregime” / t the majority of theoretical studies on DSs in BECs have can be identified around γ O(1), as also attained ex- a for simplicity been performed in the framework of the ∼ m perimentally [23]. The value of γ can be controlled ex- one-dimensional (1D) cubic NLS equation, which in this perimentally by various independent parameters, such - context is referred to as the Gross-Pitaevskii equation; d as scattering length, transverse confinement, density, or n the latter, is the commonly adopted mean-fieldtheoretic even modification of the effective mass of the system. o model describing ultracoldweakly-interacting Bose gases Motivated by the investigation of the DS dynamics c intheabsenceofthermalorquantumfluctuations. Many : in the two limiting regimes, in the present work we ob- v ofthe abovementionedtheoreticalstudies havebeen de- tainanalyticalresults,whichareconfirmedbynumerical i votedtotheanalysisofthedynamicalpropertiesofmov- X simulations, for the oscillation frequency of a DS in the ing DS, such as their oscillations [10, 11, 12, 13, 14] and r sound emission [11, 12, 13] in the presence of the ex- crossover regime for a purely 1D system. In particular, a we show that for γ O(1) the DS oscillates with a fre- ternal trapping potential. On the other hand, and in ∼ quency 2/3Ω, which is higher than the one ( 1/2Ω) the same context of the 1D Bose systems, DSs have also been studied in the framework of a quintic NLS equa- pertaining to the weakly interacting Bose gas. Such an p p increaseinthesolitonoscillationfrequencycouldserveas tion [14, 15, 16, 17], a long-wavelength model which has been proposed [15] for the opposite limit of strong in- an additional diagnostic test for the deviationfrom pure teratomic coupling [18, 19]; in this case, the collisional bosonic mean field behaviour. propertiesofthebosonicatomsaresignificantlymodified, The paper is organized as follows: In section II we with the interacting bosonic gas behaving like a system present the generalized NLS model proper and find its of free fermions [20]. Such, so-called, Tonks-Girardeau ground state, as well as its linear (sound waves) and (TG) gases have recently been observed experimentally nonlinear (dark solitons) excitations. Section III is de- as well [21]. voted to the analytical derivation of the DS oscillation 2 frequency and the discussionof relevantnumericalsimu- parameter ψ describes the “wavefunction” of a weakly- lations. Finally,insectionIVwesummarizeourfindings. interacting1DBosegasunderharmonicconfinement,for which Eq. (1) reduces to the 1D Gross-Pitaevskii equa- tion, II. THE MODEL AND ITS ANALYTICAL ∂ψ ~2 ∂2 4π~2 a CONSIDERATION i~ = +V (x)+ 3D ψ 2 ψ. ∂t −2m∂x2 ext m 2πl2 | | (cid:20) (cid:18) (cid:19) (cid:21) ⊥ (3) A. The generalized NLS equation In the opposite limit of strong interatomic coupling, which, rather counter-intuitively, corresponds to the low Importantly for our present analysis, the parameter γ density limit n n˜ , the parameter ψ satisfies the fol- c separating the regimes of weak and strong interatomic lowing quintic N≪LS equation, couplingcanbe re-expressed,fora givensystemconfigu- ∂ψ ~2 ∂2 π2~2 ration,intermsofthe inverseratioofthesystemdensity i~ = +V (x)+ ψ 4 ψ, (4) tosomecriticaldensity. Thisenablesustoperformauni- ∂t −2m∂x2 ext 2m | | (cid:20) (cid:21) fied analysis of all regimes by means of a NLS equation which has been derived by various different methods in withageneralizednonlinearity,fortheparameterψ con- [15, 28]. While the validity of Eq. (4) to discuss co- nected to the density n = ψ 2 of an ultracold confined herence properties of strongly-interacting 1D Bose gases | | 1D Bose gas. This equation takes the form has been questioned [29], the corresponding hydrody- namic equations for the density n and the phase φ (or ∂ψ ~2 ∂2 i~∂t = −2m∂x2 +Vext(x)+Φ(n) ψ, (1) tthioenautnodmericthveelMocaitdyeluvng≡tr∂axnφsf)oramriasitniognfψro=m√thnisexepq(uiφa)- (cid:20) (cid:21) are well-documented in the context of the local density wheremistheatomicmassandVext(x)=(1/2)mωx2x2 is approximation [18, 25]. An equation of the form (4), the external trapping potential (ωx being the axial con- whichexplictly includesthe so-called“quantumpressure fining frequency). term”(~2/2m)∂2ψ/∂x2,shouldhoweveronlybevalidfor The exact form of the nonlinearity Φ(n) valid in both density variations which occur on a lengthscale which is limitingregimesandthecrossoverregioniswell-knownin largerthan the Fermi healing length ξ 1/(πn ), where p the homogeneous hydrodynamic limit. While the func- n = ψ(0)2 is the peak density of th≡e gas at the trap p tionaldependenceofΦ(n)onγ(anditsanalyticalasymp- center.| Toa|voidsuchpotentialcomplications,the physi- totics)areknown[24],itsprecisevaluesinthe“crossover calanalysispresentedinthispaperisonlyconcernedwith region” can only be evaluated numerically. Such inter- the limit of shallow DSs, for which there is only a very mediate values have been tabulated by Dunjko et al. in slow density variation within the Fermi healing length. [18], and subsequently discussed by various authors in Measuring the variables x and t, and the density ψ 2, the local density approximation, see e.g. [25, 26]. Since in units of the Fermi healing length ξ, the time τ| |= 0 we are interested in this “crossover region”, we should ~/mξ2, and the density √2n respectively, Eq. (1) can p thus use an approximate expression for the nonlinearity, be rewritten in the following dimensionless form, which captures both limiting regimes exactly and pro- 1 vides a good approximation for intermediate values. At iu = u +V(x)u+g(n)u, (5) t xx −2 thesametime,however,weareconstrainedinourpresent workbytheneedforarelativelysimpleexpressionwhich where the subscripts denote partial differentiation. In will enable rather involved analytical work to be carried view of the above scalings, the normalized confining po- out. For our purpose, it is thus sufficient to use a some- tential becomes V(x) = (1/2)(ξ/lx)4x2, where lx = whatsimplifiedgeneralizednonlinearityΦ(n)oftheform ~/mω is the harmonic oscillator length in the ax- x ial direction. As the parameter ξ/l is apparently x π2~2 n2 psmall, it is convenient to define the small parameter Φ(n)= , (2) 2m 1+n/n˜c ǫ≡Ω−2/3(ξ/l)4/3 where Ω is a parameter of order O(1), that will be used in the perturbation analysis to fol- whichnonethelesscorrespondstoafairlygoodanalytical low. This way, the external potential is actually a func- approximation [26, 27] to the exact nonlinearity [18]. In tion of the slow variable X ǫ3/2x, and has the form this notation, the critical density approximately mark- V(X) = (1/2)Ω2X2, where≡Ω expresses the trap fre- ing the crossover region is given by n˜ = 4/(π2a ), quency. Finally, the nonlinearity function g(n) (with c 1D where a is the effective 1D “scattering length”. This n= u2 being the normalized density) becomes, 1D | | is defined in terms of the usual three-dimensional s- n2 2√2 1 wl av=e sca~t/tmerωingilsenthgtehhaar3mDovniiac oas1cDilla=torl⊥2l/ean3gDth, iwnhtehree g(n)= 1+n/nc, where nc = π2 a1Dnp, (6) t⊥ransverse dire⊥ction. The weakly and strongly interacting limits discussed p Equation (1) is well-documented in the high-density above respectively arise when the dimensionless param- limit n n˜ (corresponding to γ 1). In this case, the eter γ = 2/(na ) [19] obeys γ 1 or γ 1. In our c 1D ≫ ≪ ≪ ≫ 3 presentnotation,n/n =(π2/2)γ 1, sothatourapprox- resultthatthedensity profileis parabolic,withn (X)= c − 0 imate crossover region actually corresponds to a value n 1[µ V(X)], inthe regimen n andelliptic, with γ =2/π2 0.2. For such a relatively smallvalue of γ, it n−c(X)0=− µ V(X), in the reg≫imecn n . Also, it ≈ 0 0− ≪ c is still reasonable to use Eqs. (1)-(2) [or Eqs. (5)-(6)] to is noticed that from Eq. (9), and for the harmonic trap p describe this physical system. underconsideration,theaxialsizeofthegasis2L,where StudiesofDSsbasedongeneralized1DNLSequations L=√2µ /Ω is the TF radius. 0 first appeared in the literature for homogeneous systems We now consider the propagation of small-amplitude to deal with saturable nonlinearities appearing in the linearexcitations(e.g.,soundwaves)ofthegroundstate, context of nonlinear optics [3]; in this case, the nonlin- by seeking solutions of Eqs. (7)-(8) of the form n = earitybecomescubicatlowdensities,ratherthanathigh n (X) + ǫn˜(x,t) and φ = µ t + ǫφ˜(x,t), where the 0 0 densities which occurs for ultracold pure 1D Bose gases. functions n˜ and φ˜ describe−the linear excitations. In- Morerecently,DSsingeneralized1DNLSequationswere serting this ansatz into Eqs. (7)-(8), to order O(1) we also considered in the BEC context, but only as effec- recover the TF approximation, while to order O(ǫ) we tivetheoriesforweakly-interactingelongated3Dconden- obtain a system of linear equations for the linear excita- sates. These equations contained either a cubic-quintic tions. Assumingplanewavesolutionsofthissystem,i.e., nonlinearity with constant coefficients [30], or a gener- (n˜,φ˜) exp[i(kx ωt)], wereadilyobtainthe dispersion alized non-polynomial nonlinearity depending explicitly relatio∼n ω2 = g˙ n−k2+k4/4, where g˙ (dg/dn) . onthe trap aspectratio [31, 32]. The aim ofthis work is 0 0 0 ≡ |n=n0 This dispersion relation has the form of a Bogoliubov- ratherdistinct,namelytocalculatethesolitonoscillation type excitation spectrum, but with the excitation fre- frequency in the crossoverbetween weakly- and strongly quency ω being a function of the slow variable X. The interacting 1D Bose gases. The validity of our work is speedofsoundislocal,duetothepresenceoftheexternal thus restricted to the pure 1D regime, with the precise potential, and is given by experimental conditions needed for such a crossover dis- cussed in detail in [18, 33]. C = g˙ n , (10) 0 0 Below we apply the reductive perturbation method Note that Eq. (10) showspthat the speed of sound is (see, e.g., [34, 35]), valid in the limit of shallow soli- tons, to analytically obtain the soliton oscillation fre- given by C = √ncn0 for n nc, and C = √2n0 in the ≫ opposite regime n n . quency close to the critical density n , which approxi- c c ≪ Next we analyze the evolutionof the nonlinear excita- mately marksthe crossoverbetween the regimes ofweak tionsontopofthegroundstate,employingthereductive and strong interatomic coupling. perturbation method [34, 35] (see also [11] and [16] for relevant studies in Bose gases). As the system of Eqs. (7)-(8) is inhomogeneous,we introduce a new slow time- B. Ground state, linear and nonlinear excitations variable T = ǫ1/2 t xC 1(x)dx , and the following − 0 − ′ ′ asymptotic expansions for the density n and phase φ, Starting off from the generalized NLS Eq. (5), we use (cid:0) R (cid:1) theMadelungtransformation,u=√nexp(iφ),toobtain n = n (X)+ǫn (X,T)+ǫ2n (X,T)+ , 0 1 2 ··· the following set of hydrodynamic equations, φ = µ t+ǫ1/2φ (X,T)+ǫ3/2φ (X,T)+ .(11) 0 1 2 − ··· nt+(nφx)x =0, (7) Substituting the expansions (11) into Eqs. (7)-(8), we 1 1 obtain the following results: First, to order O(1), Eq. φ +g(n)+ φ2 n 1/2(n1/2) +V(X)=0.(8) t 2 x− 2 − xx (8) leads to the TF approximation [see Eq. (9)]. Then, to the first-order of approximation in ǫ [i.e., to orders The above equations are similar to the ones that have O(ǫ) and O(ǫ3/2)], Eqs. (8) and (7) yield the equation, beenemployedto discussthe crossoverfromTGto BEC regime [18, 25]. The ground state of the system can T φ (X,T)= g˙ (X) n (X,T )dT , (12) be obtained upon assuming that the atomic velocity 1 − 0 1 ′ ′ Z0 v φ = 0 (i.e., no flow in the system) and φ = µ x t 0 (d≡imensionlesschemicalpotential). Then,asEq. (7)−im- connectingtheunknownfunctionsn1 andφ1. Finally,to the second order of approximation [to order O(ǫ2) and pliesthatn=n istime-independentinthegroundstate, 0 O(ǫ5/2)],Eqs. (8)and(7)leadtothefollowingnonlinear we assume that n =n (X). Thus, to leading order in ǫ 0 0 evolution equation for n , [i.e., to O(1)] Eq. (8) yields, 1 (3g˙ +n g¨ ) 1 0 0 0 g(n0)=µ0−V(X), (9) n1X − 2C3 n1n1T + 8C5n1TTT d in the region where µ0 > V(X) and n0 = 0 out- = ln(C g˙0)1/2 n1, (13) −dX | | side. Equation (9) determines the density profile in h i the so-called Thomas-Fermi (TF) approximation; note where g¨ (d2g/dn2) . Equation (13) has the form 0 ≡ |n=n0 that for the typical case of the harmonic trap, e.g., of a Korteweg-deVries (KdV) equation with variable co- V(X) = (1/2)Ω2X2, Eq. (9) recovers the well-known efficients, which has been used in the past to describe 4 shallow water-waves over variable depth, or ion-acoustic Note that the above procedure is general, and does solitons in inhomogeneous plasmas [35]. Moreover, such not rely on the ratio of the parameter n/n , although it c KdV equations have recently been used to analyze the has been shown to yield the correct results in both lim- dynamics of DSs in Bose gases both in the weakly- its n n [11] and n n [16]. In this work, we use c c ≫ ≪ interacting[11]andthestrongly-interacting[16]regimes. the above general results for the evolution of the soli- As the inhomogeneity-induced dynamics of the KdV ton parameters, to derive the equation of motion of the solitonshasbeenstudiedanalyticallyinthepast[36],we DSandfindits oscillationfrequencyin the “nonlinearity may employ these results to analyze the coherent evolu- crossoverregime” n n . c ≈ tion of DSs in the Bose gas under consideration. Thus, introducing the transformations χ = (8C) 5dX and − n1 = (3/2)(3g˙0+n0g¨0)−1C−2υ(χ,T),Rwe first put Eq. III. SOLITON OSCILLATION FREQUENCY (13) into the form, Confining ourselves to the “crossover regime” n n , c υ 6υυ +υ =λ(χ)υ, (14) ≈ χ T TTT we may use a Taylor expansion of the function g(n) − aroundn=n ,namelyg(n) g(n )+g˙(n )(n n )+ , where λ(χ) (d/dχ)ln n3/4g˙1/4(3g˙ +n g¨ ) . In the c ≈ c c − c ··· ≡ 0 0 0 0 0 toderivetheapproximateexpressiong0 nc(3n0 nc)/4; ≈ − case λ = 0, i.e., for a hhomogeneous gas within0(X) = thelatter,alongwiththeTFapproximation[seeEq. (9)], n = const., Eq. (14) is the completely integrable KdV leads to the following density profile, p equation, which possesses a single-soliton solution of the following form [37], n = 4√µ0 1+ 1δ2 X 2 , (18) 0 υ = 2κ2sech2Z, Z =κ[T ζ(χ)], (15) 3δ " 4 −(cid:18)L(cid:19) # − − where ζ(χ) = 4κ2χ + ζ is the soliton center (with where δ nc/√µ0 (it is reminded that L √2µ0/Ω 0 ≡ ≡ dζ/dχ = 4κ2 being the soliton velocity in the T-χ ref- defines the axial size of the gas). Based on Eq. (18), it is now possible to derive the equation of motion of the erence frame), while κ and ζ are arbitrary constants 0 DS as follows. First, we find the solitonphase, which, to presentingthesoliton’samplitude(aswellasinversetem- order O(ǫ3/2), reads, poral width) and initial position respectively. Equation (15) describes a density notch on the backround density dX 9 3/2κ2(0) X np, with a phase jump across it [see Eq. (12), which Z =ǫ1/2κ(X) t ′ ǫ . implies that φ1 tanhZ] and, thus, it represents an ap- " −Z C(X′) − µ20Ωp(4+δ2)5/2 (cid:18)L(cid:19)# ∼ proximate DS solution of Eq. (5). (19) On the other hand, in the general case of the inho- Then,lookingalongthecharacteristiclinesofsolitonmo- mogeneous gas [i.e., in the presence of V(X)], soliton tion,itispossibletoshowthatthepositionofthesoliton dynamics can still be studied analytically, provided that satisfies the following equation of motion, the right-handside of Eq. (14) canbe treatedas a small dX C perturbation. As λ(χ) is apparently proportional to the = . (20) density gradient, such a perturbative study is relevant dt 1+ǫ(9√3/2)κ2(0)(4µ+δ2)−5/2C in regions of small density gradients (e.g., near the trap Forsufficientlysmallǫthe secondterminthe denomina- center for a harmonic trapping potential), which is con- tor can be neglected; in this case, Eq. (20) shows that sistent with the use of the local density approximation. the velocity of sufficiently shallow DSs is approximately Inthiscase,employingtheadiabaticperturbationtheory the same as the speed of sound given by Eq. (10), i.e., for solitons [38], we may seek for the soliton solution of dX/dt √g˙ n . Thus, Eq. (20) can be approximated 0 0 Eq. (14) in the form of Eq. (15), but with the soliton ≈ by the separable differential equation, parameters κ and ζ being now unknown functions of χ. n +n The respective evolution equations for the soliton’s am- 0 c dX =dt, (21) plitudeandcentercanbesolvedanalytically[36]andthe n 2n2+n n 0 c 0 c results, expressed in terms of the slow variable X, read: which can readilypbe integrated. In particular, taking σ(X) 2/3 into accountEq. (18), we find that Eq. (21) leadsto the κ(X)=κ(0) , (16) following result: σ(0) (cid:18) (cid:19) 1 X σ(X ) 4/3 3 X 2 ζ(X)= 2κ2(0)Z0 C−5(X′)(cid:18) σ(0)′ (cid:19) dX′ 4√13h(δ)lnw(X)+arcsin(cid:18)L˜(cid:19)= Ωr3!t, (22) 2/3 + 1 1 σ(X) − , (17) where h(δ)=δ 1+ 1δ2 −1/2, 2κ(0)" −(cid:18) σ(0) (cid:19) # 4 (cid:0) L˜(cid:1)2 X2+ √13h(δ)X whereκ(0)andσ(0)arethevaluesoftherespectivefunc- w(X)= − 2 , (23) tions at X =0. pL˜2 X2 √13h(δ)X − − 2 p 5 and L˜ L 1+ 7δ2. It can be seen that in the regions ≡ 2 −30 of small denqsity gradient where X/L is sufficiently small [which is consistent with the assumption that the per- −20 0.2 turbation λ(χ) in the KdV Eq. (14) is weak], and for ∼ sufficientlysmallvaluesoftheparameterδ,thefirstterm −10 on the left-hand side of Eq. (22) can safely be neglected 0.1 [39]. Then, it is readily seen that Eq. (22) is reduced to x 0 the following equation, 0 2 10 X =Lsin Ωt . (24) 3 r ! 20 −0.1 Equation (24) demonstrates that a shallow DS displays an oscillatory motion in the harmonic trap V(X) = 30 (1/2)Ω2X2 inthe“nonlinearitycrossoverregime”n n 0 50 100 150 200 250 ≈ c t of Eq. (2), with an oscillation frequency Ω given by osc 2 FIG.1: Spatio-temporalevolution ofthereducedcondensate Ωosc =Ω . (25) density(thegroundstatedensityminustheactualdensity)for r3 theNLSequationwithg(n)=n2/(1+n)andV(x)=Ω2x2/2, This finding has also been verified through appropri- with Ω=0.0707. The dashed line shows the theoretical pre- ately crafted numerical experiments. As a typical ex- diction which agrees well with the full numerical result over ample, we show, in particular, the evolution of a shal- (at least) thefirst oscillation period. low gray soliton (originally located at the origin) with initial speed C = 0.9 on the background of a potential V(x)=Ω2x2/2, with Ω=0.0707; n was chosen to be 1 predicted soliton oscillation frequency from the regime c in this case. Figure 1 shows the evolution of the space- n nc to n nc. ≫ ≪ time contour plot of the reduced density (the ground Assuming that the temperature is low enough for the statedensityminustheactualdensity)fortheNLSequa- solitonto performmany oscillations before decaying due tion, alongside our theoretical prediction of Eq. (24). to additional dissipative mechanisms excluded from the It is clear that during the first period, where the soli- NLSequation,anobservationofchangeintheoscillation tary wave does not significantly interact with the emit- frequency can be treated as a possible diagnostic tool of ted “radiation”, the agreement between the theoretical thesystembeinginaparticularinteractionregime. Arel- prediction and the numerical results is very good. Sub- evantexperimentmightthus beto createaBosegasofa sequently, as expected given the above interaction, this certainunknowninteractionstrength,andphaseimprint agreement deteriorates. The approximate initial condi- a DS in such a system. Measurement of the oscillation tion is produced by imposing a NLS gray soliton on top frequency of this soliton will then provide important in- of the ground state of the system for the nonlinearity formationon the system parameterregime,with anyde- of interest. Finally, we note in passing that our ana- viation from the oscillation frequency of Ω/√2 denoting lytical considerations are, strictly speaking, valid away aregimeofsufficientlystrongcorrelationswhich,inturn, fromtheturningpoints(wherethewavespeedvanishes). indicate that the weak-interaction model is no longer However,ournumericalresults,aswellasalternativeap- an adequate description of the system. Alternatively, proaches such as the ones of [13] and [14], illustrate that one could create a dark soliton in a weakly-interacting the range of validity of our results is, in fact, wider than 1D BEC,andgradually increasethe effective interaction what may be expected based on the mathematical limi- strength, as done in some experiments. In this case, one tations of the method. should be able to observe a gradual monotonic increase Note that the established values of the oscillation fre- oftheoscillationfrequency,eventhoughthelongitudinal quencies in the two limiting regimes of interatomic cou- confinement remains unaffected. plingcanalsobefoundintheframeworkofthepresented analysis, upon utilizing Eq. (21) in the relevant limits n/n 1 and n/n 1 and using the respective den- IV. CONCLUSIONS c c ≫ ≪ sity profiles. In the limit n n coresponding, e.g. to a c ≫ weakly-interactingBose-Einsteincondensate,the oscilla- In summary, we have discussed the dynamics of both tion frequency is Ωosc = Ω/√2 (or Ω 2/4), whereas, in linearandnonlinearexcitationswithinageneralizednon- the opposite regime n nc it has recently been found linear Schr¨odinger equation motivated by considerations ≪ p [14, 16, 22] that the oscillation frequency is Ωosc = Ω ofthe behaviour ofultracoldatomic 1D Bosegases. The (or Ω 2/2). The oscillation frequency in Eq. (25) is consideredmodel differs from relevant ones appearing in thus predicted to lie between the above mentioned lim- the context of optics [3] in that the linear behaviour in p iting cases. This suggestes a continuous change in the the density dominates at high densities, with quadratic 6 behaviour in the density dominating at lower densities. arenotrestrictedtothe detailsofthis particularsystem. Withinthisgeneralisedmodel,discussedhereforthefirst time in relation to dark solitons in 1D ultracold Bose gases, we have studied the dynamics of dark solitons in Acknowledgments the“crossoverregime”intheeffectiveatomicinteraction strength, which is approximately marked by a critical density. Our main conclusion, stemming from analyt- It is a pleasure to acknowledge discussions with Luis ical considerations and confirmed by numerics, is that Santos and Joachim Brand on the form of the general- the soliton oscillation frequency in this regime lies be- ized nonlinearity, and with Vladimir Konotop regarding tween the known values arising in the two limiting cases theapplicabilityofthe analyticalapproach. Theworkof of weakly and strongly interacting gases, indicating a D.J.F. was partially supported by the Special Research continuous change between these two regimes. Finally, Account of the University of Athens. PGK gratefully we note that although motivated by a particular physi- acknowledges support from NSF-DMS-0204585, NSF- calsystem,ourmodel andanalysisare quite generaland DMS-0505663and NSF-CAREER. [1] Ph. Emplit et al. Opt.Commun. 62, 374 (1987). T. Wenger, and D.S. Weiss, Science 305, 1125 (2004). [2] D.R.Andersen et al. Opt.Lett. 15, 783 (1990). [22] Th. Busch and G. Huyet,J. Phys.B 36, 2553 (2003). [3] Yu.S.KivsharandG.P.Agrawal, Optical Solitons: From [23] H. Moritz et al., Phys. Rev. Lett. 91, 250402 (2003); B. Fibers to Photonic Crystals (Academic Press, 2003). LaburtheTolraetal.,Phys.Rev.Lett.92,190401(2004). [4] B. Denardo et al.,Phys. Rev.Lett. 64, 1518 (1990). [24] E.H.LiebandW.Liniger,Phys.Rev.130,1605(1963). [5] B. Denardo et al.,Phys. Rev.Lett. 68, 1730 (1992). [25] P. O¨hberg and L. Santos, Phys. Rev. Lett. 89, 240402 [6] M. Chen et al.,Phys. Rev.Lett.70, 1707 (1993). (2002); [7] F. Dalfovo et al. Rev. Mod. Phys. 71, 463 (1999). [26] J. Brand, J. Phys. B 37, S287 (2004). [8] S. Burger et al., Phys. Rev. Lett. 83, 5198 (1999); J. [27] J. Brand and L. Santos (PrivateCommunication). Denschlag et al., Science 287, 97 (2000); B.P. Anderson [28] E. B. Kolomeisky and J. P. Straley, Phys. Rev. B 46, et al.,Phys.Rev.Lett.86,2926 (2001); Z.Duttonet al. 11749 (1992); R.K. Badhuri et al., J. Phys. A 34, 6553 Science 293, 663 (2001). (2001); M.D.Leeetal.,Phys.Rev.A65,043617(2002). [9] A.G¨orlitz et al., Phys.Rev.Lett. 87, 130402 (2001). [29] M.D. Girardeau and E.M. Wright, Phys. Rev. Lett. 84, [10] Th. Busch and J. R. Anglin, Phys. Rev. Lett. 84, 2298 5691 (2000). (2000);D.J.Frantzeskakisetal.,Phys.Rev.A66,053608 [30] A. E. Muryshev et al., Phys. Rev. Lett. 89, 110401 (2002);V.V.KonotopandL.Pitaevskii,Phys.Rev.Lett. (2002). 93, 240403 (2004); G. Theocharis et al., Phys. Rev. A [31] L.Salasnich, A.Parola and L.Reatto,Phys.Rev.A65, 72, 023609 (2005). 043614 (2002). [11] G. Huang, J. Szeftel, and S. Zhu, Phys. Rev. A 65, [32] Asubsequentgeneralisationof[31],presentedinL.Salas- 053605 (2002). nich, A. Parola and L. Reatto, Phys. Rev.A 72, 025602 [12] N.G. Parker et al., Phys. Rev. Lett. 90, 220401 (2003); (2005), could also be used to numerically probe the de- N.P.Proukakisetal.,Phys.Rev.Lett.93,130408(2004). sired crossover regime, but would not be amenable to [13] D.E. Pelinovsky, D.J. Frantzeskakis, and P.G. analytical considerations of theform presented here. Kevrekidis, Phys.Rev.E 72, 016615 (2005). [33] C. Menotti and S. Stringari, Phys. Rev. A 66, 043610 [14] V.A.Brazhnyi,V.V.Konotop,andL.P.Pitaevskii,Phys. (2002). Rev.A 73, 053601 (2006). [34] T. Taniuti, Prog. Theor. Phys.Suppl. 55, 1 (1974). [15] E.B.Kolomeiskyetal.,Phys.Rev.Lett.85,1146(2000). [35] N. Asano, Prog. Theor. Phys.Suppl. 55, 52 (1974). [16] D.J.Frantzeskakis,N.P.Proukakis,andP.G.Kevrekidis, [36] V.I.KarpmanandE.M.Maslov, Phys.Fluids25, 1686 Phys.Rev.A 70, 015601 (2004). (1982). [17] M. O¨gren, G.M. Kavoulakis and A.D. Jackson, Phys. [37] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Rev.A 72, 021603(R) (2005). Evolution Equations and Inverse Scattering (Cambridge [18] V.Dunjko,V.Lorent,andM.Olshanii,Phys.Rev.Lett. University Press, Cambridge, England, 1991). 86, 5413 (2001). [38] V. I. Karpman and E. M. Maslov, Zh. Eksp. Teor. Fiz. [19] D.S.Petrov,G.V.Shlyapnikovand J. T. M.Walraven, 75, 504 (1978) [Sov.Phys.-JETP 48, 252 (1978)]. Phys.Rev.Lett. 85, 3745 (2000). [39] In fact, this term becomes significant only near therims [20] L. Tonks, Phys. Rev. 50, 955 (1936); M. Girardeau, J. of thegas (i.e., near theturningpointsX ≈±L), where Math. Phys.(N.Y.) 1, 516 (1960). the perturbation theory fails. [21] B. Paredes et al.,Nature429, 277 (2004); T. Kinoshita,