Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion, and dissipation
Manas Kulkarni1,2 and Alexander G. Abanov3
1Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A72Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 3H6
3Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA(Received 13 June 2012; published 13 September 2012)
Dynamics of interacting cold atomic gases have recently become a focus of both experimental and theoreticalstudies. Often cold-atom systems show hydrodynamic behavior and support the propagation of nonlineardispersive waves. Although this propagation depends on many details of the system, great insight can be obtainedin the rather universal limit of weak nonlinearity, dispersion, and dissipation. In this limit, using a reductiveperturbation method we map some of the hydrodynamic models relevant to cold atoms to well-known chiralone-dimensional equations such as the Korteweg–de Vries (KdV), Burgers, KdV-Burgers, and Benjamin-Onoequations. These equations have been thoroughly studied in the literature. The mapping gives us a simple wayto make estimates for the original hydrodynamic equations and to study the interplay between nonlinearity,dissipation, and dispersion which are the hallmarks of nonlinear hydrodynamics.
Recently, there has been a flurry of experiments that studythe collective behavior of systems of particles with bothbosonic [1–4] and fermionic [5–12] statistics. There havealso appeared experimental realizations of cold-atom systemswith long-range interatomic interactions (such as dipolarbosons [13–15], dipolar fermions [16], ions, and Rydbergatoms). Investigation of the dynamics of such systems isinteresting and important for many reasons. The dynamics canprovide important information about the nature and strengthof particle interactions. In many cases the nonlinear dynamicalevolution leads to the formation of shock waves and solitons.Out-of-equilibrium dynamics in the presence of defects anddisorder (see, e.g., Ref. [17]) can shine light on dissipationand localization phenomena. One of the most intriguing andimportant directions is the search for universality in dynamicalproperties of systems governed by different microscopicHamiltonians [18].
The dynamics of cold Bose gases is being extensivelyinvestigated both theoretically [19,20] and experimentally.Experiments on Bose-Einstein condensates (BECs) includesound-velocity measurements [3] and collisions of atomicclouds, which lead to the observation of dispersive shockwaves and soliton trains [4,21]. Dissipative transport in BECshas been recently investigated in Ref. [17]. The propagationof matter-wave soliton trains in BECs has been realizedin Ref. [22]. The strongly interacting limit of interactingone-dimensional (1D) bosons, in other words, a Tonks gasanalogous to that for free fermions has been realized in Ref.[23].
The dynamics of Fermi systems [24] has also been ofgreat interest recently [10,25,26]. For instance, in Ref. [25]the spin transport was studied by colliding two oppositelyspin-polarized clouds of fermions. In Refs. [10,11] someaspects of nonlinear hydrodynamics (such as shock waves) inunitary Fermi gases were investigated. These experiments havealso motivated new numerical studies of nonlinear dynamicsin Fermi systems [27,28].
It is also worth mentioning that in addition to conventionalFermi and Bose many-body systems there exists a remarkablefamily of models interpolating between fermionic and bosonicsystems. This is the family of Calogero models. The exactnonlinear collective description for these models is known[29–32] and has a form of equations of hydrodynamic type.These hydrodynamic equations are integrable and exhibitfeatures such as solitons and dispersive shock waves. The equa-tions are related to a known integro-differential equation—the Benjamin-Ono equation [30–32]. Unlike other integrablemodels, the Calogero model and its hydrodynamic descriptionretain integrability even in the presence of an externalharmonic potential. In particular, solutions of multisoliton typehave been found for these models in Ref. [33].
Because of the variety of systems and models usedin studying nonlinear dynamics of cold-atom systems, theunderstanding of major effects resulting from the interplaybetween nonlinearity, dispersion, and dissipation within asimple unifying description would be very useful. Fortunately,this description is well known in the limit of weak nonlinearity,dispersion, and dissipation. It is summarized by the Korteweg–de Vries–Burgers (KdVB) equation [see Eq. (25) below]. Themain goal of this paper is to connect this universal pictureto some particular models used in collective descriptions ofcold-atom systems. For simplicity we concentrate here onsimple Galilean-invariant fluids, although many features ofnonlinear dynamics described in this paper can be found inmore complex systems as well.
The following is a brief outline of this paper. We startby constructing a rather general one-dimensional model of asimple Galilean-invariant fluid in Sec. II. We linearize thismodel in Sec. III and proceed to the derivation of the effectiveKdVB equation using the reductive perturbation method inSec. IV A. We describe the effects of nonlinearity, dispersion,and dissipation for the effective KdVB equation and introducecorresponding scales and limits. These scales and limits arepresented in the triangle phase diagram in Fig. 1. We concludethe qualitative description of KdVB dynamics in Sec. Vdescribing two different scenarios for shock-wave formation
MANAS KULKARNI AND ALEXANDER G. ABANOV PHYSICAL REVIEW A 86, 033614 (2012)
in the presence of both dissipation and dispersion. Finally,in Sec. VI we give relations between the coefficients of theeffective KdVB equation and parameters of some models usedin cold-atom studies. We conclude with some open questionsand possible generalizations of our results in Sec. VII anddescribe non-KdVB universal behaviors for some systems withlong-range interactions in the Appendix.
II. HYDRODYNAMIC MODEL
The goal of this section is to construct a hydrodynamicmodel (hydrodynamic equations) describing a rather generalone-dimensional motion of a simple fluid. An effectiveone-dimensional motion usually describes a class ofsolutions of three-dimensional hydrodynamics obtained by arequirement on the hydrodynamic fields ρ(x,y,z; t) = ρ(x,t),etc., or through more complicated procedures of dimensionalreduction, e.g., due to the presence of a quasi-one-dimensionaltrap in a cold-atom system (see, e.g., Sec. VI and Refs. [10,11]).An effective “truly” 1D model can also appear as a result ofthe transverse quantization in a quantum system [19,34]. Inthe following we assume that the reduction to one dimensionis already performed and outline other assumptions that weuse to construct an effective one-dimensional hydrodynamicmodel.
(a) A fluid has one component. We do not consider heremore complicated theories of mixtures of fluids. In particular,we do not consider the two-fluid hydrodynamics of superfluidswith nonvanishing normal component. If the system underconsideration is in the superfluid regime, we assume here thatthe normal component can be “integrated out,” resulting in thehydrodynamics of an effective single-component fluid withsome additional terms (such as viscous terms) generated bythe normal component.
(b) Entropy generation is very small. This assumptionallows us to consider closed equations for the density andvelocity fields only and assume that the fluid motion isisentropic. Combined with the first assumption, this meansthat the only relevant degrees of freedom can be de-scribed by one-dimensional velocity v(x,t) and density ρ(x,t)fields.
(c) Locality. We assume that the energy functional is localin the density and velocity fields and the equations of motioncontain only local values of the fields and their derivatives. Thisconstraint can be relaxed (see the examples in the Appendix).
(d) Galilean invariance. In the presence of Galileaninvariance the energy density of the fluid is given by ρv2/2with other terms either independent of velocity or proportionalto velocity gradients. Systems without this invariance cannotbe described by the simple hydrodynamic Hamiltonian (1). Wefocus on Galilean-invariant systems in this work for simplicitybut do expect that chiral differential equations of the form(25) still provide a rather universal effective description ofcollective dynamics even in the absence of Galilean invariance(e.g., for the effective description of the XXZ spin chain in anexternal magnetic field [35]).
(e) The energy functional contains density derivatives upto the second order. This approximation is known as theBoussinesque approximation. In the absence of dissipationthis assumption combined with previous assumptions leads to
the following hydrodynamic Hamiltonian:
H =∫
dx
[ρv2
2+ ρε(ρ) + A(ρ)
(∂xρ)2
4ρ
]. (1)
Here A(ρ) is some function of density. We notice here thatif A = const the gradient term becomes the Madelung termwell known in superfluid hydrodynamics. The hydrodynamicHamiltonian (1) equipped with Poisson’s brackets
{ρ(x),v(x ′)} = ∂xδ(x − x ′) (2)
generates the hydrodynamic equations of an ideal fluid[36–38].
(e) The dissipative function is quadratic in velocity gradi-ents. The positivity of the dissipative function does not allowfor terms linear in velocity gradients and we have genericallyfor the (Rayleigh) dissipative function
F = 1
2
∫dx ηB(ρ)(∂xv)2, (3)
where ηB(ρ) is the bulk viscosity of the 1D fluid. In thepresence of dissipation the hydrodynamic system is notHamiltonian, and the Hamilton equations following from (1)and (2) should be supplemented by dissipative terms encodedin (3). We arrive at the following equations of motion:
∂tρ + ∂x(ρv) = 0, (4)
∂tv + ∂x
(v2
2+ w − (∂ρA)(∂x
√ρ)2 − A
∂2x
√ρ√
ρ
)
= 1
ρ∂x (ηB∂xv) , (5)
where w = ∂ρ(ρε(ρ)) is the specific enthalpy (the same as thechemical potential at zero temperature) of the fluid and theright-hand side (RHS) of the second equation is obtained as− 1
ρδFδv
. The rate of the dissipation of energy is then given by2F .
Let us rewrite these equations in terms of density andcurrent (momentum density) j = ρv. We have
∂tρ + ∂xj = 0, (6)
∂tj + ∂x(T + T ′) = 0, (7)
j = ρv, (8)
T = ρv2 + P − 12ρ
√A∂x(
√A∂x ln ρ), (9)
T ′ = −ηB∂xv. (10)
Here the pressure P (ρ) is given by ∂ρP = ρ∂ρw. The last termof (9) is called the “quantum pressure term” in the context ofquantum fluids. The viscous part T ′ of the stress tensor is linearin the gradient of velocity and is obtained from ∂xT
′ = δF/δv
[39]Many models of recent interest can be cast in the form of
Eqs. (4) and (5). The expressions for w(ρ), A(ρ), and ηB(ρ)for some of those models are listed in Table I.
Equations (4) and (5) are nonlinear. The nonlinearity resultsin the steepening of the density and velocity profiles duringthe evolution and in shock waves. The quantum pressure
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TABLE I. Hydrodynamic descriptions for some cold-atom systems. Mapping of the parameters in the hydrodynamic equations (4) and (5)to the coefficients of the effective KdVB equation.
Riemann-Hopf [Eq. (33)]Plane-wave unitary gas (1 + β) h2
2m2 (3π 2ρ)2/3 1/2 ∼ρb 23 w0
23
c
ρ0
h2
8m2c∼1
KdVB [Eq. (25)]
Quasi-1D unitary gas h2
2m2l2⊥(1 + β)
[15π
2ρl⊥1+β
]2/59/20 ∼ρb 2
5 w03c
5ρ0
9h2
80m2c∼1
KdVB [Eq. (25)]
Calogero gas h2λ2π2ρ2
2m2 + h2λ2πρHx
m2 λ2/2 0h2π2λ2ρ2
0m2
πhλ
mαBO = hλ
2m0
Benjamin-Ono [Eq. (A6)]Quasi-1D dipolar BEC gA1Dρ
ml2⊥+ gB
m∂2
x ρU 1/2 0 h2
m2l2⊥γA1D
34
c
ρ0αD = γB h2
2m2c0
Nonlocal KdV [Eq. (A20)] α = h2
8m2c
aIn experiments, many atomic gases which to a good approximation behave like a Lieb-Liniger gas might still have mechanisms of dissipation.The dissipation can be included phenomenologically by adding νuξξ to the right-hand side of the effective differential equation, which makesit a KdVB (dense limit) or Burgers (Tonks limit) equation.bSee the discussion of viscosity in a unitary Fermi gas in Sec. VI B for more details.
terms [terms containing A(ρ)] depend on the density gradients.They typically give rise to dispersive oscillations of densitiesand velocities. The viscosity term [the RHS of (5)] containsthe gradient of the velocity field. It results in damping.All these terms play rather different roles in fluid dynamicsand the understanding of their interplay is of great interestboth theoretically and experimentally.
III. LINEARIZATION
Let us start by studying the system (4),(5) in the limit ofsmall deviations from the uniform solution ρ(x) = ρ0, v(x) =0. We consider this solution as a background configurationof fields and linearize equations in δρ(x) = ρ(x) − ρ0 andδv(x) = v(x). We obtain
δρ + ρ0δv′ = 0, (11)
δv + (∂ρP )0
ρ0δρ ′ − A0
2ρ0δρ ′′′ = ηB0
ρ0δv′′. (12)
Here the subscript “0” means that the quantity is calculated atρ = ρ0. The linearized equations give the dispersion equation(for δρ ∼ e−iωt+ikx , etc.)
ω2 = c2k2 + 12A0k
4 − 2iν0ωk2, (13)
where c2 = ∂ρP |ρ0 is the linear sound velocity and ν0 = ηB0
2ρ0is
proportional to the kinematic bulk viscosity at ρ = ρ0. Solving(13) and expanding up to k3, we arrive at
ω(k) ≈ ±ck
(1 + A0 − 2ν2
0
4c2k2
)− iν0k
2. (14)
For the wave propagating to the right we have
ω(k) ≈ ck − iν0k2 + A0
4ck3. (15)
Here we kept only leading terms representing different effects(dissipation ν0 and dispersion A0). We consider the limit ofsmall dissipation and dispersion and, therefore, neglect thecorrection to the dispersion quadratic in ν0, assuming it issmaller than the corresponding A0 contribution. We keep theterm A0k
3, which can be comparable in magnitude to the termiν0k
2 at small (but finite) k if the viscosity ν0 is very small.The dispersion (15) can be reproduced by the following linearequation:
ut + cux − αuxxx = ν0uxx, (16)
where α = A0/(4c). Here one can substitute either δρ or δv
instead of u as δρ and δv are related linearly by (11). Toobtain an effective equation for the left-moving linear waveone should just change x → −x in (16).
In the linear approximation an arbitrary initial profile ofdensity and velocity is split into right and left moving waves.These waves move with velocities ±c respectively and slowlydisperse and decay. If the amplitude of the initial profileis small but finite one should add nonlinear corrections tothe chiral (right and left) equations (16). In addition to this,the nonlinear terms also couple the equations for left- andright-moving waves. However, as the right and left profilespass each other with finite velocity (with the relative velocity2c) they interact only for a short time. In the limit of weaknonlinearity the coupling between equations is small andwill not significantly change the solutions. Therefore, it canbe neglected [40], and most important nonlinear correctionsshould enter the chiral wave equation (16) itself.
The goal of the next section is to justify the above argumentsand to derive (16) together with the corresponding nonlin-ear corrections using the well-known reductive perturbationmethod [41–44].
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IV. KdVB EQUATION VIA REDUCTIVEPERTURBATION METHOD
The linear equation (16) suggests that in the long-waveapproximation k → 0 the dissipative ν0 term always winsover the dispersive α term. This is true unless the viscositycoefficient ν0 is small (or zero) for some reason and weare interested in the regime of small but finite k. Then thecomparison of the first and third terms on the LHS of (16)suggests that we have to treat all derivatives as small, withtheir relative values determined by the scaling ∂t = O(ε3) and∂x = O(ε), where ε is some auxiliary “counting” parametertreated as a small parameter. This scaling was introduced inRef. [45] in the derivation of the Korteweg–de Vries equationfor water waves. The only modification we make here is thatwe consider the viscosity coefficient ν0 = O(ε) which willallow us to have the RHS of (16) of the same order as ut anduxxx . The aim of this section is to include a weak nonlinearityinto this scaling. This goal is achieved by using the so-calledreductive perturbation method (or a proper power-countingscheme in modern language).
A. Korteweg–de Vries–Burgers equation
Let us introduce the following scaling scheme:
δρ → ε2f (εξ,ε3t), (17)
δv → ε2g(εξ,ε3t), (18)
ηB → εηB, (19)
where
ξ = x − ct − x0. (20)
The scaling laws εξ , ε3t , and εηB have already been motivatedon the basis of the linear equation (16) at the beginning of thissection. The ε2 scaling of the amplitudes can be obtainedby comparing the relative values of, say, the term ut ∼ ε3u
of (16) with a foreshadowed nonlinear correction uux ∼ εu2.Then according to the general reductive perturbation methodthe scaling scheme (17),(19) should be supplemented by anappropriate perturbation scheme
f = f (0) + ε2f (1) + · · · , (21)
g = g(0) + ε2g(1) + · · · . (22)
Substituting ρ = ρ0 + ε2f (εξ,ε3t) and v = ε2g(εξ,ε3t) with(21), (22), and (19) into the system (4),(5), we obtain a systemof coupled equations which can be analyzed in each orderof ε separately (a similar scheme has been used recentlyin the context of a quasi-one-dimensional BEC [46]). Thefirst nonvanishing order is O(ε3) and gives the following twoequations:
g(0)
c= f (0)
ρ0, f (0)w′
0 = cg(0). (23)
Compatibility of these equations determines the sound velocityc, giving the well-known thermodynamic relation
c2 = ρ0w′0. (24)
Equation (23) gives a linear relation between f (0) and g(0).
The next nonvanishing order is O(ε5). It gives
∂tf(0) + ∂ξ (f (0)g(0)) = ∂ξ (cf (1) − ρ0g
(1)),
∂tg(0) + ∂ξ
(g(0)2
2+ w′′
0
2f (0)2
)− A0
2ρ0∂3ξ f (0) − ηB0
ρ0∂2ξ g(0)
= ∂ξ (cg(1) − w′0f
(1)).
We use the relation (23) to exclude g(0) from these equations.Then the difference of these two equations gives a closedequation for u = f (0):
ut + ζuuξ − αuξξξ = νuξξ (KdVB), (25)
with
ζ = 1
2
(3c
2ρ0+ w′′
0ρ0
2c
)= 1
2
(c
ρ0+ ∂c
∂ρ0
), (26)
α = A0
4c, (27)
ν = ν0 = ηB0
2ρ0. (28)
A similar equation for g(0) can be obtained from (25) using(23). Equation (25) is known as the Korteweg–de Vries–Burgers equation. It describes the correct far-field behaviorof solutions of (4),(5) in the limit of weak nonlinearity,dispersion, and dissipation. A linearization of (25) gives (16)as expected.
We note here that in the conventional form of the KdVBequation (and the KdV equation, see below) the α term isusually written with a plus sign. There is, however, a simpletransformation
u(x) → −u(−x) (29)
which maps the solutions of equations with opposite signs ofα to each other. Using this transformation we can borrow anysolutions of the conventional KdVB and KdV equations to findthe corresponding solutions of (25).
B. Scales and phase diagram of the KdVB equation
The KdVB equation has been extensively studied (see,e.g., Refs. [47–51]). It is a nonintegrable equation and onlyfew traveling-wave solutions of the KdVB equation areknown analytically [47]. The KdVB equation has a reachdynamics which includes an intricate interplay of nonlinearity,dispersion, and dissipation. An example of numerical analysisof the KdVB equation can be found in Ref. [50].
The effective KdVB equation (25) with (26), (27), and (28)describes the universal limit of weak nonlinearity, dispersion,and dissipation of one-dimensional solutions for a large classof hydrodynamic systems. Depending on initial conditionsdifferent terms of (25) play different roles at different times ofevolution. The goal of this section is to summarize differentregimes of hydrodynamic behavior and to show how to makesimple estimates for the onsets of these regimes.
To understand qualitatively the role of different effects, letus consider an initial profile u(x,t = 0) which is characterizedjust by two scales: a typical width of the profile W and a typicalamplitude of the initial profile U . One can think of this initialprofile as the steplike function u(ξ ) shown in the inset of Fig. 1
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W
V U
Solitons
Riemann-Hopf
Dispersion Dissipa�on
Classical Dispersive Shock Waves
Diffusion Dispersion
Nonlinearity
Linear Equa�on Edge
Conven�onal Shock
FIG. 1. (Color online) A schematic phase diagram summarizingvarious hydrodynamic regimes of the KdVB equation. We consider atypical step profile shown in the left inset characterized by a typicalwidth W and an amplitude U . The directions towards the corners ofthe triangle correspond to increase of the corresponding ν, α, andζ terms of the KdVB equation (25) evaluated on the typical profile.Corresponding to the strength of those terms, the diagram is dividedinto regions corresponding to diffusive, dispersive, and shock-waveregimes with dominating dissipation, dispersion, and nonlinearityterms, respectively. The edges of the triangle correspond to the exactlysolvable equations with one of the terms vanishing. See Secs. IV andV for details. The right inset (x and y axes in units of μm and μm−1,respectively) shows the evolution (time snapshot) of an initial profileevolving according to the KdVB, KdV, and Burgers equations.
or as a single lump of density (velocity). We use the parametersU and W together with the values of the coefficients ζ , α, andν to form characteristic times corresponding to the differentterms of the KdVB equation. We characterize the strengthof nonlinearity, dispersion, and dissipation by the followinginverse times (characteristic frequencies):
�ζ = ζUW−1, �α = αW−3, �ν = νW−2, (30)
respectively. The relative values of these frequencies definethe regime of evolution of the initial profile described by theKdVB equation. Different regimes are summarized in Fig. 1.It is also convenient to introduce the spatial scales
Wζν = ν
ζU, Wζα =
√α
ζU, Wνα = α
ν. (31)
These scales are defined so that, e.g., when W ∼ Wζν the ζ andν terms of the KdVB equation are of the same order, �ζ ∼ �ν .
The center of the triangle in Fig. 1 represents an initialprofile for which all three scales are of the same order, i.e.,nonlinearity, dispersion, and dissipation are equally importantat the beginning of the evolution. Generally, different initialprofiles correspond to different points of the triangle so thatthe corresponding parameters grow from zero at the respectiveside to infinity at the respective vertex of the triangle.
In the absence of nonlinearity (the bottom part of thetriangle, �ζ � �α,ν) the dynamics is approximately linearand is described by (16). It is either diffusive (left part ofthe triangle, �α � �ν) or initially dispersive (right part,
�ν � �α). The dispersive evolution, even if dominant at thebeginning of the evolution, �α �ν , leads to growth of thewidth of the initial profile and decrease of the gradient terms.The dispersion term will become of the same order as thediffusion term at the time
tαν ∼ �α�−2ν = αν−2W. (32)
At that time the width of the profile becomes of the orderof Wνα (31) and after that moment the dispersion becomessubdominant to the diffusion.
The evolution is much more interesting and complicatedwhen the nonlinear term dominates for an initial profile. In thiscase the evolution is initially described by the Riemann-Hopf(or inviscid Burgers) equation
ut + ζuuξ = 0 (Riemann-Hopf), (33)
which can be easily solved for any initial profile u(x,t = 0) =f (x) giving u(x,t) implicitly, as a solution of u = f (x − ζut),where the unknown u enters both left- and right-hand sides.For our typical initial UW profile this solution is well definedat small times, becoming multiply valued for times after
tc ∼ �−1ζ = W/(ζU ). (34)
The time tc is known as the time of “gradient catastrophe”and is defined as the time at which ux becomes infinite. Theclassical problem (33) is ill posed at larger times and hasto be regularized by higher gradient corrections, which in ourcase are either dispersive [α term of (25)] or diffusive (ν term).According to the relative strength of the subdominant diffusiveand dispersive terms in the regime of strong nonlinearity(the upper part of the triangle phase diagram), one hasformation of either classical dissipative shock waves (leftpart) or dispersive shock waves (right part). We discuss theseshock-wave regimes separately in Sec. V. Before going to thisdiscussion we consider important limits of the KdVB equation(25) corresponding to the left and right sides of the trianglephase diagram of Fig. 1.
C. Dissipative limit: Burgers equation
If the dispersion does not play a significant role in theevolution (�α � �ν), one can neglect the α term in (25) andarrive at the well-known Burgers equation
ut + ζuuξ = νuξξ (Burgers). (35)
The regime described by (35) corresponds to the left side ofthe triangle phase diagram of Fig. 1. The different points of theleft side of the triangle correspond to different relative valuesof the nonlinear and dissipative terms of (35).
We note here that one can arrive at the Burgers equation(35) more formally starting directly from (4) and (5). Insteadof (19) one should take ηB = O(ε0). Then the scaling scheme(17),(18) is not consistent and should be replaced by theBurgers scheme [41]
δρ = εf (εx,ε2t), (36)
v = εg(εx,ε2t) (37)
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with the corresponding perturbative expansion
f = f (0) + εf (1), (38)
g = g(0) + εg(1) (39)
instead of (21) and (22). Expanding in ε, we arrive at therelation (23) and at the Burgers equation (35) as a closedequation for u = f (0) with parameters given by (26) and (28).
The Burgers equation (35) is often used as a modeldescribing classical dissipative shock waves which result fromthe interplay between nonlinearity and dissipation [20]. Itcan be analytically solved via Cole-Hopf transformation u =− 2ν
ζ∂ξ ln φ which results in a diffusion equation φt = νφxx
[52,53].There is a simple exact solution of (35) given by a steplike
profile traveling with a constant velocity V to the right:
u(ξ,t) = U
2{1 − tanh[W−1(ξ − V t)]}. (40)
Here the parameters of the step are related to the velocity V as
W = 2ν/V = 4Wζν, (41)
U = 2V/ζ. (42)
The width of the solution is given by (41), i.e., essentiallyby the scale Wζν . In the limit ν → 0 this width goes to zeroand the solution (40) describes a discontinuous shock front.The solution (40) is defined when �u = uξ→−∞ − uξ→+∞ =U > 0.
D. The limit of no dissipation: KdV equation
In the limit when dissipation is absent or very small at agiven wavelength the ν term of the KdVB equation (25) canbe dropped and we arrive at the celebrated Korteweg–de Vriesequation
ut + ζuuξ − αuξξξ = 0 (KdV) (43)
with parameters determined by (26) and (27). It is a purelydispersive equation. It is integrable [54,55] and possessesinfinitely many conserved quantities. The first three of theseintegrals are given explicitly by
I0 =∫
dξ u, (44)
I1 =∫
dξu2
2, (45)
I2 =∫
dξ
[ζ
u3
3+ αu2
ξ
]. (46)
The integrals I0,1,2 are related to the total number of particles,total momentum, and total energy of the system (4),(5) asfollows:
N − N0 = I0, (47)
P = 2c
ρ0I1 + cI0, (48)
E − E0 = c
ρ0I2 + 2c2
ρ0I1 + w0I0. (49)
The higher integrals of motion of the KdV equation are relatedto more complicated symmetries of the problem and areusually destroyed by small corrections to the KdV equationwhich destroy integrability. We do not need their exact form inthe following discussion. The conserved quantities (47), (48),and (49) on the other hand are related to fundamental space-time-gauge symmetries and, therefore, play an important rolein the hydrodynamic approach.
The KdV equation has a solitary-wave solution (soliton)moving to the left with velocity V [56]:
u(ξ,t) = −U cosh−2[W−1(ξ + V t)]. (50)
This solution corresponds to a local depletion of the particledensity [minus sign in (50)] and is known as the dark soliton.The width W of the soliton and the amplitude U of thedepletion in (50) are defined by its velocity V as
W = (4α/V )1/2 =√
12 Wζα, (51)
U = 3V/ζ. (52)
While the parameters of the soliton (50) are given exactlyby (51) and (52), they could also be estimated from thecondition that the nonlinearity and dispersion scales of thesoliton solution should be of the same order, �α ∼ �ζ (exactlywe have �ζ = 12 �α) [57].
The solution (50) corresponds to the special point on theright side of the triangle phase diagram in Fig. 1. Thesesolutions (like the effective KdV equation itself) are to betrusted only when the relative depletion U/ρ0 ∼ V/c is small,i.e., V � c.
The total number of depleted particles in the soliton solution(50),(51),(52) is given by the zeroth integral of motion (44),
n =∫
u(ξ,t) dξ = −2UW = −12
ζ
√αV . (53)
The values of other integrals of motion in the KdV solitonsolution (50) are
I1 = 2
3U 2W = 12
V√
αV
ζ 2= −V
ζn, (54)
I2 = − 4
15ζU 3W = −72
5
V 2
ζ 2
√αV = 6
5
V 2
ζn. (55)
Using the above expressions and (47), (48), (49) we obtain thedispersion of the soliton,
E(P ) − E0 − nw0 = c (P − cn) + (P − cn)2
2m∗ , (56)
with the “effective mass” m∗ given by
m∗ = 5cn
3ζρ0. (57)
The form (56) is similar to the dispersion of particles witha quadratic spectrum except that the effective mass (57) isvelocity (and momentum) dependent [see Eq. (53)].
Another important exact solution of the KdV equation isthe periodic traveling-wave solution given by
u(ξ,t) = −Ucn2(W−1(ξ + V t)|m). (58)
In Eq. (58), cn(y|m) is the Jacobi elliptic function of modulusm (0 < m < 1). The modulus defines the period of the solution
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in ξ [the period L itself is L = WF (π/2,m)]. In the limitm → 1 the period L → ∞ and the Jacobi elliptic functionreduces to a periodic array of well-separated solitons (50).
So far we discussed very special solutions of the KdVequation corresponding to the “Solitons” point in Fig. 1. Whatwill happen to the initial profile with typical dimensions suchthat the nonlinear term is dominant, �ζ �α? In this casethe profile will initially evolve according to the Riemann-Hopfequation (33). However, at the gradient catastrophe time (34)large gradients will develop and the dispersive term willbecome of the order of the nonlinear term. Details of theevolution after that point do depend on the particular shapeof the initial profile. However, generically, the modulatedperiodic solutions of the KdV equation will be generatedproviding oscillating (i.e., dispersive) shock fronts [58]. Thesephenomena are referred to as dispersive shock waves (DSWs)in contrast to conventional or dissipative shock waves whichoccur in equations of Burgers type. Without dissipationthe steep profiles described by the KdV equation generateoscillations and eventually decay into trains of well-separatedsolitons.
V. SHOCK WAVES IN THE KdVB EQUATION
So far we have discussed the dynamical regimes of theKdVB equation corresponding to the sides of the trianglephase diagram, Fig. 1. These regimes are realized when one ofthe three scales of the KdVB equation (30) can be neglectedcompared to the two others. If all three scales are significantthe different ordering of these scales roughly divides the phasediagram of Fig. 1 into six parts. Up to this point we treated thethree scales as equal, but to understand the KdVB dynamicsof a typical profile even better one should realize that thethe nonlinearity, dissipation, and dispersion enter in a veryasymmetric way. Indeed, we have already seen that in the casewhen the nonlinearity dominates the initial evolution (�ζ is thelargest scale), the evolution itself will result in large gradientsin finite time which is of the order of the gradient catastrophetime (34). At that point either dissipation or dispersion willbecome of the order of nonlinearity and our discussion offurther evolution should start from a different part of thetriangle phase diagram [59]. Another example of this type isgiven by the linearized KdVB equation (16) with dominatingdispersive term (�α �ν). The corresponding initial profilewill disperse almost nondissipatively up to a time of the orderof tαν (32), at which point the dissipation term will become ofthe order of the dispersion term and further evolution will bediffusionlike. In contrast to these examples, if the dissipationterm is dominant in the beginning it remains dominant and theevolution is diffusionlike at all times.
In the following we apply the understanding of theasymmetry of different KdVB terms to describe two distinctshock-wave regimes of the KdVB equation. We consider aninitial profile of the step form shown in the inset of Fig. 1.Shock waves appear in the regime when nonlinearity is themost important scale in the problem, e.g., if the width of theinitial profile W is sufficiently bigger than Wζν and Wζα (31).Then the nature of the shock waves depends on the relativestrength of the dispersive and dissipative terms.
The goal of this section is to describe the major features ofclassical and dispersive shock waves and show how to estimatetypical scales of these shock-wave solutions.
A. Classical shock waves
To have a clear separation of scales we consider thefollowing relation between the width of the initial steplikeprofile and the scales of the KdVB equation (31)
W Wζν Wζα,Wνα, (59)
which means �ζ �ν √�α�ζ . In this case the dominant
term of the KdVB equation for the initial profile is thenonlinear ζ term and the evolution of this profile is determinedby the competition between the dominant nonlinear and thesubdominant dissipative ν term.
Up to a time of the order of tc (34), the nonlinear termwill dominate the evolution which will be described by theRiemann-Hopf equation (33). Then the decreasing width ofthe profile (the width of the wave front) will reach the scaleWζν (30) and an approximate balance between nonlinearity anddissipation will be achieved. The profile will become stationaryand will have a width of the order of
WCSW ∼ Wζν. (60)
The quasistationary profile can be roughly described by thesolution (40). The ratio of the dispersion scale to the dissipativescale for this profile is given by �α/�ν = Wνα/W and willremain very small for all times due to (59). Therefore, weexpect that the dispersive effects are not important in theregime (59) and might result only in small oscillations on topof the stationary classical shock-wave (CSW) solution (40).
B. Dispersive shock waves
The case of the initial steplike profile characterized by thewidth
W,Wνα Wζα Wζν (61)
is, probably, the most interesting. The inequalities (61) meanthat the evolution of the initial profile is defined in thebeginning by an interplay between the dominant nonlinearand subdominant dispersion terms with dissipation playingsome role at very large times only. As both ν and α terms aresubdominant to the nonlinearity, the scale Wνα does not playany role at the beginning of the evolution and its relation tothe initial profile width W is not very important. We note alsothat the second of the inequalities (61) is a consequence of thefirst one, and the smallest scale Wζν does not play a major rolein this regime.
As in the case of classical shock waves, for t < tc theevolution is governed by the Riemann-Hopf equation (33) andthe width of the step decreases, reaching Wζα in a time ofthe order of tc (34). At this time oscillations develop at thetrailing edge of the step profile [60]. These oscillations growin amplitude, with the largest amplitude becoming of the orderof the size of the step U . At this point the typical wavelength ofoscillations is given by the scale Wζα from (31). The number ofoscillations and the spatial extent of the oscillating part of theshock front continue to grow. In the absence of dissipation (the
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KdV regime) this growth continues forever, with oscillationsevolving into a train of well-separated solitons [58].
In the case of nonvanishing dissipation the front of a DSWbecomes stationary at large times. The width of the stationaryprofile can be estimated in the following simple way. Asthe width of the oscillating shock profile is determined bythe dissipation, we evaluate the magnitude of the dissipativeterm �ν (30) at the gradients developed due to oscillations,i.e., �ν |Wζα
∼ ν
W 2ζα
= ναζU . On a stationary profile this scale
should be of the order of the nonlinear scale evaluated atthe overall width of the profile WDSW , i.e., �ζ |WDSW
∼ ζU
WDSW.
Equating these scales, we obtain the estimate
WDSW ∼ W 2ζα
Wζν
= α
ν= Wνα. (62)
The formation of the stationary shock-front profile of DSWstakes a time of the order of tDSW ∼ Wνα/(ζU ), and the numberof oscillations in the stationary profile can be estimated as
N ∼ Wνα
Wζα
∼(
αζU
ν2
)1/2
1. (63)
This condition can be considered as necessary for observingoscillations in the stationary shock-wave profile.
We have to remark here that the presented picture capturesonly major scales of dispersive shock-wave formation. Forexample, while we assumed for our estimates that the typicalamplitude of oscillations is U , oscillations at all amplitudesranging from 0 to ∼U are present in the shock front. As aresult even after formation of the stationary shock-wave profilefor the major scale U , the small-amplitude oscillations keeppropagating at the leading edge of the shock-wave profile.The reader is referred to the seminal Ref. [61] for details. InRef. [61] Gurevich and Pitaevskii used the large parameterseparating the scales Wνα/Wζα 1 to describe the formationof the DSW profile of the KdVB equation analytically, usingthe Whitham modulation theory. They described the oscillatingpart of the profile by a modulated periodic solution of the KdVequation (58) (see also Ref. [20] for a recent discussion ofDSWs in KdV and cold-atom dynamics).
We conclude that, in striking contrast to the classicalshock wave the dispersive shock-wave profile has an internalstructure—oscillations with the typical wavelength Wζα , andthat while the width of the conventional shock wave isproportional to ν (60), the overall width of the DSW isproportional to ν−1 (62).
The main steps in the formation of DSWs are illustratedin Fig. 2. Although the KdVB equation itself is consideredin this work only as an approximation to the more precisehydrodynamics (4), (5), it is important that it gives a verygood qualitative and often even quantitative understanding ofthe latter. Below, in Sec. VI B, using the numerical methodof smoothed particle hydrodynamics, we present the resultsof numerical solutions of hydrodynamic equations written fora cold unitary Fermi gas (see Fig. 3). The similarity betweenthese solutions and the dispersive shock waves of the KdVBequation described in this section are evident. Using the resultsgiven in this section we will make estimates (in Sec. VI B) fordispersive shock-wave scales in a unitary Fermi gas system forgas parameters similar to those in the recent experiment [10].
10 10 20 30 40 50x
1.0
0.5
0.5
U x,t
FIG. 2. (Color online) The evolution of the initial steplike profile(blue dotted) governed by the KdVB equation (25) in the dispersiveshock-wave regime. The dimensions of x and y axes are μm andμm−1, respectively. The direction of time evolution is blue (dotted),purple (dash-dotted), red (dashed), and black (solid).
VI. INTERACTING COLD ATOMS
In this section we consider several one-dimensional modelswhich have recently attracted a lot of interest in connectionwith cold-atom systems. These are systems of bosons withcontact interaction in both the weak- and strong-couplinglimits and the Fermi gas at unitarity.
It is important to note here that the one-dimensionalityof these models is of totally different origin. The model ofbosons with contact interaction is strictly one dimensional. Itappears in the context of cold atoms due to the strong transversequantization. The applicability of the hydrodynamic approachto this model requires separate discussion, which is presentedin Sec. VI A. The model of a Fermi gas at unitarity consideredin Sec. VI B is essentially three dimensional. Here we restrictourselves to describing one-dimensional solutions of this 3Dhydrodynamics so that the one-dimensionality in this case isa property of the class of solutions rather than an intrinsicproperty of the original 3D system. This class of solutionsis especially important in the geometry of elongated trapsaddressed in Sec. VI B2.
These models do respect Galilean invariance and thereforecan be described by the general form of (4) and (5) in thehydrodynamic regime. The goal of this section is to relate thechemical potential w(ρ) and the values of the coefficients A,ηB
in (5) to the parameters (26),(27)(,(28)) of the effective KdVBdescription (25). We delegate the analogous discussion ofseveral models with long-range interactions to the Appendix.The results are summarized in the Table I.
A. 1D bosons with contact interaction
One-dimensional bosons with contact interaction can bedescribed by the Lieb-Liniger model [62]
H = − h2
2m
N∑i=1
∂2
∂x2i
+ g∑i<j
δ(xi − xj ). (64)
The background density of bosons ρ0 and the coupling constantg define the dimensionless coupling γ as
γ = m
h2ρ0g. (65)
The value of γ determines whether the system is in a weak-coupling (γ � 1) or a strong-coupling (γ 1) regime.
The model (64) is integrable by the Bethe ansatz methodfor all values of the coupling γ . In particular, a general form of
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HYDRODYNAMICS OF COLD ATOMIC GASES IN THE . . . PHYSICAL REVIEW A 86, 033614 (2012)
00.8
1.2
1.6
2.
00.8
1.2
00.8
1.2
0 50 100 150 2000.8
1.
1.2
1.4
x m
x10
3m
1
FIG. 3. (Color online) Numerical density evolution (based onthe smoothed-particle approach [10,11,72,73]) of a Gaussian initialprofile (with profile scales similar to the experiment in Ref. [10]) in aunitary Fermi gas in cylindrical geometry (see Sec. VI B3 for details).We use the symmetry of the plot and show only its positive-x-axispart for visual clarity. The red (dashed) plot depicts a situation whenthe dissipative term dominates the dispersive one [10,11] (αη = 1)and the black (solid) plot represents a situation when both termsare important (αη = 0.1). Albeit in highly nonlinear regimes, theplots shown can be qualitatively described by the KdVB physics ofFig. 1. Indeed, the black (solid) plot is similar to the one of Fig. 2while the red (dashed) one is reminiscent of the evolution governedby the Burgers equation. See Sec. VI B for related discussions.
w(ρ) can be found implicitly through the solution of the Betheansatz integral equations [62]. For a general value of γ thissolution can be found only numerically. Here we discuss onlythe limits of weak and strong coupling in which analyticalformulas can be obtained using expansions in γ and 1/γ ,respectively.
As the model (64) is integrable it does not include anydissipation mechanism. However, in more realistic modelingsof experiments the dissipative effects might be significant. Inthe simplest cases these effects can be incorporated into the ef-fective one-dimensional hydrodynamic description by adding
the Burgers terms νuξξ with ν treated as a phenomenologicalparameter.
The model (64) is a quantum one-dimensional integrablemodel. As an integrable model it possesses an infinitenumber of conservation laws related to hidden symmetriesof the problem. Generally the hydrodynamic approach is notapplicable here as there is no relaxation in such systems dueto additional conservation laws. Instead, one can talk aboutequations of hydrodynamic type describing particular sectorsof coherent states of the quantum model (64) or assume thatthe integrability is broken by different corrections to (64)coming from more realistic interatomic potentials or due tothe coupling to the environment. In the latter case the smallcorrections to the model (64) might destroy its integrabilityand result in the relaxation of higher (more fragile) integralsof motion. The conservation of the number of particles,momentum, and energy, on the other hand are based on gauge,space, or time symmetries and are robust. Therefore, at timescales much larger than the time scales for relaxation ofhigher integrals of motion one can still use the hydrodynamicapproach. In this paper we assume that the hydrodynamicapproach developed below can be justified in either of thescenarios described above.
1. Weak-coupling (high-density) limit γ � 1
It is well known that a collective description of (64) inthe high-density limit γ � 1 is given by the Gross-Pitaevskiequation [63] (GPE) (see, e.g., Refs. [63,64])
ih∂tψ(x,t) ={− h2
2m∂xx + g|ψ(x,t)|2
}ψ(x,t). (66)
Using a “hydrodynamic” change of variables,
ψ = √ρ ei(m/h)
∫ x
0 v(x ′) dx ′, (67)
the GPE can be cast in the form (4), (5) with w, A, and ηB
given by
w(ρ) = g
mρ, (68)
A = h2
2m2, (69)
ηB = 0. (70)
Then from (24), (26), and (27) we have
c = hρ0
m
√γ , (71)
ζ = h
m
3
4√
γ , (72)
α = h
mρ0
1
8√
γ. (73)
As mentioned earlier, a small amount of dissipation that arisesexperimentally can be taken into account by introducing ν
as a phenomenological parameter. Then, the dynamics of themodel is described by the KdVB equation (25) with the valuesof ζ and α given by (72) and (73), respectively, and with ν asa phenomenological parameter. In the limit of no dissipationone can describe this system by the KdV equation (i.e., theKdVB equation with ν = 0).
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The parameters of the soliton solution of an effective KdVequation in this case can be found from (51), (52), and (53),
W = 1
ρ0
√c
2V γ, (74)
U = 4V
cρ0, (75)
n = −8
√V
2cγ, (76)
and the effective mass (57) is
m∗ = 20
9n. (77)
This soliton solution of the KdV equation coincides withthe soliton solution of the GPE (66) in the limit of weaknonlinearity. The latter is known to correspond to a quasiholeexcitation of the quantum Lieb-Liniger model (64) [65,66].The soliton solutions should be trusted only in the limit |n| 1and V � c. The first condition corresponds to the applicabilityof the GPE while the second one allows us to replace the GPEby the KdV equation corresponding to the weakly nonlinearlimit of the GPE.
Note that here the velocity V is defined in the referenceframe moving with the sound velocity c so that the weak-nonlinearity limit is applicable only for waves propagatingwith velocities close to the sound velocity.
2. Strong-coupling (low-density) limit γ � 1
In the limit γ 1 the chemical potential is given by thefollowing expansion in 1/γ [62,67,68]:
w(ρ) = π2h2ρ2
2m2
[1 − 4
3γ+ 20
γ 2+ 64(π2
15 − 1)
γ 3+ · · ·
], (78)
where γ (ρ) is understood as a function of the local density ρ
[cf. Eq. (65)],
γ (ρ) = m
h2ρg. (79)
Using (24) and (26) and replacing γ (ρ) by its backgroundvalue γ = γ (ρ0) (65), we obtain
c = h
mπρ
[1 − 1
γ+ · · ·
], (80)
ζ = h
mπ
[1 − 3
2γ+ · · ·
], (81)
where for the sake of brevity we kept only the first two termsof the corresponding expansions.
A derivation of the dispersive term A(ρ) requires somewhatmore involved analysis and will be considered elsewhere. Thevalue A vanishes in the limit γ → ∞ (see below).
The limit of an extremely strong coupling γ → ∞ is knownas the Tonks limit. This limit corresponds to impenetrablebosons which in turn can be mapped to free one-dimensionalfermions (see Refs. [64,69] for recent discussions). Thecollective description in this limit is essentially given by
A(ρ) = 0, ηB = 0, and
w(ρ) = π2h2ρ2
2m2, (82)
c = h
mπρ0. (83)
The effective KdVB equation becomes the Riemann-Hopfequation [Eq. (33)] with
ζ = h
mπ. (84)
In fact, for 1D fermions the left- and right-moving waves aredecoupled and the equation [Eq. (33)] follows directly andexactly from the Euler and continuity equations with (82).
An important point about the Tonks limit lies in the absence[19,70,71] of the quantum pressure or the zero-point fluctua-tion term (i.e., A = 0, α = 0). Therefore, in the Tonks limitone does not expect to see dispersive KdV-like oscillations. Toaccount for a dissipative mechanism that might be present inexperiments, the dissipative term term νuξξ can be added withthe effective equation becoming the Burgers equation (35).
B. Fermi gas at unitarity
Recently there have been numerous experiments studyingthe dynamics of strongly interacting cold Fermi atoms and,in particular, of the Fermi gas at unitarity [5,26,74,75]. TheFermi gas at unitarity is highly hydrodynamic even deep inthe nonlinear regime [10]. Interacting fermions can also bestudied in a quasi-one-dimensional regime where the effective1D hydrodynamics is obtained as a result of the dimensionalreduction of the 3D hydrodynamic equations [10,76].
In this section we consider the superfluid hydrodynamics ofthe Fermi gas at unitarity. We assume that the temperature islow so that the gas can be effectively described as a simple fluidinstead of two-fluid hydrodynamics relevant for superfluidsat finite temperatures. We take the presence of the normalcomponent of the fluid into account only through the effectiveshear viscosity which is of the order of ηshear ∼ h
mρ with the co-
efficient of the order of 1 under typical experimental conditions[10]. The bulk viscosity should be absent in the hydrodynamicsof the gas due to the scaling invariance at unitarity [77].
In the following we consider two types of one-dimensionalbehavior of nonlinear waves in unitary Fermi gases. The firstis the plane-wave propagation through the 3D Fermi gas(considered in Sec. VI B1) and the other is the propagationof waves in a Fermi gas confined to elongated cigar-shapedtraps, where the effective 1D reduction of 3D hydrodynamicsis used [10] (see Sec. VI B2). The results are summarized inTable I.
1. Plane waves in a 3D unitary gas
The 3D superfluid hydrodynamics of a unitary Fermi gasis given by the continuity and the Euler equations with thechemical potential fixed by the scaling invariance as
w(ρ) = (1 + β)h2
2m2(3π2ρ)2/3, (85)
where β ≈ −0.61 is the Bertsch parameter [78–80]. Here thedensity is the 3D density of the Fermi gas. In addition, we
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take into account the gradient corrections coming from theMadelung pressure term with [81]
A(ρ) = h2
2m2(86)
and the effective shear viscosity ηshear, which appear as a resultof the presence of the normal component.
We consider the plane-wave solutions of the correspondinghydrodynamic equations assuming that the fluid moves alongthe z direction and
ρ(x,y,z) = ρ(z), vz(x,y,z) = v(z). (87)
The substitution of (87) into the 3D hydrodynamic equationsresults in the 1D equations (4) and (5) with the effective 1Dbulk viscosity ηB originating from the shear viscosity of the3D fluid,
ηB = 43ηshear. (88)
We obtain the effective parameters of the KdVB equation as
c =(
1 + β
3
)1/2h
m(3π2ρ0)1/3 (89)
and
ζ = 2
3
c
ρ0, (90)
α = h2
8m2c, (91)
ν = 2
3αη
h
m. (92)
We notice here that the effective dissipation parameter ν
is defined by the shear viscosity of the 3D unitary gasηshear = αη
hm
ρ0. Here αη is the dimensionless shear viscosity.Setting αη to zero would lead to dispersive shock waves (see,e.g., Ref. [76]). However, the shear viscosity of the unitarygas has a nontrivial temperature behavior [82,83]. It has auniversal dependence αη ∼ T 3/2 at large temperatures anddiverges as αη ∼ T −5 at very low temperatures. The latterdivergence is related to the divergence of the phonon meanfree path (determined by the rate of phonon-phonon collisions)at low temperatures [84]. As a result the shear viscosity of aunitary gas is expected to develop a minimum at a temperatureclose to the temperature of the superfluid phase transition withαη ∼ 0.5 [82,83].
Substituting (90), (91), and (92) into the criterion fordispersive shock waves (63) we obtain αη � [3U/(16ρ0)]1/2,which is not possible even by extrapolating the amplitude ofthe wave into the deeply nonlinear regime U ∼ ρ0. We willsee below that the unitary gas in a quasi-1D trap gives a morefeasible way to observe dispersive shock waves.
2. 3D unitary gas in a quasi-1D trap
There is another important case when the dynamics of aunitary Fermi gas can be reduced to be one dimensional. Thisis the case of a unitary gas confined to elongated cigar-shapedtraps. In this case the longitudinal waves along the trap canbe described by the effective reduced 1D hydrodynamics [10].It is convenient to use the effective one-dimensional densityρ(x,t) equal to the 3D density integrated over the transverse
section of the trap. Then the effective hydrodynamics can becast in the form of Eqs. (4) and (5), where [10,11]
w(ρ) = h2
2m2l2⊥
(1 + β)
(15π
2
ρ l⊥1 + β
)2/5
(93)
with the transverse oscillator length given by the radial trapfrequency ω⊥,
l⊥ =√
h
mω⊥. (94)
The Madelung term of (5) is given by the reduction of the bulkMadelung term (86) and turns out to be [81]
A(ρ) = 9
20
h2
m2. (95)
The effective one-dimensional bulk viscosity coefficient ηB
originates from the normal component of the superfluid.It should be obtained in the process of averaging theeffective shear viscosity of the 3D superfluid unitary gas.It is also affected by the boundary of the trap, where thehydrodynamic approximation does not work, and a morecomplete kinetic theory should be considered. This derivationis beyond the scope of this paper and here we treat ηB as aphenomenological parameter. At the typical conditions of theexisting experiments [10] it is
ηB(ρ) = αη
h
mρ, (96)
where αη is a dimensionless parameter of the order of 1. Ithas been estimated in Ref. [10] as being in the range 1–10 forthe conditions of Ref. [10].
It was shown in Ref. [10] that the hydrodynamics (4), (5)with (93), (96) gives a very good quantitative description of thecloud collision experiment. The dispersive term was neglectedin Ref. [10] as it turns out to be inefficient under the experi-mental conditions (see below). The hydrodynamic equationscannot be solved analytically and numerical solutions wereused in Ref. [10]. In the following we use instead the mappingto the effective KdVB equation to identify important scalesin the problem and to analyze the feasibility of observing thedispersive effects in cold Fermi atoms.
Having the unitary gas dynamics in the form (4), (5), weimmediately derive the parameters of the effective KdVBequation (25) in the limit of weak nonlinearity, dispersion,and dissipation. The sound velocity is given by (24) and isequal to
c =(
1 + β
5
)1/2h
ml⊥
(15π
2
ρ0l⊥1 + β
)1/5
. (97)
Then (26), (27), and (28) give
ζ = 3c
5ρ0, (98)
α = 9h2
80m2c, (99)
ν = αη
h
m. (100)
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3. Shock waves in a unitary gas in a quasi-1D trap
Let us now use the effective KdVB equation for the unitaryFermi gas in the elongated harmonic trap to understandthe qualitative behavior and relevant scales for particularexperimental conditions of Ref. [10]. In the latter experiment atwo-component lithium gas was cooled in a trap with the radialfrequency ω⊥ = 2π × 437 Hz. The corresponding transverseoscillator length (94) can be easily found as
l⊥ = 1.8 μm, (101)
using mLi = 1.15 × 10−26 kg, corresponding to h/mLi ≈9.2 μm2/ms. The one-dimensional density in the middle ofthe trap is
ρ0 ≈ 1.1 × 103 μm−1, (102)
and can be determined from the equation of the state of theunitary Fermi gas, the trap parameters, and the total number ofparticles (N ∼ 2 × 105). The temperature in the experimentwas very low and will be considered as zero here (except inusing the effective viscosity parameter).
The experiment [10] studied the collision of two atomicclouds created within the trap using a blue-detuned laser beam.Here, instead, we make our estimates for a weakly nonlinear,dispersive, and dissipative limit presumably described by theeffective KdVB equation (25) with the speed of sound givenby (97) and other parameters by (98), (99), and (100) as
c ≈ 14.5μm
ms, (103)
ζ ≈ 8 × 10−3 μm2
ms, (104)
α ≈ 0.65μm3
ms, (105)
ν ≈ αη × 9.1μm2
mswith αη ≈ 10. (106)
Let us now assume that we created a steplike profile of thedensity of the typical scale of U ∼ ερ0 or (102)
U ∼ ε × 1.1 × 103 μm−1 , (107)
where we keep ε � 1 as a parameter. Then the relevant spatialscales (31) are estimated as
Wζν ∼ αηε−1, 1 μm, (108)
Wζα ∼ ε−1/2, 10−1 μm, (109)
Wνα ∼ α−1η , 10−1 μ m. (110)
We find that the condition (63) becomes
N = Wνα
Wζα
∼ ε1/2α−1η 1 (111)
and is definitely not valid for αη of the order of 1 orlarger. Therefore, unless the dimensionless viscosity αη issomehow experimentally suppressed, the shock waves havethe conventional dissipative character and dispersive shockwaves should not be observed.
Indeed, the steep density profiles in the experiment ofRef. [10] were identified as dissipative shock waves. Stretching
our estimates to ε ∼ 1 to take into account the highly nonlinearcharacter of the cloud collision in the experiment (of course,the KdVB equation derived in the limit of weak nonlinearitycan serve here only as a tool for estimates), we obtain thatN ∼ 0.1, and the approximate width of the dissipative shockfront should be of the order of
WCSW ∼ Wζν ∼ 10 μm. (112)
This number is indeed of the order of the one observedexperimentally [10].
To see whether it is feasible at all to observe dispersiveshock waves in cold Fermi atoms we make the following crudeestimate. We start with the condition (63) rewritten as
ν2 � αζU (113)
and use for the estimate α ∼ h2
m2c, ν ∼ αη
hm
, ζ ∼ cρ0
, and U ∼ρ0. We have
αη � 1 (114)
as a necessary condition for the observation of the oscillationsin the stationary-shock-wave profile. Is it possible to lower theeffective viscosity to achieve these conditions in an experimentsimilar to that of Ref. [10]? A straightforward estimate of themean free path for phonon-phonon collisions for the conditionsof Ref. [10] gives
λ ∼ (TF /T )9 4 × 10−8 μm. (115)
We used here the expression [83] λ = hcEF
× 2.8 × 10−5(1 +β)5( TF
T)9 with TF ≈ 1.1 μK and c = vF /
√3 ≈ 19 μm/ms
calculated at the center of the trap, corresponding to a 3Ddensity [10] n3D = 6.1 μm−3.
From (115) we see that the mean free path evaluatedat the center of the trap becomes of the order of the trapsize (∼200 μm from the center to the edge of the trap)already at T ≈ 0.1TF which is roughly the temperature ofthe cold-atom system in Ref. [10]. Therefore, we expectthat if one lowers the temperature the mean free path willsaturate at the size of the trap, and the shear viscosity of 3Dsuperfluid will be dominated by the density of the normalcomponent of the superfluid, ρn ∼ T 4 (from kinetic theoryηshear = ρnpλ, where p is the average momentum). Now itis clear that the decrease of the temperature by the factor of2 in the experiment [10] might decrease the normal densityand, therefore, the shear viscosity at the center of the trap bya factor of 20. Although the effective 1D viscosity should beobtained as a result of a complicated averaging of the two-fluidhydrodynamic equations in directions transverse to the trap,this rough estimate shows that the possibility of observingthe dispersive behavior in quasi-1D traps is indeed feasibleand requires experiments at somewhat lower temperaturethan the one in Ref. [10]. In fact, the (tunable) dissipativemechanism along with the dispersive mechanism provided bythe superfluid nature of the unitary gas causes the unitary gassystem to be one of the best experimental systems for probingthe intricate interplay between dissipation, dispersion, andnonlinearity.
To show that the presented analysis of the shock-wavephysics of KdVB equation describes qualitatively the solution
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HYDRODYNAMICS OF COLD ATOMIC GASES IN THE . . . PHYSICAL REVIEW A 86, 033614 (2012)
of the actual hydrodynamic system (4), (5), we present anumerical solution of the hydrodynamic equations in Fig. 3for the hydrodynamic system derived for a unitary gas in aquasi-1D trap (see Ref. [10] and Sec. VI B2 of this paper).We obtain numerical solutions of the continuity and Eulerequations using the technique of smoothed-particle hydrody-namics [10,11,72,73]. This numerical method is a meshlessalgorithm. The algorithm essentially consists of mapping thesystem of differential equations describing the density andvelocity fields to dynamic equations of a sufficiently largeset of moving pseudoparticles (approximating the density andvelocity fields) and then solving the molecular dynamics ofthese pseudoparticles.
The results of a numerical solution of hydrodynamicequations are shown as a red (dashed) plot in Fig. 3. Theparameters in the equations are taken similar to the conditionsof the experiment [10] discussed in Sec. VI B3. We useda constant background with the background density givenby Eq. (102) with an initial density profile of a Gaussianform of height equal to ρ0, i.e., in Eq. (107) we take ε = 1.The full width at half maximum of the Gaussian bump istaken to be about 20 μm (5% of the system size) similarlyto the experiment [10]. The red (dashed) plot in Fig. 3corresponds to αη = 1 and we clearly see that the shockwaves are of dissipative nature. Qualitatively the red (dashed)plot is very similar to the evolution described by the Burgersequation (35). We also plot (the black solid plot) the profileobtained numerically for αη = 0.1 (see the discussion of thefeasibility of obtaining small viscosities in finite systemsabove). For this value of effective dissipation we clearly seenonvanishing oscillations in the stationary profile of the shockwave.
VII. CONCLUSION
We presented a unified picture of the dynamics of a largeclass of nonlinear systems in the limit of weak nonlinearity,dissipation, and dispersion (WNDD). This picture is delineatedby the Korteweg–de Vries–Burgers equation (25) for thedeviation of the density from its background value. Wedescribed various regimes of nonlinear evolution describedby the KdVB equation and summarized these regimes in thediagram in Fig. 1. The parameters of the KdVB equationwere related to those of a generic one-dimensional fluidand then to particular parameters of several models usedin cold-atom research. These relations are summarized inTable I.
Arguably the most interesting regime in the WNDD limit isthe formation of dispersive shock waves which are known tooccur in interacting systems [19,20,85]. The analysis presentedin this paper uses well-known results in the theory of nonlinearpartial differential equations. It allows us to make quickestimates of important scales and parameters of shock waves.
It is worth mentioning that in addition to external potentialsubiquitous in cold-atom experiments (such as harmonic trapsor optical lattices), one could also study the dynamics inthe presence of more complicated potentials within the sameframework, by merely adding a force term −V ′(ξ ) to the RHSof the mapped chiral differential equations with V (ξ ) beingthe associated potential. Examples might include an external
double-well potential [86], Gaussian defects [17], and disordercreated using a laser speckle [17].
Although in this work we focused on a Galilean-invariantfluid, the general analysis can be extended to other systemsas well. The KdVB equation still remains the most universaldescription in the WNDD limit unless additional symmetriesrestrict the form of the terms of lowest order in nonlinearityand/or field gradients (e.g., the underlying particle-hole sym-metry might require the symmetry u → −u so that the lowestallowed nonlinear term has the form of u2ux , leading to theso-called modified KdV equation). The WNDD limit thereforeallows division of the complicated dynamics of nonlinear sys-tems into several distinctive “nonlinear universality classes,”giving an insight into their generic dynamical features (for arecent related discussion see Refs. [18,35,87]).
We have also discussed another important method ofgeneralizing the KdVB description appropriate for systemswith long-range interactions. For those systems locality of theeffective description is not required. The relaxation of the lo-cality requirement leads to a wider class of integro-differentialeffective equations (see the Appendix for some examples).
Another important assumption used in this paper is thepossibility of having a consistent effective description of thesystem by a single-component fluid. Future work includesstudying the presence of several components such as thenormal and superfluid components in BECs or equivalentcomponents in spinor BECs, as this is crucial for the accuratedescription of corresponding physical systems.
The described reduction of nonlinear three-dimensionaldynamics to an effective chiral one-dimensional equationobtained in the WNDD limit (25) cannot capture all thefascinating and complex effects in the collective behaviorof interacting 3D systems. For example, taking into accountthe next-order terms in the reductive perturbation expansionwill lead to an interaction between chiral sectors (left- andright-moving waves) neglected in this work. The next-orderterms in time derivatives will result in a frequency dependenceof the kinetic coefficients essential for, e.g., the physics of theunitary gas [82,88].
Finally, the description used in this work is completelyclassical (for example, there is no h in Secs. II–V. Quantumphysics enters here only through the values of the effectiveparameters of the classical hydrodynamic equations (seeSec. VI). One of the most interesting problems is to identifythe quantum effects not describable by conventional classicalphysics (quantization of the vorticity in a superfluid systemcan serve as an example of such effects).
ACKNOWLEDGMENTS
We are grateful to J. E. Thomas, J. Joseph, D. H. J. O’ Dell,J. Thywissen, M. Punk, and E. Taylor for useful discussions.We thank the Princeton Center for Theoretical Science fortheir hospitality during the workshop on “Ultracold Atoms andMagnetism,” where this work started. The work of A.G.A. wassupported by the NSF under Grant No. DMR-0906866. M.K.acknowledges the hospitality of the “Cold Atoms Meeting”at Banff, funded by the Canadian Institute for AdvancedResearch (CIFAR), where several interesting discussions tookplace.
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MANAS KULKARNI AND ALEXANDER G. ABANOV PHYSICAL REVIEW A 86, 033614 (2012)
APPENDIX: RELAXING LOCALITY
In the main part of the paper we assumed that the systemof particles is well described by a local hydrodynamics theory.The assumption of locality was important in deriving theeffective KdVB equation for those systems. In the presenceof long-range interactions between particles, our derivationsshould be modified and generally a different counting schemeshould be used in the reductive perturbation method. In thisappendix we show how the assumption of locality can berelaxed on the example of two models: the Calogero modeland the dipolar BEC model. The former is often used asa theoretical toy model as it is exactly integrable and verywell studied (the justification of the use of the hydrodynamicapproach to this model is the same as that in Sec. VI A). Thedipolar BEC model is relevant to some experimental cold-atomsystems [13,15].
1. Calogero model
The Calogero model is a model of one-dimensional par-ticles interacting via an inverse square potential [89]. TheCalogero Hamiltonian is given by
H = 1
2m
N∑j=1
p2j + h2
2m
N∑j,k=1;j �=k
λ(λ − 1)
(xj − xk)2. (A1)
Here the coupling constant λ is dimensionless. For simplicitywe consider here the quasiclassical limit corresponding to λ 1, and in the following we replace λ(λ − 1) → λ2 and neglectcorrections of the order of 1/λ. It turns out that one can writedown the collective field theory of the above model exactly[29–32,90], i.e., including all gradient and nonlinear terms.This field theory can be again cast in the form (4),(5) with nodissipation, ηB = 0, and
w(ρ) = h2λ2
m2
(1
2(πρ)2 + πρH
x
), (A2)
A(ρ) = h2λ2
2m2. (A3)
Here the superscript H stands for the Hilbert transform definedby
f H (x) = 1
πP
∫f (y) dy
y − x. (A4)
Notice that the second term in (A2) is nonlocal, reflecting thelong-range interaction between Calogero particles in Eq. (A1).
The sound velocity in the collective field theory for theCalogero model [29–32] can be obtained from (A2) using (24)and is given by
c = h
mλπρ0. (A5)
The second term in (A2) is of first order in ∂x and the Burgerscounting scheme (36)–(39) should be used in the reductiveperturbation method of Sec. IV. As a result we obtain theso-called Benjamin-Ono equation
ut + ζuuξ + αBO(uξξ )H = 0 (Benjamin-Ono) (A6)
as an effective description of Calogero hydrodynamics inthe limit of weak nonlinearity and dispersion [31,32]. Thecoefficients of (A6) are given by
ζ = πhλ
m, (A7)
αBO = hλ
2m. (A8)
Two remarks are in order. First, notice that the Benjamin-Onoterm [the last term of Eq. (A6)] dominates the conventionalKdV term uξξξ in the long-wave limit. It is of the same orderas the Burgers term uξξ . Second, although the term uH
ξξ scalessimilarly to the Burgers term, it is very different in nature as itdoes not result in any dissipation. Indeed, the Fourier transform(uH
ξξ )k = ik|k|uk , and this term results in the real correctionω ∼ αBOk|k| to the spectrum of linear waves, in contrast tothe Burgers term giving the imaginary (dissipative) correctionω ∼ −iνk2.
2. Quasi-1D dipolar BEC
Systems of bosons interacting via a long-range potentialhave been recently realized experimentally [13,15] for bosonswith dipole interactions. The quasi-1D dipolar BEC can bedescribed by a nonlocal version [14,91,92] of the Gross-Pitaevskii [63] equation. By choosing a sufficiently large radialtrap frequency it is possible to freeze the radial motion [93,94]of the BEC and thereby obtain a quasi-1D dipolar BECdescribed by the following nonlocal one-dimensional GPE(see Ref. [93] for the derivation):
ih∂tψ(x,t) =(
− h2
2m∂2x + mw(|ψ |2)
)ψ(x,t), (A9)
where
w(ρ) = g
ml2⊥
(A1Dρ + Bl2
⊥∂2x ρU
). (A10)
Here g is the 3D contact interaction strength, l⊥,z =√h/(mω⊥,z) is the transverse (axial) oscillator length, and
the dimensionless constants A1D and B are
A1D = 1
2π
[1 + 1
2εdd
(1 − 3n2
z
)], (A11)
B = 3
8πεdd
(1 − 3n2
z
). (A12)
nz is the z component of the direction of the dipole axis nand εdd is a natural dimensionless parameter for the relativestrength of dipolar and s-wave interactions [93].
In (A10) we used the dipolar integral transform
f U (x) =∫ +∞
−∞U1D(x − y)f (y) dy (A13)
defined by its kernel
U1D(x) =√
π
2 l2⊥
ex2/2l2⊥erfc
( |x|√2l⊥
), (A14)
where erfc is the complementary error function. The Fouriertransform of U1D(x) is given by
U1D(k) =∫ ∞
0dκ
e−κ/2
κ + (kl⊥)2(A15)
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HYDRODYNAMICS OF COLD ATOMIC GASES IN THE . . . PHYSICAL REVIEW A 86, 033614 (2012)
and its asymptotic behavior as k → 0 is
U1D(k) ∼ −γE − ln
((kl⊥)2
2
)+ · · · , (A16)
where γE ≈ 0.5772 is the Euler-Mascheroni constant.The hydrodynamic change of variables ψ =√
ρei(m/h)∫ x
0 v(x ′)dx ′brings (A9) to the hydrodynamic
form [94] (4), (5) with the chemical potential w(ρ) given by(A10) and
A(ρ) = h2
2m2. (A17)
The speed of sound in this system can be easily found using(24) and (A10):
c =√
gρ0
ml2⊥
A1D = h
ml⊥(γA1D)1/2, (A18)
where we introduced the dimensionless coupling constant
γ = mρ0
h2 g. (A19)
The resulting Euler equation in this case is nonlocal. Theperturbative scaling scheme [Eqs. (17), (18), (21), and (22)]
gives rise to the following non-local KdV-like equation(“dipolar-KdV”):
Here the superscript U again denotes the transform (A13). Aswe work only in the long-wavelength limit, we can think of thetransform’s kernel as the Fourier transform of (A16). Althoughtechnically in the asymptotic long-wavelength limit k → 0 theαD term will dominate the α term, we keep both of them in(A20). Indeed, the kernel (A16) is growing logarithmicallyslowly with k, and both terms become of the same order atkl⊥ ∼ exp(πα/αD), which is defined by the actual values ofthe dimensionless parameters of the dipolar system.
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